Je Remercie Aussi

Directrice Agnès Sulem

Julien Lepers Ne Serait Rien

Noufel, Pascal Kilian

Alexandre

Nicolas

Paul

Aymen Benjamin

Riadh

Keywords: Mouvement Brownien multifractionnaire, calcul stochastique par rapport aux processus gaussiens, théorie du bruit blanc, S-transform, limites de processus, chaining, quantification fonctionnelle Multifractional Brownian motion, stochastic calculus with respect to multifractional Gaussian processes, White Noise theory, S-transform, limit of processes, chaining, functional quantification

Cette thèse est le fruit d'un travail de plusieurs années. Le moment est venu de s'en détacher et donc de formuler les remerciements à ceux -et ils ne manquent pas ! -qui m'ont accompagné pendant ce périple.

Al ' o r i g i n ed em o ng o û tp o u rl e sm a t h é m a t i q u e s ,i lyam o np è r e .P h i l o s o p h ed ef o r m a t i o n ,p a s s éàl a psychanalyse et revenu à la philosophie, il s'est pris d'un goût tardif pour les mathématiques en ce qu'elles étaient l'outil indispensable des penseurs, du penseur.C'estaucoursdesheuresquej ep as s ais ,c h aqu ew e e kend, en sa compagnie à résoudre des problèmes mathématiques, que m'est venu le goût de cette discipline. Ma mère n'a pas été en reste. Psychanalyste, elle aussi accorde une grande valeur aux mathématiques qui occupent une place de choix dans l'enseignement de Lacan. Il n'y eut pas de meilleur exemple pour moi que de voir mes deux parents se confronter à des textes ardus de philosophie ou de psychanalyse. C'est le principe même de la recherche. On ne comprend pas grand-chose jusqu'à ce qu'une étincelle jaillisse et que la lumière se fasse ! Et c'est pour le plaisir de cet instant unique que l'on travaille et que l'on cherche.

Cette thèse doit aussi beaucoup à mes deux directeurs, Marc Yor et Jacques Lévy Véhel, que je tiens àr e m e r c i e rp a r t i c u l i è r e m e n t .B i e nq u ' a s s e zd i ff é r e n ts dans leur façon d'appréhender les mathématiques, ils partagent de nombreuses qualités : une curiosité insatiable non seulement pour les mathématiques mais aussi pour tous les autres domaines de la connaissance, scientifiques ou non. Je ne compte plus les discussions que nous avons eues, avec Jacques ou Monsieur Yor, qui portaient sur la littérature, la religion, la politique, le cinéma -j'en passe. L'un et l'autre ont fait preuve d'une disponibilité et d'une écoute constantes et m'ont fait profiter de leurs profondes connaissances mathématiques tout en sachant me laisser une large part d'autonomie. Leur bienveillance, leurs relectures toujours minutieuses et leurs suggestions m'ont été d'une aide incomparable. Leur enthousiasme enfin, est communicatif. Combien de fois avons nous, avec Jacques, échangés des arguments jusqu'à ce qu'un consensus émerge ! Combien profitables m'ont été les séminaires W.I.P.

Définition et propriétes

Le mouvement brownien fractionnaire, terme que nous abrégerons systématiquement par fBm dans la suite de ce travail, a été introduit par A.Kolmogorov en 1940, dans [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum[END_REF], comme moyen d'engendrer des "spirales" gaussiennes dans un espace de Hilbert. B.Mandelbrot donna le nom de coefficient de Hurst à la constante H,q u ic a r a c t é r i s et o u tm o u v e m e n tb r o w n i e nf r a c t i o n n a i r e ,e nl ' h o n n e u rd el ' h y d r o l o g i s t eb r i t a n i q u eH a r o l d E. Hurst qui, tentant de modéliser les crues annuelles du Nil, mit en évidence la corrélation à long terme des amplitudes de ces dernières au cours du temps. Il en déduisit alors qu'il était impossible de modéliser ce phénomène par un processus présentant des accroissements non corrélés au cours du temps, tel que le mouvement brownien. A partir de la fin des années 1960, B.Mandelbrot et J.Van Ness (cf. [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF]) popularisèrent le fBm en tant que modèle financier et en en étudiant les propriétés. Nous considérons fixé pour tout ce chapitre un espace de probabilité, noté (Ω, F , P).

Définition 1.1 (Mouvement brownien fractionnaire). Soit H une constante appartenant à l'intevalle ]0; 1[ et I un intervalle de R.U nm o u v e m e n tB r o w n i e nf r a c t i o n n a i r es u rI d'indice de Hurst H est un processus gaussien centré, noté B H := (B H t ) t∈I , dont la fonction de covariance, notée Σ H ,e s td o n n é e ,p o u r tout couple réel (t, s) de I 2 ,p a r:

Σ H (t, s) := γ H 2 Ä |t| 2H + |s| 2H -|t -s| 2H ä ,
où γ H est une constante positive ; de façon équivalente :

E[(B H t -B H s ) 2 ]=γ H |t -s| 2H .
L'existence du mouvement brownien fractionnaire résulte du théorème général d'existence des processus gaussiens centrés de fonction de covariance donnée (on pourra consulter [START_REF] Schoenberg | Metric spaces and positive definite functions[END_REF][START_REF] Neveu | S é m i n a i r ed eM a t h é m a t i q u e sS u p é r i e u r e s ,N o .3 4( É t é[END_REF][START_REF] Kahane | Some random series of functions,v o l u m e5o fCambridge Studies in Advanced Mathematics[END_REF][START_REF] Janson | Gaussian Hilbert spaces[END_REF] pour plus de détails).

Il est clair que lorsque H vaut 1/2 et que γ H =1 , le fBm n'est rien d'autre que le mouvement brownien fractionnaire standard sur I.P a rs o u c id es i m p l i c i t én o u sp o s e r o n sI := R + dans la suite de cette section. Par définition, le fBm jouit des propriétés suivantes.

1.B H 0 =0 p.s. Bien des résultats sont connus concernant le fBm (on pourra consulter [START_REF] Nualart | The Malliavin Calculus and related Topics[END_REF][START_REF] Yuliya | Stochastic calculus for fractional Brownian motion and related processes[END_REF][START_REF] Biagini | Stochastic calculus for fractional Brownian motion and applications[END_REF] ainsi que les références qui ys o n ti n d i q u é e s ) .O np e u ta i n s im o n t r e rq u e ,p r e s q u es û r e m e n t ,B H := (B H t ) t∈R+ admet une rectification Hölderienne de ses trajectoires pour tout ordre strictement inférieur à H.Parailleurs,pasplusquelemouvement brownien le fBm n'est différentiable en temps. Pour cette raison, une bonne mesure de la régularité du fBm est l'exposant de Hölder local. Défini en tout point t 0 de R + ,l ' e x po s a n td eH ö l d e rl oc a ld ' u np r oc e s s u s (X t ) t∈R+ est défini par :

α X (t 0 ) := sup ß α :limsup ρ→0 sup (s,t)∈B(t0,ρ) 2 |X t -X s | |t -s| α < +∞ ™ .
(1.1)

Il est prouvé dans [START_REF] Herbin | Stochastic 2 micro-local analysis[END_REF] que pour tout H de ]0; 1[,presquesûrement,pourtoutréelt 0 de R + ,onaα B H (t 0 )=H.

Les trois figures qui suivent, obtenues grâce au logiciel libre Fraclab 1 ,p r é s e n t e n td e st r a j e c t o i r e s ,t → (t, B H t (ω)),d em o u v e m e n t sb r o w n i e n sf r a c t i o n n a i r e sB H , pour différents coefficients de Hurst. Comme le résultat de régularité précédent le laissait présager, on remarque que, plus la valeur de H croît, plus la régularité augmente. Inversement, plus la valeur de H décroît plus les trajectoires sont irrégulières. Corrélation des accroissements du fBm et dépendance de long terme Le fBm ayant été, en partie, popularisé parce qu'il permettait de rendre compte de phénomènes présentant des corrélations à long terme au cours du temps il nous semble important de s'attarder un instant sur cette notion. Nous excluons d'emblée le cas H =1 /2 qui correspond au cas du mouvement Brownien dont les accroissements sont indépendants. Dès que H est différent de 1/2,l e sa c c r o i s s e m e n t sd uf B ms o n tc o r r e l é s . Plus précisément, on a pour tout couple (s, t) de R 2 + et tout réel positif r tels que s + r t et t -s = nr, où n est un entier positif,

ρ H (n) := Cov Ä B H t+r -B H t ,B H s+r -B H s ä = 1 2 r 2H [(n +1) 2H +(n -1) 2H -2n 2H ]. (1.2)
De la précédente égalité nous déduisons que les accroissements B H t+r -B H t et B H t+2r -B H t+r sont corrélés positivement lorsque H>1/2 et négativement lorsque H<1/2.I ls e m b l ed o n cp e r t i n e n td ' u t i l i s e ru n mouvement brownien fractionnaire d'indice de Hurst H strictement supérieur à 1/2 pour modéliser les phénomènes à mémoire présentant des accroissements positivement corrélés. (1.4)

On peut aussi prendre l'égalité (1.4) comme définition de la dépendance à long terme. Cette seconde définition de la dépendance à long terme d'une suite de variables aléatoires stationnaires est alors évidemment plus faible que la définition 1.2. Quelle que soit la définition choisie pour la dépendanceàlongterme,uncalculdirect montre que les accroissements X k := B H k -B H k-1 et X k+n := B H k+n -B H k+n-1 du fBm B H présentent telle dépendance lorsque H est strictement supérieur à 1/2 puisque l'on a alors :

ρ H (n) := Cov (X k+n ,X k )= 1 2 [(n +1) 2H +(n -1) 2H -2n 2H ] ∼ H n→+∞ (2H -1) n 2H-2 .
(1.5)

En revanche nous avons +∞ n=1 |ρ(n)| < +∞ lorsque H<1/2.

Les différentes représentations

Dans la suite de ce mémoire, nous utiliserons abondamment les notions de mesures gaussiennes réelles et complexes. Nous avons donc ajouté, dans l'annexe A, un court résumé de cette notion qui couvre tous les cas d'utilisation que nous en ferons dans les chapitres suivants. Nous encourageons le lecteur qui ne serait pas familier avec cette notion à s'y reporter avant de commencer la lecture de cette section.

Il existe de nombreuses représentations d'un mouvement brownien fractionnaire. Plus ou moins compliquées selon que l'on souhaite obtenir une représentation sur un compact de R ou sur R tout entier.

Représentations du fBm sur R Sur R nous avons la famille de représentations du fBm donnée, pour tout H de (0, 1) et (a + ,a -) de R 2 \{(0, 0)},p a r

Y (a + ,a -) (t) := δ(H) R Ä a + f + (t, H, u)+a -f -(t, H, u) ä W (du) (1.6)
où δ(H) est un réel non nul, W désigne une mesure gaussienne réelle et où l'on a défini :

f ± (t, H, u) := (t -u) H-1/2 ± -(-u) H-1/2 ± , (1.7)
avec la convention x γ -:= (-x) γ + , x, γ dans R et x γ + := (-x) γ , x>0, x γ + =0si x 0. On notera que si W est la mesure gaussienne complexe, construite à partir de W (pour plus de détails sur les mesures gaussiennes voir l'annexe A située à la fin de ce mémoire) alors on a, pour tout H de (0, 1) et (a + ,a -) de R 2 \{(0, 0)},l ' é g a l i t és u i v a n t e: ∀t ∈ R, ‹ Y (a + ,a -) (t)

p.s.

= Y (a + ,a -) (t), (1.8) où l'on a défini ‹ Y (a + ,a -) (t) := Γ(H+1/2) √ 2π R (e itξ -1) |ξ| H+1/2 U (a + ,a -) (ξ; H) W (dξ), Γ désignant la fonction Gamma d'Euler et où, pour tout réel ξ,o nap o s é , U (a + ,a -) (ξ; H) := a + e -i sign(ξ)(H+1/2)π/2 + a -e i sign(ξ)(H+1/2)π/2 .

On appelle représentation harmonisable du fBm d'indice de

Y (1,0) (t)= R Ä (t -u) H-1/2 + -(-u) H-1/2 + ä W (du).
Le lecteur souhaitant trouver davantage de réprésentations de fBm pourra se référer par exemple à [73,[START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF].

Représentations du fBm sur un compact

Nous nous contentons ici de donner la représentation suivante, que l'on pourra trouver dans [START_REF] Nualart | The Malliavin Calculus and related Topics[END_REF] B H t := T 0 {0 u<t T } (t, u) K H (t, u) W (du), (1.9) ] ,i le s tm o n t r éq u eI(B H , [0,T]) = 1 H pour tout H de ]0; 1[.I l s'ensuit que, pour tout réel positif p,l as u i t e(V n,p ) n∈N définie par,

K H (t, u) :=    α H ï Ä t u ä H-1/2 (t -u) H-1/2 -(H -1/2) u 1/2-H t u (v -u) H-1/2 v H-1/2 dv ò si 0 <H<1/2, Ä H(2H-1) β(2-2H,H-1/2) ä 1/2 u 1/2-H t u (v -u) H-3/2 v
V n,p := n i=1 B H i n -B H ( i-1 n )
p converge en probabilité vers 0 si pH > 1 et tend vers +∞ si pH < 1,l orsqu en tend vers +∞.Onenconclut donc que I(B H , [0,T]) = 1 H ,p o u rt o u tH.P u i s q u el ' i n d i c ed ev a r i a t i o nd ' u n es e m i m a r t i n g a l ea p p a r t i e n tà l'ensemble [0; 1] ∪{2},o ne nd é d u i tq u eB H est une semi-martingale si et seulement si H =1/2. Lorsque H =1 /2,q u ie s tl ec a sq u iv an o u si n t é r e s s e rd a n sl as u i t e ,l ec a l c u ls t o c h a s t i q u ed ' I t ôd é v e l o p p é pour les semi-martingales ne peut donc pas s'appliquer. Aussi, depuis une quinzaine d'années de nombreuses voies visant à établir un calcul stochastique relatif auf B mo n té t ée x p l o r é e s .P a r m ic e l l e s -c io np e u tc i t e r

Approches probabilistes

Approches déterministes eoifijoeooiosidfjosdijfsoijsoidjfosis eoifijoeooiosidfjosdijfsoijsoidjfosis Calcul de Malliavin Théorie des chemins rugueux L.Decreusefond, S.Ustünel, Alos, Mazet, Nualart... Lyons, Coutin, Nourdin, Gubinelli Théorie du bruit blanc Integrale de Stratonovich étendue Elliott, Van Der Hoek, Bender, Oksendal,...

Russo & Vallois Autres approchesd é t e r m i n i s t e s

Autres approchesd é t e r m i n i s t e s Zähle, Feyel & de la Pradelle Zähle, Feyel, de la Pradelle Nous renvoyons à [START_REF] Coutin | An Introduction to (Stochastic) Calculus with Respect to Fractional Brownian Motion[END_REF] et aux références qui y sont données pour une présentation détaillée de ces différentes méthodes. Nous utilisons dans le présent mémoire essentiellement la théorie du bruit blanc. Les approches déterministes sont ainsi nommées car elles correspondent à l'utilisation d'intégrales déterministes, définies ω par ω et construites en dehors de tout cadre probabiliste, pour des processus stochastiques de régularités diverses. Si l'on considère le cas de l'intégrale stochastique par rapport au fBm, cette répartition entre approches déterministes et probabilistes, pour classique qu'elle soit, semble cependant quelque peu arbitraire tant les outils de l'analyse réelle et fonctionnelle sont omniprésents dans ces différentes constructions. Nous reviendrons, au cours des prochains chapitres sur les raisons pour lesquelles nous avons choisi d'utiliser, de préférence à toute autre, la théorie du bruit blanc. Pour motiver, a priori,c ec h o i xp r é c i s o n s qu'il nous fallait une méthode d'intégration par rapport au mBm qui nous permette à la fois de résoudre des équations différentielles, sans conditions préalables sur les valeurs que peut prendre la fonction h dans ]0; 1[ ou sur la croissance des coefficients de l'E.D.S (ainsi nous ne voulions pas devoir exiger une croissance polynomiale,•••)m aisaussidedéfinirunein tégraledeW ienerdet outefonctiona y an tunerégularitéraisonnable (i.e pas nécessairement de classe C 1 ). La théorie des chemins rugueux satisfait la première condition mais pas la seconde alors que le calcul de Malliavin satisfait la seconde mais pas la première.

Les limites de validité des applications du mouvement Brownien fractionnaire

Très p opulaire et utilisé dans de nombreuses applications (financières notamment), le mouvement brownien fractionnaire présente cependant quelques inconvénients qui peuvent s'avérer gênants lorsque l'on souhaite modéliser certains phénomènes. En effet la régularité des trajectoires du fBm reste la même au cours du temps. De plus il est à accroissements stationnaires.C e c ie m pêc h ed on cd em od é lis e r,àl'aid ed 'u nf B m ,l e s phénomènes dont la régularité évolue au cours du temps ou dont les accroissements ne sont pas stationnaires comme par exemple le traffic internet, le traitement d'images, le relief des montagnes, la volatilité des cours de bourse ••• Ceci conduit donc naturellement à la question suivante : Peut on généraliser le mouvement Brownien fractionnaire à un processus gaussien dont la régularité des trajectoires varie au cours du temps ?

2L e m o u v e m e n t b r o w n i e n m u l t i f r a c t i o n n a i r e ( m B m )

La réponse, affirmative, à la question ci-dessus a été apportée par Jacques Lévy-Véhel et Romain Peltier dans [START_REF] Peltier | Multifractional brownian motion[END_REF].

Définition et propriétés

Dans ce premier article consacré au mouvement Brownien multifractionnaire (noté mBm dans la suite), ils ont défini le mBm à partir du fBm en remplaçant "simplement" le réel H par h(t),o ùh : R → (0, 1) désigne dans toute cette section une fonction continue. Le mBm originel est donc défini par,

X h t := 1 Γ(h(t)+1/2) R Ä (t -s) h(t)-1/2 + -(-s) h(t)-1/2 + ä W (ds).
(2.1)

Entre autres choses, il est établi dans [START_REF] Peltier | Multifractional brownian motion[END_REF], que pour tout intervalle I,l ep r oc e s s u sX h n'est pas stationnaire et admet une rectification hölderienne de ses trajectoires de tout ordre γ strictement inférieur à inf u∈I h(u).L e processus X h n'étant pas stationnaire, il est plus intéressant de chercher des propriétés de régularité locale plutôt que globales le concernant. L'idée qui a présidé à la définition du mBm bien équilibré est la même que celle qui a permis de définir X h .P l u sp r é c i s é m e n t ,l av e r s i o nb i e né q u i l i b r é ed um B maé t éd é fi n i ee n1 9 9 7d a n s [ 7 ] ,e np a r t a n td el a représentation harmonisable du fBm et en posant,

‹ X h t := R e itξ -1 |ξ| h(t)+1/2 W (dξ).
(2.2)

Même si les processus X h et ‹ X h ne sont pas équivalents (i.e n'ont pas la même loi), nous verrons à la section suivante qu'ils font partie d'un même ensemble de processus. Le fait que l'on ait cru pendant longtemps que X h et ‹ X h désignent le même processus (en loi) d'une part, le fait que ‹ X h est plus facile à manipuler pour effectuer des calculs d'autre part explique que les auteursai e n tp ré f é réd on n e rl e sp ropriétés de régularité et de dépendance à long terme pour le processus ‹ X h .N o u sr e s t i t u o n si c iu n ep a r t i ed el e u r sr é s u l t a t s . Bien que n'étant plus auto-similaire et bien que n'étant pas stationnaire, le mBm possède une propriété plus faible appelée localisabilité. Si la fonction h est β-hölderienne, alors pour tout t tel que h(t) <β,o na ß

X h t+ρu -X h t ρ h(t)
; u ∈ R + ™ converge en loi, lorsque ρ tend vers 0,v e r su nf B md ' i n d i c ed eH u r s th(t).

Régularité hölderiènne de ‹ X h

Il a été établi dans [START_REF] Herbin | Processus (multi-) fractionnaires à paramètres multidimensionnels et régularité.P h Dt h e s i s[END_REF] que, pour tout réel positif t, α X h (t)= β(t) ∧ h(t) p.s où β(t) désigne l'exposant de Hölder local de la fonction h en t.

Les trois figures qui suivent, toujours obtenues grâce au logiciel Fraclab, représentent des trajectoires, t → (t, X h t (ω)),d em o u v e m e n t sb r o w n i e n sm u l t i f r a c t i o n n a i r e sX h ayant différents paramètres fonctionnels. Al av u ed ec e st r o i sfi g u r e s ,o nc o n s t a t el àe n c o r eq u ep l u sl av a l e u rh(t) est grande plus les trajectoires sont régulières. Réciproquement, plus la valeur h(t) est petite plus les trajectoires sont irrégulières.

Corrélation des accroissements du mBm et dépendance à long terme

Puisque les accroissements du mBm ne sont pas stationnaires il convient de redéfinir la notion de dépendance àlongterme.Dans [5],ilestproposéunenouv elledéfinitiondedépendanceàlongtermequenousreprenons ici. Précisons d'abord que nous appelons processus du second ordre tout processus X := (X t ) t∈R tel que X t appartient à L 2 (Ω) pour tout réel t. Les deux résultats principaux, portant sur ‹ X h sont les suivants :

Proposition 2.1. [5, corollaire 1]Pourtoutefonctionh non constante, le mBm ‹ X h ,définien(2.2) présente une dépendance à long terme au sens de la définition 2.2. De plus, si pour tout s, h(s)+h(t) > 1,p o u r tout t suffisamment grand, alors le mBm ‹ X h présente également une dépendance à long terme au sens de la définition 2.1, le paramètre fonctionnel α vérifiant α(s)=h(s) -1 pour tout s in R + .

Proposition 2.2. [5, Proposition 8](Comportement asymptotique des accroissements du mBm) Soit ‹ X h le mBm défini en (2.2) et Z h := (Z h t ) t∈R+ le processus défini par Z h t := ‹ X h t+1 -‹ X h t . Lorsque t tend vers +∞,e tp o u rt o u ts 0 tel que les quatre quantités a 0,0 (t, s) := h(t)+h(s), a 1,0 (t, s) := h(t +1)+h(s), a 0,1 (t, s) := h(t)+h(s +1) et a 1,1 (t, s) := h(t +1)+h(s +1) sont toutes différentes, alors :

max (i,j)∈{1;2} 2 a i,j (t, s) < 1 ⇒ cov Z h (t, s) ≈ t→+∞ 1, max (i,j)∈{1;2} 2 a i,j (t, s) > 1 ⇒ cov Z h (t, s) ≈ t→+∞ t max (i,j)∈{1;2} 2
ai,j (t,s)-1 .

Les différentes représentations du mBm

• Sur R Comme nous l'avons indiqué précédemment, les processus gaussiens X h et ‹ X h ne sont pas égaux en loi. Pour autant, ils ont comme point commun d'être des fBm "sur lequels on fait courir une fonction h". En 2006 S.Stoev et M.Taqqu proposèrent, dans [73], une nouvelle définition de mBm, plus générale, et qui englobait les processus originels X h et ‹ X h .U n ef o n c t i o nβ-Hölderienne h étant fixée, ils proposèrent, et partant de (1.6), de définir un mBm, pour tout (a + ,a -) de R 2 \{(0, 0)},p a r = Y (a + ,a -) (t).

Y (a + ,a -) (t) := δ(h(t)) R Ä a + f + (t, h(t),u)+a -f -(t, h(t),u) ä W (du), (2.3
Cette nouvelle famille de mBm étant définie, les considérations que nous avons développées au paragraphe précédent demeurent valables. Aussi, et plutôt que de donner la fonction de covariance des processus ‹ Y (a + ,a -) , dont une forme intégrale se déduit immédiatement de (2.4), nous allons donner et étudier la fonction de corrélation des mBm définis par (2.4) ou, ce qui revient au même, (2.3). Par ailleurs, il convient de se demander dans quelle mesure les éléments de cette famille de mBm sont différents. Cette question délicate a été traitée, dans le cadre de cette nouvelle définition, dans [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF]. Nous rappelons ici les principaux résultats relatifs à la structure de covariance des mBm Y (a + ,a -) qui y sont établis puisqu'ils permettent de mieux saisir la nature intrinsèque de ces mBm. Commençons par la fonction de covariance.

Proposition 2.3 ([75]

). Pour tout couple (a + ,a -) dans R 2 \{(0, 0)} et tout réel t,n o u sa v o n s

Var(Y (a + ,a -) (t)) = C 2 h (t) |U (a + ,a -) (ξ; h(t))| 2 pour tout ξ =0, (2.5)
où

C 2 h (t) := |t| 2h(t) cos(πh(t))Γ 2 (1/2+h(t))Γ(2-2h(t)) πh(t)(1-2h(t)) si h(t) =1/2 1 si h(t)=1/2 (2.6)
Nous ne nous intéressons plus, à partir de maintenant, qu'aux processus (Y (a + ,a -) (t)) t∈R ,t e l sq u el ' o na i t a + = a -ou bien h(t) =1/2. Définissons en outre la fonction α : R → [0, 2π[,p a r α(t) := Arg(a + e -i(h(t)+1/2)π/2 + a -e i(h(t)+1/2)π/2 ),

où Arg désigne la détermination principale de l'argument sur [0; 2π[.L af o n c t i o nd ec o r r é l a t i o nd ec ep r ocessus, notée ρ (a + ,a -) ,e s td é fi n i e ,p o u rt o u tc o u p l e(a + ,a -) de R 2 \{(0, 0)} par

ρ (a + ,a -) (t, s) := E[Y (a + ,a -) (t)Y (a + ,a -) (s)] »
Var(Y (a + ,a -) (t)) Var(Y (a + ,a -) (t)) .

(2.8)

Nous avons alors le résultat suivant :

cos(∆α t,s )πh t,s sign(t -s)|t -s| 2ht,s ã , où ∆α t,s := α(t)α(s), α ayant été définie en (2.7) D h (t, s) := Γ(h(t)+1/2) Γ(h(s)+1/2)

πC h (t) C h (s) Γ(2-2ht,s) 2ht,s(1-2ht,s) .
-s ih t,s =1/2, alors

ρ (a + ,a -) (t, s)= Γ(h(t)+1/2) Γ(h(s)+1/2) πC h (t) C h (s) (π/2cos(∆α t,s )(|t| + |s|-|t -s|)) -sin(∆α t,s )(t ln |t|-s ln |s|-(t -s) ln(|t -s|)) (2.10)
Comparaison entre mBm de paramètres (a + ,a -) différents

Nous allons voir ici que la famille de mBm définie par (2.3) est en fait suffisamment riche pour contenir des processus de fonctions de corrélation différentes. • Les différentes représentations du mBm sur un intervalle Pour obtenir un mBm sur un compact, par exemple [0,T],i ls u ffi td ec o n s i d é r e r(Y (a + ,a -) (t)) t∈[0;T ] .C ependant, dans [3], il est indiqué une méthode d'intégration stochastique, utilisant le calcul de Malliavin, par rapport à tous les processus gaussiens pouvant s'écrire comme intégrale sur [0,T] d'un noyau K(t, .) ayant certaines propriétés. Dans l'article [START_REF] Boufoussi | Local time and Tanaka formula for a Volterra-type multifractional Gaussian process[END_REF] une intégrale stochastique par rapport au processus gaussien

ρ (a + ,a -) (t, s)=ρ (b + ,b -) (t, s) (b) Si a + b -= a -b + et si h est continue alors, pour tout s =0fixé, il existe, pour tout χ>0,u n ef o n c t i o n h χ : R → (0, 1),v é r i fi a n t , sup t∈R |h(t) -h χ (t)| <χ, telle que ρ (a + ,a -) (t, s; h χ ) = ρ (a + ,a -) (t, s; h χ ), pour tout ρ (a + ,a -) (t, s; h χ ) = ρ (b + ,b -) (t, s; h χ ), pour tout t ∈ ® (s, s + ε) si s>0 (s -ε, s) si s<0,
Y h := (Y h t ) t∈[0,T ] ,d é fi n ip o u rt o u tt par Y h t := t 0 K h(t) (t, u) W (du) où K H (., .
) aé t éd é fi n i ej u s t ea p r è s (1.9), est étudiée, à partir des propriétés de l'intégrale stochastique établies dans [3]. Le processus Y H (correspondant au cas où h est constante et égale à H)étantunfBm,ilestnatureldesedemandersileprocessus Y h est un mBm au sens de la définition 2.3. Si tel était le cas, il existerait un couple (a + ,a -) de R 2 et une fonction δ tels que l'on ait l'égalité

T 0 K h(t) (t, u) K h(s) (s, u) du = δ(h(t)) δ(h(s)) R Ä a + f + (t, h(t),u) -a -f -(t, h(t),u) äÄ a + f + (s, h(s),u) -a -f -(s, h(s),u) ä du.
(2.11)

Nous donnons dans le deuxième chapitre du présent mémoire des raisons qui nous font croire qu'une telle égalité a peu de chances d'être vérifiée. De plus, que la dernière égalité soit vérifiée ou non, accepter la définition 2.3 comme définition des mBm revient à considérer qu'un mBm sur un intervalle I quelconque de R est un processus gaussien dont la fonction de covariance doit nécessairement appartenir à l'ensemble de fonctions

{(t, s) → E[Y (a + ,a -) (t)Y (a + ,a -) (s)]; (a + ,a -) ∈ R 2 }.
Ainsi, tout processus gaussien (X(t, h(t))) t∈I tel que (X(t, H)) t∈I est un fBm de coefficient de Hurst H pour tout H de ]0; 1[,m a i st e lq u e(X(t, H)) t∈I ne puisse pas s'écrire sous la forme (1.6), ne pourrait pas être considéré comme un mBm. Outre l'arbitraire de la définition 2.3, son principal inconvénient est de ne pas tenir compte du fait qu'un mBm "devrait être un fBm dans lequel on a changé H en h(t)", car c'est bien là l'idée originelle et originale de [START_REF] Peltier | Multifractional brownian motion[END_REF].

Ceci nous amène à penser que la définition 2.3, efficace en ce qu'elle permettait de généraliser la notion de mBm, est à la fois trop arbitraire et pas assez générale. Nous proposons au chapitre 2 du présent mémoire, qui est extrait de [START_REF] Herbin | Stochastic integration with respect to multifractional brownian motion via tangent fractional brownian motion[END_REF], une nouvelle définition de mBm qui pallie ces différents inconvénients.

Calcul stochastique par rapport au mBm

Le calcul stochastique par rapport au mBm faisant l'objet de cette thèse, nous renvoyons à la description des travaux du présent manuscrit, située au prochain paragraphe, pour un plan détaillé.

3P r é s e n t a t i o n d e s t r a v a u x d e t h è s e

L'objectif de cette thèse était de construire, développer et étudier un calcul stochastique (et plus particulièrement une intégrale stochastique) par rapport au mouvement brownien multifractionnaire (dans sa version harmonisable). Le choix de la méthode ou plus exactement des outils à employer n'étant pas fixé a priori, notre choix s'est porté sur la théorie du bruit blanc, qui généralisait, dans le cas du fBm, le calcul stochastique au sens de Malliavin.

Chapitre 2

Dans le deuxième chapitre de cette thèse nous donnons une construction ainsi que les principales propriétés de l'intégrale stochastique par rapport au mBm harmonisable. Y sont également établies des formules d'Itô et une formule de Tanaka pour l'intégrale stochastique par rapport à ce mBm. Plus précisément, à partir des travaux de [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF], [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF] et [START_REF] Bender | An S-transform approach to integration with respect to a fractional Brownian motion[END_REF] nous montrons explicitement que l'ensemble des éléments déterministes qui admettent une intégrale de Wiener par rapport au fBm d'indice de Hurst H est exactement l'espace de Sobolev d'ordre 1/2 -H,n o t éL 2 H (R),e td é fi n ip a r

L 2 H (R) := {u ∈ S ′ (R): u = T f ; f ∈ L 1 loc (R) and ||u|| H < +∞}, (3.1)
où T f désigne la transformée de Fourier au sens des distributions, ||u||

H := 1 c 2 H R |ξ| 1-2H |" u (ξ)| 2 dξ est la norme issue du produit scalaire défini, sur L 2 H (R) × L 2 H (R),p a r: <u,v> H := 1 c 2 H R |ξ| 1-2H " u (ξ) " v (ξ) dξ, (3.2) où c x := Å 2cos(πx)Γ(2-2x) x(1-2x) ã 1 2 2
pour tout x de (0, 1).N o t o n sq u eL 2 H (R) est aussi l'image par un opérateur bijectif M H de l'espace L 2 (R) (cf. théorème 3.7 du chapitre 2 pour plus de précisions). Après avoir introduit, dans la partie 3, et étudié les principales propriétés de l'opérateur ∂MH ∂H ,d é fi n ic om m e étant la dérivée, au sens de L 2 (R),del'opérateurM H ,nousconstruisonsparunepremièreméthodel'intégrale de Wiener par rapport au mBm. Pour ce faire, nous établissons (cf. proposition 4.1) que la forme bilinéaire, notée <, > h ,d é fi n i es u rE(R) ×E(R) (où E(R) désigne l'espace des fonctions en escalier) par,

< 1 [0,t] , 1 [0,s] > h = R h (t, s)
est un produit scalaire. Nous ne considérons plus, à partir de la section 5,q u ' u n ef o n c t i o nh de classe C 1 .N o u sd é fi n i s s o n sd ' a b o r d la dérivée, au sens des distributions stochastiques, du mouvement Brownien multifractionnaire par :

W (h) (t) := +∞ k=0 ï d dt Å t 0 M h(t) (e k )(s) ds ãò < ., e k >, (3.3)
où la famille (e k ) k∈N désigne la famille orthonormée des fonctions de Hermite (définies au cf. chapitre 2 égalité (2.4)). Nous donnons ensuite le Théorème-Définition 3.1. Si la fonction h ′ est bornée, le processus W (h) := (W (h) (t)) t∈R defini par la formule (3.3) est un (S) * -processus qui vérifie dans (S) * ,l ' é g a l i t és u i v a n t e:

W (h) (t)= +∞ k=0 M h(t) (e k )(t) < ., e k > +h ′ (t) +∞ k=0 Å t 0 ∂MH ∂H (e k )(s) H=h(t) ds ã < ., e k >. (3.4)
De plus le processus

B (h) est (S) * -differentiable sur R et vérifie, dans (S) * , dB (h) dt (t)=W (h) (t)= d dt [Λ(t, h(t))]. (3.5)
Nous définissons enfin l'intégrale stochastique par rapport au mBm de la façon suivante, désignant le produit de Wick,

Définition 3.1 (Intégrale de Wick-Itô multifractionnaire). Soit Y : R → (S) * un processus tel que t → Y (t) W (h) (t) est (S)
* -integrable sur R.O nd i tq u el ep r o c e s s u sY est dB (h) -intégrable ou intégrable sur R par rapport au mBm B (h) .L ' i n t e g r a l ed eY par rapport à

B (h) est définie par R Y (s) dB (h) (s) := R Y (s) W (h) (s)ds. (3.6)
Pour tout intervalle

I de R,o nd é fi n i t I Y (s)dB (h) (s) := R 1 I (s) Y (s)dB (h) (s).
Le reste de la section est consacré à la résolution de l'équation différentielle stochastique dont la solution est un mBm géométrique ainsi qu'à la définition d'une seconde intégrale de Wiener par rapport au mBm, issue de la définition 3.1. Il appert alors que l'ensemble des intégrales de Wiener par rapport au mBm définies àl as e c t i o n3 et l'ensemble des intégrales d'éléments deterministes obtenues à partir de la définition 3.1 coincident.

Le paragraphe 6 est consacré à l'obtention de formules d'Itô dans l'espace (L 2 ) mais aussi dans l'espace des distributions stochastiques (S) * .P a r m ic e l l e s -c i ,c i t o n sl e sd e u xr é s u l t a t ss u i v a n t s .

F (b, B (h) (b)) -F (a, B (h) (a)) = b a ∂F ∂t (s, B (h) (s)) ds + b a ∂F ∂x (s, B (h) (s)) dB (h) (s) + 1 2 b a Å d ds [R h (s, s)] ã ∂ 2 F ∂x 2 (s, B (h) (s)) ds.
Théorème 3.2. Soit T>0 et h : R → (0, 1) une fonction de classe C 1 telle que h ′ est bornée sur R et soit f une fonction de classe C 1,2 ([0,T] × R, R).S u p p o s o n sq u ' i le x i s t ed e u xc o n s t a n t e sC 0 et λ<

1 4m a x t∈[0,T ] t 2h(t) telles que pour tout (t, x) de [0,T] × R, max t∈[0,T ] ß |f (t, x)| , ∂f ∂t (t, x) , ∂f ∂x (t, x) , ∂ 2 f ∂x 2 (t, x) ™ Ce λx 2 .
Alors pour tout t de [0,T],l ' é g a l i t és u i v a n t ee s tv é r i fi é ed a n s(L 2 ) :

f (T,B (h) (T )) = f (0, 0) + T 0 ∂f ∂t (t, B (h) (t)) dt + T 0 ∂f ∂x (t, B (h) (t)) dB (h) (t) + 1 2 T 0 Å d dt [R h (t, t)] ã ∂ 2 f ∂x 2 (t, B (h) (t)) dt.
L'obtention d'une formule de Tanaka ainsi que l'étude de deux mBm de paramètre fonctionnel h particulier achèvent le chapitre 2.

Chapitre 3

Le point de départ du chapitre 3 est la question suivante : Peut-on approximer un mBm de paramètre fonctionnel h par une suite de sommes de fBm de coefficient de Hurst égaux à certaines valeurs prises par la fonction h ? Nous interrogeant alors sur ce que devrait être (compte tenu des arguments avancés dans les deux soussections qui précèdent) un mBm, nous rédéfinissons le mBm à partir de la notion de champ fractionnaire.

Le champ fractionnaire (B(t, H)) (t,H)∈R×(0,1) est un champ gaussien tel que pour tout H dans (0, 1),l e processus (B H t ) t∈R défini par B H t := B(t, H) est un mouvement brownien fractionnaire de coefficient de Hurst H.U nm o u v e m e n tb r o w n i e nm u l t i f r a c t i o n n a i r ed ep a r m è t r ef o n c t i o n n e lh étant alors défini comme étant le processus (B(t, h(t))) (t,H)∈R×(0,1) .C e t t ed é fi n i t i ongén é ral i s et ou t e sl e sd é fi n i t i on sd em B mc on n u e s àc ej o u r . Un intervalle [a, b] de R étant fixé, nous construisons, à partir du champ fractionnaire B,l as u i t ed es o m m e de fBm par morceaux, pour tout t dans [a, b] et n dans N,p a r:

B hn t := B(t, h n (t)) = qn-1 k=0 [x (n) k ,x (n) k+1 ) (t) B(t, h(x (n) k )) + {b} (t) B(b, h(b)). (3.7)
L'hypothèse de régularité suivante permet de répondre à la question posée au début de ce paragraphe.

(H 1 ):∀[a, b] ⊂ R, ∀[c, d] ⊂ (0, 1), ∃(Λ,δ) ∈ (R * + ) 2 , such that E[(B(t, H) -B(t, H ′ )) 2 ] Λ |H -H ′ | δ , pour tout (t, H, H ′ ) in [a, b] × [c, d] 2 .
En effet, on montre le théorème d'approximation suivant :

) → E[B(t, H) B(s, H ′ )] est conti- nue sur R 2 × (0, 1) 2 ,a l o r sl as u i t ed ep r o c e s s u s(B hn ) n∈N converge dans L 2 (Ω) vers B h ,i.e ∀t ∈ [a, b], lim n→+∞ E Ä B hn t -B h t ä 2 =0. 2. Si B satisfait l'hypothèse (H 1 ) et si h is β-Hölderienne pour un certain β>0,alorslasuitedepr o c essus (B hn ) n∈N * converge (i) en loi, i.e {B hn t ; t ∈ [a, b]} loi -----→ n→+∞ {B h t ; t ∈ [a, b]}. (ii) presque sûrement, i.e P Å {∀t ∈ [a, b], lim n→+∞ B hn t = B h t } ã =1.
La question qui se pose ensuite naturellement est las u i v a n t e:P e u t -o nàp a r t ir du précédent théorème inférer une méthode d'intégration stochastique par rapport au mBm à partir des méthodes d'intégration stochastiques existant pour le fBm ? Une méthode (M) d'intégration par rapport au fBm étant fixée (Chemins rugueux, calcul de Malliavin, Théorie du bruit blanc, par exemple) il convient alors de donner la définition suivante : Définition 3.2 (intégrale par morceaux par rapport à une somme de fBm). Soit Y := (Y t ) t∈[0,1] un processus sur [0; 1] intégrable par rapport à tout fBm d'indice de Hurst H appartenant à h([0; 1]),a usensd elamétho de (M). On appelle intégrale par morceaux par rapport à une somme de fBm au sens de la méthode (M) :

1 0 Y t d (M) B hn t := qn-1 k=0 1 0 [x (n) k ,x (n) k+1 ) (t) Y t d (M) B h(x (n) k ) t ,n ∈ N. ( 3
1 0 Y t d (M) B h t := lim n→∞ 1 0 Y t d (M) B hn t :=Ln(Y ) +Φ (M) h (Y ), (3.9)
où la limite dans (3.9) est prise au sens de L 2 (Ω) ou de (S) * selon la méthode (M) employée et où

Φ (M) h (Y ) := 1 0 h ′ (t) ϕ (M) (Y t , ∂B ∂H (t, h(t))) dt,
avec ϕ (M) (x, y) := x y ou xy selon la méthode employée ( désignant le produit de Wick). Le champ fractionnaire B étant, quant-à lui, supposé vérifier l'hypothèse (H 2 ) suivante :

(H 2 ):∀[a, b] × [c, d] ⊂ R × (0, 1),H → B(t, H) est de classe C 1 ,a us e n sd eL 2 (Ω),d e(0, 1) dans L 2 (Ω) pour tout t de [a, b] et ∃(∆,α,λ) ∈ (R * + ) 3 s.t. E Ä ∂B ∂H (t, H) -∂B ∂H (s, H ′ ) ä 2 ∆ Ä |t -s| α + |H -H ′ | λ ä ,p o u r tout (t, s, H, H ′ ) de [a, b] 2 × [c, d]
H E := α∈h([0,1]) Y ∈ E [0,1] : [0,1] Y t d (M) B α t existe
0 k n-1 a k ) -1 =0.
Supposons de plus que la suite (q n ) n∈N ,d e fi n i ep a rq 0 =1et q n+1 = a n q n pour tout n dans N soit telle que l'on puisse trouver une norme sur H E ,n o t é e HE ,t e l l eq u ' i le x i s t eM>0 et telle que pour toute famille de boréliens disjoints A 1 ,...,A n de [0, 1],

Y. A1 HE + •••+ Y. An HE M Y HE .

Supposons de plus que la fonction

I : H E × (0, 1) → F ,d é fi n i ep a r ∀ Y ∈H E , ∀α ∈ (0, 1), I(Y, α) := [0,1] Y t d (M) B α t ,
est θ-Hölderienne par rapport à α uniformément en Y pour un certain réel θ,i . e .i le x i s t eθ>0 et K>0 tels que

∀(α, α ′ ) ∈ (0, 1) 2 , sup Y H E 1 I(Y, α) -I(Y, α ′ ) F K |α -α ′ | θ .
Alors la suite de fonctionnelles (L n ) n∈N définie en 3.9 converge simplement vers une fonction L : H E → F .

Dans les parties

Chapitre 4

Le troisième et dernier chapitre de ce mémoire, extrait de [START_REF] Corlay | Multifractional volatility models[END_REF], concerne une application aux mathématiques financières du calcul stochastique développé dans les deuxpremierschapitres.Plusprécisement, nous partons du modèle à volatilité stochastique proposé dans [START_REF] Comte | Long memory continuous-time stochastic volatility models[END_REF], qui s'écrit,

ß dF t = µ(t, F t )dt + F t σ t dW t , d ln(σ t )=θ (µ -ln(σ t )) dt + γdB H t ,σ > 0, (3.10)
où (F t ) t∈[0;T ] désigne le processus de prix forward d'une action, W un mouvement brownien et (B H t ) t∈[0;T ] un fBm de coefficient de Hurst H,i n d e p e n d a n td eW . L'introduction d'un fBm dans (3.10) tient au fait que les auteurs souhaitaient rendre compte du phénomène de dépendance à long terme du processus de volatilité observé en pratique mais dont le modèle de Hull et White standard (i.e (3.10) où un brownien est mis à la place de B H )nerendpascompte.Malheureusemen t , et comme nous l'avons indiqué dans le paragraphe de cette introduction consacré aux propriétés du fBm, la dépendance à long terme n'existe pour les accroissements du processus B H que si H>1/2.Depluslemodèle (3.10) ne permet pas de modéliser de façon satisfaisante des smiles de volatilité de différentes maturités. Pour pallier ces deux inconvénients l'idée est de substituer au fBm B H un mBm B h dans (3.10). Le modèle àv o l a t i l i t és t o c h a s t i q u ed e v i e n ta l o r s

   dF t = F t σ t dW t , d ln(σ t )=θ (µ -ln(σ t )) dt + γ h d B h t + γ σ dW σ t ,σ 0 > 0, d W, W σ t = ρdt, (3.11) où d B h
t désigne la différentielle prise au sens de la théorie du bruit blanc. Ceci nous permet ainsi de considérer un processus de volatilité dont les accroissements ne sont pas stationnaires et présentent une dépendance à long terme y compris lorsque la régularité Hölderienne de la volatilité est inférieure à 1/2. La résolution de cette equation différentielle stochastique est exacte en ce qui concerne le processus de volatilité qui est donné par,

σ t =exp ln(σ 0 )e -θt + µ Ä 1 -e -θt ä + γ σ t 0 e θ(s-t) dW σ s + γ h t 0 e θ(s-t) d B h s .
En revanche, la solution donnant (F

(R + ) 2 , E N Ä B h , |•| L p ([0,T ]) ä = O log(N ) -(H1∧ βδ 2 ) , où E N Ä B h , |•| L p ([0,T ])
ä désigne l'erreur de quantification, au sens de la norme L p (R) du mBm.

Suivant la méthode proposée, par exemple dans [START_REF] Pagès | Functional quantization for numerics with an application to option pricing[END_REF], le préconditionnement par rapport à la tribu F σ,h T nous permet d'écrire :

E Å F T F τ -K ã + = E E Å F T F τ -K ã + F σ,h T :=ψ(W σ ,B h )
, Fractional Brownian motion (fBm) [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum[END_REF][START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF] is a centered Gaussian process with features that make it a useful model in various applications such as financial and teletraffic modeling, image analysis and synthesis, geophysics and more. These features include self-similarity, long range dependence and the ability to match any prescribed constant local regularity. Fractional Brownian motion depends on a parameter, usually denoted H and called the Hurst exponent, that belongs to (0, 1).I t sc o v a r i a n c ef u n c t i o nR H reads:

pour
R H (t, s) := γ H 2 (|t| 2H + |s| 2H -|t -s| 2H ), (1.1)
where γ H is a positive constant. A normalized fBm is one for which γ H =1.O b v i o u s l y ,w h e nH = ,f B m reduces to standard Brownian motion. Au s e f u lr e p r e s e n t a t i o no ff B mB (H) of exponent H is the so-called harmonizable one:

B (H) (t)= 1 c H R e itu -1 |u| H+1/2 W (du), (1.2)
where

c x := Å 2cos(πx)Γ(2-2x) x(1-2x) ã 1 2 = Å 2π Γ(2x+1) sin(πx) ã 1 2
for x in (0, 1) and W denotes the complex-valued Gaussian measure which can be associated in a unique way to W ,a ni n d e p e n d e n t l ys c a t t e r e ds t a n d a r d Gaussian measure on R (see [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF] p.203-204 and [73] p.325-326 for more information on the meaning of R f (u) W (du) for a complex-valued function f ). From (1.1) and Gaussianity, it is not hard to prove that fBm is H-self-similar. The fact that most of the properties of fBm are governed by the single number H restricts its application in some situations. Let us give two examples. The longt e r mc o r r e l a t i o n so ft h ei n c r e m e n t so ff B md e c a ya s k (2H-2) ,w h e r ek is the lag, resulting in long range dependence when H>1/2 and anti-persistent behavior when H<1/2.A l s o ,a l m o s ts u r e l y ,f o re a c ht,i t sp o i n t w i s eH ö l d e re x p o n e n ti se q u a lt oH.S i n c eH rules both ends of the Fourier spectrum, i.e.t h eh i g hf r e q u e n c i e sr e l a t e dt ot h eH ö l d e rr e g u l a r i t ya n dt h el o w frequencies related to the long term dependence structure, it is not possible to have at the same time e.g. a very irregular local behavior (implying H close to 0) and long range dependence (implying H>1/2). As a consequence, fBm is not adapted to model phenomena which display both these features, such as Internet traffic or certain highly textured images with strong global organization. Another example is in the field of image synthesis: fBm has frequently been used for generating artificial mountains. Such a modeling assumes that the regularity of the mountain is everywhere the same. This is not realistic, since it does not take into account erosion or other meteorological phenomena which smooth some parts of mountains more than others. Multifractional Brownian motion (mBm) [START_REF] Peltier | Multifractional brownian motion[END_REF]7] was introduced to overcome these limitations. The basic idea is to replace in (1.2) the real H by a function h(t).M o r ep r e c i s e l y ,w ew i l lu s et h ef o l l o w i n gd e fi n i t i o n of mBm: Definition 1.1 (Multifractional Brownian motion). Let h : R → (0, 1) be a continuous function and α : (0, 1) → R be a C 1 function. A multifractional Brownian motion with functional parameters h and α is defined as:

B (h,α) (t)=α(h(t)) R e itu -1 |u| h(t)+1/2 W (du).
(

Its covariance function reads [5]:

R (h,α) (t, s)=α(h(t)) α(h(s)) c 2 ht,s ï 1 2 Ä |t| 2ht,s + |s| 2ht,s -|t -s| 2ht,s ä ò , (1.4)
where h t,s := h(t)+h(s) 2 and c x has been defined in (1.2). It is easy to check that mBm is a zero mean Gaussian process, the increments of which are in general neither independent nor stationary. For T in R * + ,w ew i l la g a i nc a l l(h, α)-multifractional Brownian motion on [0,T] the centered Gaussian process whose covariance function is equal to

R (h,α) on [0,T] × [0,T]. When α = α c : x → 1
cx ,w eg e tt h a t :

R (h,αc) (t, s)= c 2 ht,s c (h(t)) c (h(s)) ï 1 2 Ä |t| 2ht,s + |s| 2ht,s -|t -s| 2ht,s ä ò . (1.5)
As a consequence, if h is the constant function equal to the real H,t h e nB (H,αc) is a normalized fBm. For this reason, we will call B (h,αc) an o r m a l i z e dm B m . S i n c ei nt h es e q u e lwe will consider only normalized mBm, we simplify the notation and write from now on B (h) for B (h,αc) and R h for R (h,αc) . One can show [START_REF] Herbin | From n-parameter fractional Brownian motions to n-parameter multifractional Brownian motions[END_REF][START_REF] Herbin | Stochastic 2 micro-local analysis[END_REF] that the pointwise Hölder exponent at any point t of B (h) is almost surely equal to h(t) ∧ β h (t),w h e r eβ h (t) is the pointwise Hölder exponent of h at t.I na d d i t i o n ,t h ei n c r e m e n t so fm B m display long range dependence for all non-constant h(t) (the notion of long range dependence must be redefined carefully for non-stationary increments, see [5]). Finally, at least when h is C 1 ,f o ra l lu ∈ R,m B m locally "looks like" fBm with exponent h(u) in the neighbourhood of u in the following sense [START_REF] Peltier | Multifractional brownian motion[END_REF]:

lim r→0+ B (h) (u + rt) -B (h) (u) r h(u) = B (h(u)) (t), (1.6)
where the convergence holds in law. These properties show that mBm is a more versatile model that fBm: in particular, it is able to mimic in a more faithful way local properties of financial records, Internet traffic and natural landscapes [START_REF] Bianchi | Pathwise identification of the memory function of multifractional brownian motion with application to finance[END_REF][START_REF] Li | Towards describing multi-fractality of traffic using local Hurst function[END_REF][START_REF] Echelard | Terrain modelling with multifractional Brownian motion and self-regulating processes[END_REF] by matching their local regularity. This is important e.g. for purposes of detection or real-time control. The price to pay is of course that one has to deal with the added complexity brought by having a functional parameter instead of a single number. Because of applications, in particular in finance and telecommunications, it has been an important objective in recent years to define a stochastic calculus with respect to fBm. This was not a trivial matter, as fBm is not a semi-martingale for H = 1 2 .S e v e r a la p p r o a c h e s h a v eb e e np r o p o s e d ,b a s e dm a i n l yo nM a l l i a v i n calculus [START_REF] Decreusefond | Stochastic analysis of the fractional Brownian motion[END_REF]3], pathwise approaches and rough paths ( [START_REF] Zähle | On the link between fractional and stochastic calculus[END_REF][START_REF] Coutin | An Introduction to (Stochastic) Calculus with Respect to Fractional Brownian Motion[END_REF][START_REF] Friz | Multidimensional Stochastic Processes as Rough Paths: Theory and Applications[END_REF] and references therein), and white noise theory [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF][START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF][START_REF] Bender | An S-transform approach to integration with respect to a fractional Brownian motion[END_REF]. Since mBm seems to be a more flexible, albeit more complex, model than fBm, it seems desirable to extend the stochastic calculus defined for fBm to it. This is the aim of the current work. In that view, we will use a white noise approach, as it offers several advantages in our frame. The main task is to define a multifractional white noise as a Hida stochastic distribution, which generalizes the fractional white noise of, e.g.,[ 3 4 ,1 1 ] . F o rt h a tp u r p o s e ,w eu s et h ep r o p e r t i e so ft h eG a u s s i a nfi e l d(B (H) (t)) (t,H)∈R×(0,1) . In particular, it is a crucial fact for us that the function H → (B (H) (t)) is almost surely C ∞ .T h i se n t a i l s that multifractional white noise behaves essentially as fractional white noise, plus a smooth term. We obtain an Ito formula that reads:

f (T,B (h) (T )) = f (0, 0) + T 0 ∂f ∂t (t, B (h) (t)) dt + T 0 ∂f ∂x (t, B (h) (t)) dB (h) (t) + 1 2 T 0 Å d dt [R h (t, t)] ã ∂ 2 f ∂x 2 (t, B (h) (t)) dt,
where the meaning of the different terms will be explained below. The remaining of this paper is organized as follows. In section 2, we recall basic facts about white noise theory. We study a family of operators, noted (M H ) H∈(0,1) ,w h i c ha r ei n s t r u m e n t a lf o rc o n s t r u c t i n gt h e stochastic integral with respect to mBm in section 3.S e c t i o n4d e fi n e st h eW i e n e ri n t e g r a lw i t hr e s p e c tt o mBm. We build up a stochastic integral with respect to mBm in section 5. Various instances of Ito formula are proved in section 6. Finally, section 7 provides a Tanaka formula, along with the study of two particular h functions that give notable results. Readers familiar with white noise theory may skip the next section.

2W h i t e n o i s e t h e o r y

We recall in this section the standard set-up for classical white-noise theory. We refer e.g. to [START_REF] Kuo | White Noise Distribution Theory[END_REF][START_REF] Hida | White Noise. An infinite dimensional calculus[END_REF] for more details.

White noise measure

Let S (R) := {f ∈ C ∞ (R):∀(p, q) ∈ N 2 , lim |x|→+∞ |x p f (q) (x)| =0 } be the Schwartz space. A family of functions (f n ) n∈N of (S (R))
N is said to converge to 0 as n tends to +∞ if for all (p, q) in N 2 we have

lim n→+∞ sup x∈R |x p f (q) n (x)| =0 .T h et o p o l o
g yh e n c eg i v e no nS (R) is called the usual topology. Let S ′ (R) denote the space of tempered distributions, which is the dual space of S (R).T h eF o u r i e rt r a n s f o r mo fa function f which belongs to L 1 (R) ∪ L 2 (R) will be denoted f or F (f ):

F (f )(ξ) := f (ξ) := R e -ixξ f (x)dx, ξ ∈ R.
(2.1) Define the probability space as Ω := S ′ (R) and let F := B(S ′ (R)) be the σ-algebra of Borel sets. The Bochner-Minlos theorem ensures that there exists a unique probability measure on Ω,d e n o t e dµ,s u c ht h a t :

S ′ (R) e i<ω,f > µ(dω)=e -1 2 ||f || 2 L 2 (R) , ∀f ∈ S (R), (2.2)
where <ω,f >is by definition ω(f ), i.e the action of the distribution ω on the function f .F o rf in S (R) the map, noted < ., f >,fromΩ to R defined by < ., f > (ω)=<ω,f >is thus a centered Gaussian random variable with variance equal to ||f || 2 L 2 (R) under the probability measure µ,w h i c hi sc a l l e dt h ewhite-noise probability measure.I no t h e rw o r d s ,E

[< ., f >]=0and E[< ., f > 2 ]=||f || 2 L 2 (R)
for all f in S (R).B e s i d e s , for a measurable function F ,f r o mS ′ (R) to R,t h ee x p e c t a t i o no fF with respect to µ is defined, when it exists, by E[F ] := E µ [F ] := Ω F (ω)µ(dω).E q u a l i t y( 2 . 2 )e n t a i l st h a tt h em a pζ defined on S (R) by

ζ :( S (R),<,> L 2 (R) ) → (L 2 (Ω, F ,µ),<,> (L 2 (Ω,F ,µ) ) f → ζ(f ) := < ., f > (2.3)
is an isometry. Thus, it extends to L 2 (R) and we still note ζ this extension. For an arbitrary f in L 2 (R),w e then have < ., f >:= lim n→+∞ < ., f n > where the convergence takes place in L 2 (Ω, F ,µ) and where (f n ) n∈N is a sequence of functions which belongs to S (R) and converges to f in L 2 (R).I np a r t i c u l a r ,d e fi n ef o ra l l t in R,t h ei n d i c a t o rf u n c t i o n1 [0,t] by

1 [0,t] (s)=      1 if 0 s t, -1 ift s 0 except if t = s =0 0 otherwise,
Then the process ( ‹ B t ) t∈R , defined for t ∈ R,onΩ by ‹ B t (ω) := ‹ B(t, ω) := <ω,1 [0,t] > is a standard Brownian motion with respect to µ.I tt h e na d m i t sac o n t i n u o u sv e r s i o nw h i c hw i l lb ed e n o t e dB.T h ep r e v i o u se q u a l i t y shows that, for all functions f in L 2 (R),

I 1 (f )(ω)=<ω,f>= R f (s)dB s (ω) µ -a.s.

Properties of Hermite functions and space S ′ (R)

For every n in N,d e fi n e e n (x) := (-1)

n π -1/4 (2 n n!) -1/2 e x 2 /2 d n dx n (e -x 2
) the n th Hermite function.

(

We hence have

e n (x)=π -1/4 (2 n n!) -1/2 H n (x) e -x 2 /2 ,
where H n denotes the nth Hermite polynomial, which is defined by H n (x) := (-1) n e x 2 d n dx n (e -x 2 ).W ew i l l need the following properties of the Hermite functions: Theorem 2.1.

1. The family (e k ) k∈N belongs to (S (R))

N and forms an orthonormal basis of L 2 (R) endowed with its usual inner product.

2. There exists a real constant ‹ C such that, for every k in N, max

x∈R |e k (x)| ‹ C (k +1) -1/12 .M o r e
precisely, there exist positive constants C and γ such that, for every k in N,

|e k (x)| ® C (k +1) -1/12 if |x| 2 √ k +1, Ce -γx 2 if |x| > 2 √ k +1. (2.5)
See [START_REF] Thangavelu | Lectures of Hermite and Laguerre expansions[END_REF] for proofs.

In order to study precisely S (R) and its dual S ′ (R) it is desirable to have a family of norms on the space S (R) which gives us the usual topology. It is well known (see [START_REF] Kuo | White Noise Distribution Theory[END_REF]) that the Schwartz space S (R) is the projective limit of the sequence (S p (R)) p∈N and that the space S ′ (R) of tempered distributions is the inductive limit of the sequence (S -p (R)) p∈N .S i n c e , for any p in N,t h ed u a ls p a c eS ′ p (R) of S p (R) is S -p (R),w ew i l lw r i t eS -p (R) in the sequel to denote the space S ′ p (R). Finally one can show that the usual topology of the space S (R) and the topology given by the family of norms (|| p ) p∈N are the same (see [START_REF] Hida | Brownian Motion.S p r i n g e r -V e r l a g[END_REF] appendix A.3 for example). Moreover, convergence in the inductive limit topology coincides with both convergence in the strong and the weak * topologies of S ′ (R). In view of definition 2.1, it is convenient to have defined on S (R) whose eigenfunctions are the sequence (e n ) n∈N and eigenvalues are the sequence (2n +2) n∈N .I ti se a s yt oc h e c kt h a tt h eo p e r a t o rA,denslydefined on L 2 (R),b yA := -d 2 dx 2 + x 2 +1 verifies these conditions. Note moreover that A is invertible and that its inverse A -1 is a bounded operator on L 2 (R).L e t u s note |g|

(2k +2) 2p <f,e k > 2 L 2 (R) , ∀(p, f ) ∈ Z × L 2 (R). ( 2
f := +∞ k=0 <f,e k > L 2 (R) e k in Dom(A q ),theequalityA q f = +∞ k=0 (2k +2) q <f,e k > L 2 (R) e k holds. Hence, |f | 2 q = |A q f | 2 0 = +∞ k=0 (2k +2) 2q <f,e k > 2 L 2 (R) , ∀q ∈ Z.
(2.7)

Space of Hida distributions

From now on we will denote as is customary (L 2 ) the space L 2 (Ω, G,µ) where G is the σ-field generated by

(< ., f >) f ∈L 2 (R)
.N e i t h e rB r o w n i a nm o t i o nn o rf r a c t i o n a lB r o w n i a nm o t i o n ,w h a t e v e rt h ev a l u eo fH,a r e differentiable (see [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF] for a proof). However, it occurs that the mapping t → B (H) (t) is differentiable from R into a space, noted (S) * ,c a l l e dt h es p a c eo fH i d ad i s t r i b u t i o n s ,w h i c hc o n t a i n s(L 2 ).I nt h i ss e c t i o nw e recall the construction of (S) * . For every random variable Φ of (L 2 ) there exists, according to the Wiener-Itô theorem, a unique sequence

(f n ) n∈N of functions f n in L 2 (R n ) such that Φ can be decomposed as Φ= +∞ n=0 I n (f n ),w h e r e L 2 (R n )
denotes the set of all symmetric functions f in L 2 (R n ) and I n (f ) denotes the n th multiple Wiener-Itô integral of f defined by

I n (f ) := R n f (t)dB n (t)=n! R ( tn -∞ ••• Ä t2 -∞ f (t 1 , ••• ,t n )dB(t 1 ) ä dB(t 2 ) •••dB(t n )),
with the convention that I 0 (f 0 )=f 0 for constants f 0 .F u r t h e r m o r ew eh a v et h ei s o m e t r y

E[Φ 2 ]= +∞ n=0 n! ||f n || 2 L 2 (R n ) .
For any Φ := I n (A ⊗n f n ),w h ereA ⊗n denotes the n th tensor power of the operator A (see [START_REF] Janson | Gaussian Hilbert spaces[END_REF] appendix

Ef o rm o r ed e t a i l sa b o u tt e n s o rp r o d u c t so fo p e r a t o r s ) . The operator Γ(A) is densely defined on (L 2 ) and is called the second quantization operator of A.I ts h a r e s al o to fp r o p e r t i e sw i t ht h eo p e r a t o rA.I np a r t i c u l a ri ti si n v e r t i b l ea n di t si n v e r s eΓ(A) -1 is bounded (see [START_REF] Kuo | White Noise Distribution Theory[END_REF]). Let us denote ||ϕ||

||Φ|| p := ||Γ(A) p Φ|| 0 = ||Γ(A) p Φ|| (L 2 ) , ∀p ∈ Z, ∀Φ ∈ (L 2 ) ∩ Dom(Γ(A) p ). (2.8)
For any p in N,l e t(S p ) := {Φ ∈ (L 2 ): Γ ( A) p Φ exists and belongs to (L 2 )} and define (S -p ) as being the completion of the space (L 2 ) with respect to the norm || || -p .

As in [START_REF] Kuo | White Noise Distribution Theory[END_REF], we let (S) denote the projective limit of the sequence ((S p )) p∈N and (S) * the inductive limit of the sequence ((S -p )) p∈N .T h i sm e a n st h a tw eh a v et h ee q u a l i t i e s(S)= ∩ and that convergence in (S) (resp. in (S) * )m e a n sc o n v e r g e n c ei n(S p ) for every p in N (resp. convergence in (S -p ) for some p in N ). The space (S) is called the space of stochastic test functions and (S) * the space of Hida distributions. As previously one can show that, for any p in N,t h ed u a ls p a c e(S p ) * of S p is (S -p ). Thus we will write (S -p ),i nt h es e q u e l ,t od e n o t et h es p a c e(S p ) * .N o t ea l s ot h a t(S) * is the dual space of (S).W ew i l ln o t e< <, > > the duality bracket between (S) * and (S).I fΦ belongs to (L 2 ) then we have the equality < <Φ,ϕ> > = < Φ,ϕ > (L 2 ) = E[Φ ϕ].F u r t h e r m o r e ,a so n ec a nc h e c k ,t h ef a m i l y(|f | p ) p∈Z is an increasing sequence for every f in S (R).T h u st h ef a m i l y(|| < ., f > || p ) p∈Z is an increasing sequence for every f in S (R). Since we have defined a topology given by a family of norms on the space (S) * it is possible to define a derivative and an integral in (S) * (see [START_REF] Hille | Functional Analysis and Semi-Groups,v o l u m e3 1 . A m e r i c a nM a t h e m a t i c a l Society[END_REF] chapter 3 for more details about these notions). Let I be an interval of R (which may be equal to R).

Definition 2.3 (stochastic distribution process). Af u n c t i o nΦ:I → (S)

* is called a stochastic distribution process, or an (S) * -process, or a Hida process. Definition 2.4 (derivative in (S) * ). Let t 0 ∈ I. A stochastic distribution process Φ:I → (S) * is said to be differentiable at t 0 if the quantity lim r→0 r -1 (Φ(t 0 + r) -Φ(t 0 )) exists in (S)

* .W en o t e dΦ dt (t 0 ) the (S) * -derivative at t 0 of the stochastic distribution process Φ. Φ is said to be differentiable over I if it is differentiable at t 0 for every t 0 in I.

The process Φ is said to be continuous,

C 1 , ••• ,C k , ••• in (S) * if the (S) * -valued function Φ is, continuous, C 1 , ••• ,C k , •••.W ea l

s os a yt h a tt h es t o c h a s t i cd i s t r i b u t i o np r o c e s sΦ is (S)

* -continuous and so on. It is also possible to define an (S)

* -valued integral in the following way ( [START_REF] Kuo | White Noise Distribution Theory[END_REF][START_REF] Hille | Functional Analysis and Semi-Groups,v o l u m e3 1 . A m e r i c a nM a t h e m a t i c a l Society[END_REF]). We first recall that L 1 (R,dt) denotes the set of measurable complex-valued functions defined on

R such that ||f || L 1 (R) := R |f (t)| dt < +∞.
Theorem-Definition 2.1 (integral in (S) * ). Assume that Φ:R → (S) * is weakly in L 1 (R,dt),i . ea s s u m e that for all ϕ in (S),t h em a p p i n gu →< < Φ(u),ϕ > > from R to R belongs to L 1 (R,dt).T h e nt h e r ee x i s t sa n unique element in (S) * ,n o t e d R Φ(u)du such that

< < R Φ(u)du, ϕ > > = R < < Φ(u),ϕ > >d u for all ϕ in (S).
(2.9)

We say in this case that Φ is (S) * -integrable on R in the Pettis sense.I nt h es e q u e l ,w h e nw ed on o ts p e c i f y an a m ef o rt h ei n t e g r a lo fa n(S) * -integrable process Φ on R,w ea l w a y sr e f e rt ot h ei n t e g r a lo fΦ in Pettis' sense. See [START_REF] Kuo | White Noise Distribution Theory[END_REF] p.245-246 or [START_REF] Hille | Functional Analysis and Semi-Groups,v o l u m e3 1 . A m e r i c a nM a t h e m a t i c a l Society[END_REF] def. 3.7.1 p.77 for more details.

S-transform and Wick product

For η in S (R),t h eWick exponential of < ., η >,d e n o t e d: e <.,η> :, is defined as the element of (S) given by : e <.,η> : def = +∞ k=0 k! -1 I k (η ⊗k ) (equality in (L 2 )). More generally, for f ∈ L 2 (R),w ed e fi n e: e <.,f > : as the (L 2 ) random variable equal to e <.,f >-1 2 |f | 2 0 (see [START_REF] Janson | Gaussian Hilbert spaces[END_REF] theorem 3.33). We will sometimes note exp < ., f > instead of : e <.,f > :.T h i sr a n d o mv a r i a b l eb e l o n g st oL p (Ω,µ) for every integer p 1.W en o wr e c a l lt h e definition of the S-transform of an element Φ of (S * ),n o t e dS(Φ) or S[Φ]. S(Φ) is defined as the function from S (R) to R given by ∀η ∈ S (R),S (Φ)(η) := < <Φ,: e <.,η> :> > .

(2.10)

Note that SΦ(η) is nothing but E[Φ : e <.,η> :] = e -1 2 |η| 2 0 E[Φ e <.
,η> ] when Φ belongs to (L 2 ).F o l l o w i n g [START_REF] Bender | An S-transform approach to integration with respect to a fractional Brownian motion[END_REF] , formula (6) and (7),d e fi n ef o rη in S (R) the probability measure Q η on the space (Ω, F ) by its Radon-Nikodym derivative given by dQη dµ def =: e <.,η> :. The probability measures Q η and µ are equivalent. Then, by definition,

∀Φ ∈ (L 2 ),S (Φ)(η)=E Qη [Φ]. ( 2
Φ k k! converges in (S) * ,d efineth eelementexp Φ of (S) * by exp Φ := +∞ k=0 Φ k k! . For f in L 2 (
R) and Φ :=< ., f >,i ti se a s yt ov e r i f yt h a texp Φ given by definition 2.5 exists and coincide with : e <.,f > : defined at the beginning of this section.

Remark 2.6. If Φ is deterministic then, for all Ψ in (S) * , Φ Ψ=ΦΨ.M o r e o v e r ,l e t(X t ) t∈R be a Gaussian process and let H be the subspace of (L 2 ) defined by

H := vect R {X t ; t ∈ R} (L 2 ) . If X and Y are two elements of H then X Y = XY -E[XY ].
We refer to [START_REF] Janson | Gaussian Hilbert spaces[END_REF] chapters 3 and 16 for more details about Wick product. The following results on the S-transform will be used in the sequel. See [START_REF] Kuo | White Noise Distribution Theory[END_REF] It is useful to have a criterion for integrability in (S) * in term of the S-transform. This is the topic of the next theorem (theorem 13.5 in [START_REF] Kuo | White Noise Distribution Theory[END_REF]).

Theorem 2.8. Let Φ:R → (S)

* be a stochastic distribution process satisfying:

(i) The map t → S[Φ(t)](η),f r o mR to R,i sm e a s u r a b l ef o ra l lη in S (R).

(ii) There is a natural integer p,ar e a la and a function L in L 1 (R,dt) such that for all η in S (R),

|S(Φ(t))(η)| L(t) e a|η| 2 p . Then Φ is (S) * -integrable over R.
Lastly, when the stochastic distribution process is an (L 2 )-valued process, the following result holds (see [START_REF] Bender | An S-transform approach to integration with respect to a fractional Brownian motion[END_REF]):

Theorem 2.9. Let X : R → (L 2 ) be such that the function t → S(X t )(η) is measurable for all η in S (R)

and that t → ||X t || 0 is in L 1 (R,dt).T h e nX is (S) * -integrable over R and R X t dt 0 R ||X t || 0 dt.
3T h e o p e r a t o r s M H and their derivatives

Study of M H

Let us fix some notations. We will still note u or F (u) the Fourier transform of a tempered distribution u and we let L 1 loc (R) denote the set of measurable functions which are locally integrable on R.W ea l s oi d e n t i f y , here and in the sequel, any function f of L 1 loc (R) with its associated distribution, also noted T f .W ew i l ls a y that a tempered distribution v is of function type if there exists a locally integrable function f such that v = T f (in particular, <v,φ>= R f (t) φ(t) dt for φ in S (R)). Let H ∈ (0, 1).F o l l o w i n g [ 3 4 ] ,w ew a n tt od e fi n ea no p e r a t o r ,d e n o t e dM H , which is specified in the Fourier domain by

◊ M H (u)(y) := √ 2π cH |y| 1/2-H u(y),y ∈ R * . (3.1)
This operator is well defined on the homogeneous Sobolev space of order 1/2 -H, L 2 H (R):

L 2 H (R) := {u ∈ S ′ (R): u = T f ; f ∈ L 1 loc (R) and ||u|| H < +∞}, (3.2)
where ||u||

2 H := 1 c 2 H R |ξ| 1-2H |" u (ξ)|
2 dξ derives from the inner product on L 2 H (R),d e fi n e db y :

<u,v> H := 1 c 2 H R |ξ| 1-2H " u (ξ)" v (ξ)dξ, (3.3)
and c H has been defined right after formula (1.4) (the normalization constant √ 2π

cH will be explained in remark 3.5). It is well known -see [START_REF] Chemin | Analyse harmonique et équation des ondes et de Schrödinger[END_REF] p.13 for example -that (L 2 H (R),<,> H ) is a Hilbert space. The nature of the spaces L 2 H (R) when H spans (0, 1) is described in the following lemma, the proof of which can be found in [START_REF] Chemin | Analyse harmonique et équation des ondes et de Schrödinger[END_REF] p15,t h e o r e m1.4.1 and corollary 1.4.1.

Lemma 3.1. If H is in (0, 1/2],t h es p a c eL 2 H (R) is continuously embedded in L 1/H (R).W h e nH is in [1/2, 1),t h es p a c eL 1/H (R) is continuously embedded in L 2 H (R).
Since ◊ M H (u) belongs to L 2 (R) for every u in L 2 H (R), M H is well defined as its inverse Fourier transform, i.e.:

M H (u)(x) := 1 2π F ◊ M H (u) (-x),
for almost every x in R.

(3.4)

The following proposition is obvious in view of the definition of M H :

Proposition 3.2. M H is an isometry from (L 2 H (R),<,> H ) to (L 2 (R),<,> L 2 (R) )
. Let E(R) denote the space of simple functions on R,w h i c hi st h es e to fa l lfi n i t el i n e a rc o m b i n a t i o n so f functions 1 [a,b] (.) with a and b in R.I ti se a s yt oc h e c kt h a tb o t hS (R) and E(R) are subsets of L 2 H (R). It will be useful in the sequel to have an explicit expression for M H (f ) when f is in S (R) or in E(R).T o compute this value, one may use the formulas for the Fourier transform of the distributions || α , α in (-1, 1), given for instance in [START_REF] Chilov | Les distributions,v o l u m e1[END_REF] (chapter 1, § 3). This yields, for almost every x in R,

M H (1 [a,b] )(x)= √ 2π 2cH Γ(H+1/2) cos( π 2 (H-1/2)) [ b-x |b-x| 3/2-H - a-x |a-x| 3/2-H ]. (3.5)
By the same method, for f in S (R) one gets, for almost every real x:

M H (f )(x)=γ H < |y| -(3/2-H) ,f(x + y) > for almost every real x, (3.6)
with γ H :=

√ 2π 2cH Γ(H-1/2) cos( π 2 (H-1/2)) = Ä Γ(2H+1) sin(πH) ä 1 2 2Γ(H-1/2) cos( π 2 (H-1/2)
) and where we have written, by abuse of notation, |y| -(3/2-H) for the tempered distribution y →|y| -(3/2-H) and f (x + y) for the map y → f (x + y). Note moreover the following useful equality when f belongs to S (R) ,g i v e ni n [ 3 4 ] ( u pt oac o n s t a n t ) ,f o r almost every x in R,

M H (f )(x)=α H d dx ï R (t -x)|t -x| H-3/2 f (t)dt ò (3.7)
where

α H := -γ H (H -1/2) -1 = - √ 2π Ä 2c H Γ(H +1/2) cos( π 2 (H -1/2)) ä -1
. In order to extend the Wiener integral with respect to fBm to an integral with respect to mBm (in section 4.2) we will need the following equality:

Proposition 3.3. E(R) <,> H = L 2 H (R)
. This is a straightforward consequence of the following lemma: Lemma 3.4. Let σ : R → C be a measurable function, continuous on R * ,s u c ht h a t|σ| 2 is locally integrable at 0 and that x → σ(x)

x 2 is locally integrable at +∞.D e fi n e L 2 σ (R) := {u ∈ S ′ (R):

u = T f ; f ∈ L 1 loc (R) such that ||u|| σ < +∞} where <u,v> σ := R |σ(ξ)| 2 u(ξ) v (ξ) dξ. If E(R) ⊂ L 2 σ (R), define E(R)
<,> σ as the completion of E(R) for the norm || || σ .T h e n ,t h es p a c e(L 2 σ (R),<,> σ ) is a Hilbert space which also verifies E(R)

<,> σ = L 2 σ (R). Proof. The fact that (L 2 σ (R),<,> σ )
is a Hilbert space is obvious. One needs only to show that the orthogonal space of E(R) for the norm

|| || σ is equal to {0 E(R) }.L e tu in L 2 σ (R) be such that <u,v> σ =0for all v in E(R).I np a r t i c u l a r ,f o ra l lt in R, R |σ(ξ)| 2 u(ξ) ' 1 [0,t] (ξ) dξ =0 .F o ra l lψ in S (R), R ψ ′ (t) Å R |σ(ξ)| 2 u(ξ) ' 1 [0,t] (ξ)dξ ã dt =0,
where ψ ′ denotes the derivative of ψ.T h a n k st ot h ea s s u m p t i o n so n|σ| 2 and x → σ(x)

x 2 ,F u b i n it h e o r e m applies. Moreover, an integration by parts yields

0=- R |σ(ξ)| 2 iξ u(ξ) Å R ψ ′ (t)(1 -e iξt )dt ã dξ = R |σ(ξ)| 2 u(ξ) ψ(ξ) dξ.
Thus < |σ| 2 u, ψ >=0for all ψ in S (R).S i n c eξ →|σ(ξ)| 2 u(ξ) belongs to L 1 loc (R),i ti se a s yt od e d u c et h a t u is equal to 0. Remark 3.5. 1. Because the space S (R) is dense in L 2 H (R) for the norm || || H (see [START_REF] Chemin | Analyse harmonique et équation des ondes et de Schrödinger[END_REF] p.13), it is also possible to define the operator M H on the space S (R) and extend it, by isometry, to all elements of L 2 H (R). This is the approach of [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF] and [START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF] (with a different normalization constant). This clearly yields the same operator as the one defined by (3.1).H o w e v e rt h i sa p p r o a c hd o e sn o tl e n di t s e l ft oa ne x t e n s i o nt ot h ec a s e where the constant H is replaced by a function h,w h i c hi sw h a tw en e e df o rm B m . 2. For the same reasons as in 1. it is possible to define the operator M H on the space E(R) and extend it, by isometry, to all elements of L 2 H (R).A g a i n ,t h i se x t e n s i o nc o i n c i d ew i t h(3.1).W ew i l lu s et h i si d e ai n section 4.2.

In view of (3.3), we find that < 1

[0,t] , 1 [0,s] > H = 1 c 2 H R (e itξ -1)(e -isξ -1) |ξ| 2H+1
dξ = R H (t, s).T h u s ,a si nt h ec a s e of standard Brownian motion, one deduces that the process ( ‹ B (H) (t)) t∈R ,d e fi n e df o ra l l(t, ω) in R × Ω by:

‹ B (H) (t)(ω) := ‹ B (H) (t, ω) := <ω,M H (1 [0,t] ) >, (3.8
E[B (H) (t)B (H) (s)] = E[< ., M H (1 [0,t] ) >< ., M H (1 [0,s] ) >]=< 1 [0,t] , 1 [0,s] > H = R H (t, s). (3.9)
Remark 3.6. The reason of the presence of the constant

√ 2π
cH in formula (3.1) is now clear since this constant ensures that, for all H in (0, 1),t h ep r o c e s sB (H) defined by (3.8) is a normalized fBm.

Because our operator M H is defined on a distribution space, we can not apply the considerations of [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF] p.323ffaboutthelinksbet w eentheoperatorM H and Riesz potential operator. However it is crucial for our purpose that

M H is bijective from L 2 H (R) into L 2 (R): Theorem 3.7 (properties of M H ). 1. For all H in (0, 1),t h eo p e r a t o rM H is bijective from L 2 H (R) into L 2 (R).
2. For all H in (0, 1) and

(f, g) in (L 2 (R) ∩ L 2 H (R)) 2 ,w eh a v e<f,M H (g) > L 2 (R) = <M H (f ),g > L 2 (R) .
Moreover The last equality remains true when f belongs to L 1 loc (R) ∩ L 2 H (R) and g belongs to S (R) (in this case this equality reads <f ,M H (g) >= <M H (f ),g > L 2 (R) ,w h e r e<, > denotes the duality bracket between S ′ (R) and S (R)).

3. There exists a constant D such that, for every couple

(H, k) in (0, 1) × N * , max x∈R |M H (e k )(x)| D cH (k +1) 2/3 .
Proof. 1. Since M H is an isometry, we just have to establish the surjectivity of M H ,f o ra l lH in (0, 1). The case H =1 /2 being obvious, let us fix H in (0, 1)\{1/2}, g in L 2 (R) and define the complex-valued function

w g H on R by w g H (ξ)= cH √ 2π |ξ| H-1/2 ĝ(ξ) if ξ belongs to R *
(x)=f (-x) for all x.W es h a l lp r o v et h a tv g H belongs to L 2 H (R) and that M H (v g H )=g.N o t efi r s tt h a tf o ra l lu in S ′ (R), u =2 πǔ.I ti sc l e a rt h a t " v g H = w g H and that " v g H belongs to L 1 loc (R).M o r e o v e r ,t h a n k st of o r m u l a( 3 . 3 ) ,w es e et h a t ||v g H || 2 H = 1 cH 2 R |ξ| 1-2H | " v g H (ξ)| 2 dξ = 1 2π R | g(ξ)| 2 dξ = ||g|| 2 L 2 (R) < +∞. This shows that v g H belongs to L 2 H (R).W ec a nt h e nc o m p u t e ÿ M H (v g H ) and obtain, for almost every ξ in R, ÿ M H (v g H )(ξ)= √ 2π cH |ξ| 1/2-H " v g H (ξ)= √ 2π cH |ξ| 1/2-H w g H (ξ)= g(ξ).T h i ss h o w st h a tM H (v g H ) is equal to g in L 2 (
R) and then establish the surjectivity of M H .

2. See equality (3.12) of [START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF]. The case where f belongs to

L 1 loc (R) ∩ L 2 H (R) is obvious, in view of 2 of theorem 3.7, using the density of S (R) in L 2 H (R). 3. is shown in lemma 4.1 of [34].
Of course if we just consider functions in L 2 H (R) instead of all elements of L 2 H (R),t h em a pM H is not bijective any more.

Study of ∂M H

∂H

We now study the op erator ∂MH ∂H .I tw i l lp r o v ei n s t r u m e n t a li nd e fi n i n gt h ei n t e g r a lw i t hr e s p e c tt om B m in section 5. Heuristically, we wish to differentiate with respect to H the expression in definition (3.1), i.e.

differentiate the map H → ◊ M H (u)(y) on (0, 1) for (u, y) in L 2 H (R) × R * ,
H∈VH 0 L 2 H (R).
H o w e v e r ,a sw i l lb e c o m ea p p a r e n t ,t h e formula giving the derivative makes sense without this restriction. In order to define in a rigorous manner the operator ∂MH ∂H ,w es h a l lp r o c e e di naw a ya n a l o g o u st ot h eo n e that allowed to define M H in the previous subsection. It will be shown in remark 3.9 that this construction effectively defines the derivative, inac e r t a i ns e n s e ,o ft h eo p e r a t o rM H . We will note c ′ H the derivative of the analytic map H → c H where c H has been defined in (1.4) and set

β H := c ′ H
cH .L e tH belong to (0, 1). Define:

Γ H (R)={u ∈ S ′ (R): u = T f ; f ∈ L 1 loc (R) and ||u|| δH (R) < +∞}, (3.10)
where the norm || || δH (R) derives from the inner product on Γ H (R) defined by

<u,v> δH := 1 c 2 H R (β H + ln |ξ|) 2 |ξ| 1-2H " u (ξ) " v (ξ) dξ. (3.11)
By slightly adapting lemma 3.4, it is easy to check that (Γ H (R),<,> δH (R) ) is a Hilbert space which verifies the equality

Γ H (R)=S (R) <,> δ H = E(R) <,> δ H .N o t ef u r t h e r m o r et h a t ,f o re v e r yH in (0, 1),t h ei n c l u s i o n Γ H (R) ⊂ L 2 H (R) holds. We may now define the operator ∂MH ∂H from (Γ H (R),<,> δH (R) ) to (L 2 (R),<,> L 2 (R)
), in the Fourier domain, by:

ÿ ∂MH ∂H (u)(y) := -(β H + ln |y|) √ 2π
cH |y| 1/2-H u(y), for every y in R * .

(3.12)

In particular, one can check that, for

f in S (R), ÿ ∂MH ∂H (f )(y)= ∂ ∂H ⁄ M H (f )(y) for almost every real y.S i n c e ÿ ∂MH ∂H (u) belongs to L 2 (R) for every u in Γ H (R), ∂MH
∂H is well defined and given by its inverse Fourier transform from

(Γ H (R),<,> δH (R) ) to (L 2 (R),<,> L 2 (R) ): ∂MH ∂H (u)(x)= 1 2π F Å ◊ ∂MH (u) ∂H ã (-x), for almost every x in R.
As in the previous subsection it will be useful to compute ∂MH ∂H (f ) for f in S (R).W es u m m a r i z e ,i nf o l l o w i n g proposition, the main properties of ∂MH ∂H .

Proposition 3.8. ∂MH ∂H is an isometry from (Γ H (R),<,> δH (R) ) to (L 2 (R),<,> L 2 (R) ) which verifies: ∀f ∈ Γ H (R), ||f || δH = || ∂MH ∂H (f )|| L 2 (R) , (3.13)
∀(f, g) ∈ (Γ H (R) ∩ L 2 (R)) 2 ,< ∂MH ∂H (f ),g > L 2 (R) = <f, ∂MH ∂H (g) > L 2 (R) , (3.14)
∀f ∈ S (R) ∪E(R), and for a.e. x ∈ R,

∂MH ∂H (f )(x)= ∂ ∂H [M H (f )(x)]. (3.15)
Proof. Equality (3.13) results immediately from the definition of ∂MH ∂H and from (3.12). For any couple of functions

(f, g) in (Γ H (R) ∩ L 2 (R)) 2 , < ∂MH ∂H (f ),g > L 2 (R) = 1 2π < ÿ ∂MH ∂H (f ), g> L 2 (R) = 1 2π R -(β H + ln |y|) √ 2π cH |y| 1/2-H f (y) g(y)dy = 1 2π < f, ◊ ∂MH ∂H (g) > L 2 (R) = <f, ∂MH ∂H (g) > L 2 (R) .
It just remains to prove (3.15). Since we will not use (3.15) in the sequel for f in E(R),w ew i l lj u s te s t a b l i s h it here on S (R).L e tf be in S (R) and H in (0, 1).T h a n k st o3 . 6a n du s i n gt h e o r e m5 . 1 7w h i c hi sg i v e n in subsection 5.5 below, we may write

∂ ∂H [M H (f )(x)] = γ ′ H < |y| -(3/2-H) ,f(x + y) > + γ H < |y| -(3/2-H) ln |y|,f(x + y) >. (3.16)
Furthermore, for almost every real x,w em a yw r i t e :

∂MH ∂H (f )(x)= 1 2π ÿ ÿ ∂MH ∂H (f )(-x)= 1 2π F (y →-(β H + ln |y|) √ 2π cH |y| (1/2-H) f (y))(-x) = -β H M H (f )(x) -1 cH √ 2π F (y →|y| (1/2-H) ln |y| f (y))(-x).
Define, for every real x, I(-x)

:= F (y →| y| (1/2-H) ln |y| f (y))(-x) and, for every H in (0, 1), ν H := 2Γ ′ (3/2 -H) sin( π 2 (1/2 -H)) + πΓ(3/2 -H)cos( π 2 (1/2 -H). Using [21, p.173 -174]w eg e t ,a f t e rs o m ec o m p u t a t i o n s ,I(-x)=( -c H √ 2π)γ H < |y| -(3/2-H) ln |y|,f(x + y) > -ν H < |y| -(3/2-H) ,f(x + y) >.W efi n a l l yh a v e ,f o ra l m o s te v e r yr e a lx, ∂MH ∂H (f )(x)=( -γ H c ′ H c H + ν H c H √ 2π ) < |y| -(3/2-H) ,f(x + y) > +γ H < |y| -(3/2-H) ln |y|,f(x + y) > which is nothing but (3.16) since -γH c ′ H cH + νH cH √ 2π = γ ′ H . Remark 3.9. Define Σ H,r (R) := r∈(0,min(H,1-H))
Γ H (R)

H ′ ∈[H-r,H+r] L 2 H ′ (R) . It is possible to show that, for all H in (0, 1) and f in Σ H,r (R) that ∂MH ∂H (f )(.) (resp. ÿ ∂MH ∂H (f )(.) )i st h ed e r i v a t i v e ,i nt h e L 2 (R)-sense, of M H (f ) (resp. of ◊ M H (f ) ).
4W i e n e r i n t e g r a l w i t h r e s p e c t t o m B m o n R

Wiener integral with respect to fBm

Similarly to what is performed in [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF] and [START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF] (in these works this is done only for functions of L 2 H (R)), it is now easy to define a Wiener integral with respect to fractional Brownian motion. Indeed, for any element

g in L 2 H (R),d e fi n eJ H (g) as the random variable < ., M H (g) >.I no t h e rw o r d s ,f o ra l lc o u p l e s(ω, g) in Ω×L 2 H (R): J H (g)(ω) :=<ω,M H (g) >= R M H (g)(s) dB(s)(ω), (4.1)
where the Brownian motion B has been defined just below formula at the end of subsection 2.1. We call the random variable J H (g) the Wiener integral of g with respect to fBm. Once again, when g is a tempered distribution which is not a function, g(s) does not have a meaning for a fixed real s and J H (g) is just a notation for the centered Gaussian random variable < ., M H (g) >.

Wiener integral with respect to mBm

We now consider a fractional Brownian field

Λ := Ä Λ(t, H) ä (t,H)∈R×(0,1)
,d e fi n e d ,f o ra l l(t, H) in R × (0, 1)

and all ω in Ω,b yΛ(t, H)(ω) := B (H) (t, ω) := <ω,M H (1 [0,t] ) >.
We wish to generalize the previous construction of the Wiener integralw ithre s pe c ttof Bmt ot h ec as eofm B m . Th isam ou n t st ore p lacin gth e constant H by a continuous deterministic function h,r a n g i n gi n(0, 1).M o r ep r e c i s e l y ,l e tR h denote the covariance function of a normalized mBm with function h (see definition 1.5). Define the bilinear form

<, > h on E(R) ×E(R) by < 1 [0,t] , 1 [0,s] > h = R h (t, s)
.O u rc o n s t r u c t i o no ft h ei n t e g ral of deterministic elements with respect to mBm requires that <, > h be an inner product:

Proposition 4.1. <, > h is an inner product for every function h.

Proof. See appendix X.B.

Define the linear map

M h :( E(R),<,> h ) → (L 2 (R),<,> L 2 (R) ) 1 [0,t] → M h (1 [0,t] ) := M h(t) (1 [0,t] ) := M H (1 [0,t] )| H=h(t) ,
and the process ‹

B (h) (t)=< ., M h (1 [0,t] ) >, t ∈ R.A sK o l m o g o r o v ' sc r i t e r i o na n dt h ep r o o fo ff o l l o w i n g lemma show,
this process admits a continuous version which will be noted B (h) .Aw o r do nn o t a t i o n :w e write B (.) both for an fBm and an mBm. This should not cause any confusion since an fBm is just an mBm with constant h function. It will be clear from the context in the following whether the "h"i sc o n s t a n to r not. Note that a.s., forall real t, B (h) (t)=B (H) (t) | H=h(t) .I nv i e wo fp o i n t2. in remark 3.5 we may state the following lemma.

Lemma 4.2. (i) The process B (h) is a normalized mBm. (ii) The map M h is an isometry from (E(R),<,> h ) to (L 2 (R),<,> L 2 (R) ). Proof. The process B (h) is clearly a centered Gaussian process. Moreover, for all (s, t) in R 2 , E[B (h) (t)B (h) (s)] = E[< ., M h (1 [0,t] ) >< ., M h (1 [0,s] ) >]= 1 2π < ¤ M h(t) (1 [0,t] ), ¤ M h(s) (1 [0,s] ) > L 2 (R) = 1 c h(t) c h(s) R (1 -e itξ )(1 -e -isξ ) |ξ| 1+2ht,s dξ = R h (t, s)=< 1 [0,t] , 1 [0,s] > h .
By isometry, it is possible to extend M h to the space E(R) <,> h and we shall still note M h this extension.

Define the isometry J

h := ζ • M h on E(R) <,> h , i.e: J h :( E(R) <> h ,<,> h ) M h → (L 2 (R),<,> L 2 (R) ) ζ → ((L 2 ),<,> (L 2 ) ) 1 [0,t] → M h (1 [0,t] ) → < ., M h (1 [0,t] ) >.
We can now define the Wiener integral with resp ect to mBm in the natural following way: Definition 4.1. Let B (h) be a normalized multifractional Brownian motion. We call Wiener integral on R of an element u in E(R) <> h with respect to B (h) ,t h ee l e m e n tJ h (u) of (L 2 ) defined thanks to the isometry J h given just above.

Remark 4.3. It follows from definition 4.1 that the Wiener integral of a finite linear combination of functions

1 [0,t] is J h ( n k=1 α k 1 [0,t k ] )= n k=1 α k B (h) t k . Moreover, for any element u in E(R)
<>h (which may be a tempered distribution), the Wiener integral of u with respect to mBm, still denoted J h (u),i sg i v e nb y

J h (u) def = lim n→+∞ J h (u n ),f o ra n ys e q u e n c eo ff u n c t i o n s(u n ) n∈N in E(R) N which converges to u in the norm
|| || h and where the convergence of J h (u n ) holds in (L 2 ).

Since we now have a construction of the Wiener integral with respect to mBm, it is natural to ask which functions admit such an integral. In particular, we do not know so far whether S (R) ⊂ E(R) <> h .T h en e x t section contains more information about the space E(R) <>h .

5S t o c h a s t i c i n t e g r a l w i t h r e s p e c t t o m B m

Fractional White Noise

The following theorem will allow us to give a concrete example of a derivative of an (S) * -process.

Theorem 5.1. 1. For any real H in (0, 1),t h em a pM

H (1 [0,.] ):R → S ′ (R) defined by M H (1 [0,.] )(t) := M H (1 [0,t] ) is differentiable over R and its derivative, noted d dt [M H (1 [0,t] )],i se q u a lt o +∞ k=1 M H (e k )(t) e k ,
w h e r et h ec o n v e r g e n c ei si nS ′ (R). 2. For any interval I of R and any differentiable map F : I → S ′ (R),t h ee l e m e n t< ., F (t) > is a differentiable stochastic distribution process which satisfies the equality

d dt < ., F (t) >=< ., d dt F (t) >.
Proof. The proof of point 1 is a mere re-writing of the one of lemma 2.15 of [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF] by replacing M H ± by the operator M H .P o i n t2 is theorem 2.17 of [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF].

Let H ∈ (0, 1).T h ep r o c e s s(B (H) (t)) t∈R defined in (3.8) is an fBm, and M H (1 [0,t] ) belongs to L 2 (R) for every real t.H e n c e ,u s i n ge q u a l i t y2 of theorem 3.7, we may write, for every real t and almost surely:

B (H) (t)= < ., M H (1 [0,t] ) > = < ., +∞ k=0 <M H (1 [0,t] ),e k > L 2 (R) e k > = +∞ k=0 < 1 [0,t] ,M H (e k ) > L 2 (R) < ., e k > = +∞ k=0 t 0 M H (e k )(u)du < ., e k >.
(5.1)

(5.1) and the previous theorem lead to the definition of fractional white noise [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF][START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF]:

Example 5.2 (Fracti o nal whi te no i s e ). Let:

W (H) (t) := +∞ k=0 M H (e k )(t) < ., e k >. (5.2)
Then (W (H) (t)) t∈R is a (S) * -process and is the (S) * -derivative of the process (B (H) (t)) t∈R . The proof of this fact is simple: for any integer p 2, using remark 2.3, the mean value theorem and the dominated convergence theorem,

J p,r (t) := B (H) (t+r)-B (H) (t) r -W (H) (t) 2 -p = +∞ k=0 Ä 1 r t+r t M H (e k )(u) du -M H (e k )(t) ä < ., e k > 2 -p = +∞ k=0 (2k +2) -2p Ä 1 r t+r t (M H (e k )(u) -M H (e k )(t)) du ä 2 -→ r→0 0.
Remark 5.3. In particular we see that for all (t, H) in R × (0, 1), W (H) (t) belongs to (S -p ) as soon as p 2.

Remark 5.4. There are several constructions of fBm. In particular, operators different from M H may be considered. [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF] uses an operator denoted M H + on the grounds that fBm as defined here is not adapted to the filtration generated by the driving Brownian motion as soon as H =1 /2.W h i l et h i si si n d e e dad r a w b a c k , the crucial property for our purpose is that the same probability space (S ′ (R), G,µ) is used for all parameters H in (0, 1).T h i sa l l o w st oc o n s i d e rs i m u l t a n e o u s l ys e v e r a lf r a c t i o n a lB r o w n i a nm o t i o n sw i t hH taking any value in (0, 1), which is necessary when one deals with mBm. We choose here to work with M H rather than with M H + and M H -of [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF] as its use is simpler. M H + and M H -would nevertheless allow for a more general approach encompassing the whole family of mBm at once. This topic will be treated in a forthcoming paper.

Multifractional White Noise

The main idea for defining a stochastic integral with respect to mBm is similar to the one used for fBm. We will relate the pro cess B (h) to Brownian motion via the family of operators (M H ) (H∈(0,1)) .T h i sw i l l allow to define a multifractional white noise, analogous to the fractional white noise of example 5.2. From ah e u r i s t i cp o i n to fv i e w ,m u l t i f r a c t i o n a lw h i t en o ise is obtained by differentiating with respect to t the fractional Brownian field Λ(t, H) (defined at the beginning of section 4.2) along a curve (t, h(t)).A s s u m i n g that we may differentiate in the sense of (S) * (this will be justified below), the differential of Λ reads:

dΛ(t, H)= ∂Λ ∂t (t, H) dt + ∂Λ ∂H (t, H) dH = dB (H) dt (t) dt + ∂Λ ∂H (t, H) dH = W (H) (t)dt + ∂Λ ∂H (t, H) dH, (5.3)
where the equality will be shown to hold in (S) * .W i t had i ff e r e n t i a b l ef u n c t i o nh in place of H,thisformally

yields dΛ(t, h(t)) = Ä W (h(t)) (t)+h ′ (t) ∂Λ ∂H (t, H) H=h(t) ä dt. (5.4)
In view of the definition of the stochastic integral with respect to fBm given in [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF], [START_REF] Holden | Stochastic Partial Differential Equations, A Modeling, White Noise Functional Approach[END_REF] and [START_REF] Bender | An S-transform approach to integration with respect to a fractional Brownian motion[END_REF], it then seems natural to set the following definition for the stochastic integral with respect to mBm of a Hida process

X : R → (S) * : R X(s)dB (h) (s) := R X(s)dΛ(s, h(s)) := R X(s) Ä W (h(s)) (s)+h ′ (s) ∂Λ ∂H (s, H)| H=h(s) ä ds. (5.5)
We shall then say that the pro cess X is integrable with respect to mBm if the right hand side of (5.5) exists in (S) * .R e m a r kt h a tw h e nt h ef u n c t i o nh is constant we recover of course the integral with respect to fBm. In order to make the above ideas rigorous, we start by writing the chaos expansion of B (h) .S i n c eM H (g) belongs to L 2 (R) for all (g, H) in S (R) × (0, 1),w em a yd e fi n e ,f o ra l lH in (0, 1),

M H : S ′ (R) → S ′ (R), by <M H (ω),g > = <ω,M H (g) >, for µ -a.e. ω in Ω=S ′ (R).
(

Moreover, in view of remark 3.5, we may extend (5.6) tot h ec a s ew h e r eg belongs to L 2 H (R) by writing, for all g in L 2

H (R) and almost every ω in Ω,

<M H (ω),g >:= lim n→+∞ <M H (ω),g n > = lim n→+∞ <ω,M H (g n ) > = <ω,M H (g) >, (5.7)
for every sequence (g n ) n∈N of functions of S (R) which converges to g in the norm || || L 2 H (R) .F o ra l lr e a lt and integer k in N,d e fi n et h ee l e m e n to fL 2 h(t) (R):

d (t) k := M -1 h(t) (e k ). (5.8)
It is clear that, for all t in R,t h ef a m i l yo ff u n c t i o n s(d

(t) k ) k∈N forms an orthonormal basis of L 2 h(t) (R)
.L e t us now write the chaos decomposition of mBm. For almost every ω and every real t we get, (using theorem 3.7 and (5.7)),

B (h) (t)(ω)=<ω,M h (1 [0,t] ) >=<M h(t) (ω), 1 [0,t] >=<M h(t) (ω), +∞ k=0 < 1 [0,t] ,d (t) k > L 2 H (R) d (t) k > =<M h(t) (ω), +∞ k=0 <M h(t) (1 [0,t] ),e k > L 2 (R) d (t) k >= +∞ k=0 <M h(t) (1 [0,t] ),e k > L 2 (R) <ω,M h(t) (d (t) k ) >.
We get finally:

a.s, ∀t ∈ R,B (h) (t)= +∞ k=0 t 0 M h(t) (e k )(s)ds < ., e k >.
(5.9)

We would then like to define multifractional white noise as the (S * )-derivative of B (h) ,w h i c hw o u l db e formally defined by:

W (h) (t) := +∞ k=0 ï d dt Å t 0 M h(t) (e k )(s) ds ãò < ., e k >, (5.10)
assuming h is differentiable. The following theorem states that, for all real t,t h er i g h th a n ds i d eo f( 5 . 1 0 ) does indeed belong to (S) * and is exactly the (S) * -derivative of B (h) at t.

Theorem-Definition 5.1. Let h : R → (0, 1) be a C 1 function such that the derivative function h ′ is bounded. The process W (h) := (W (h) (t)) t∈R defined by formula (5.10) is an (S) * -process which verifies the following equality in (S) * :

W (h) (t)= +∞ k=0 M h(t) (e k )(t) < ., e k > +h ′ (t) +∞ k=0 Å t 0 ∂MH ∂H (e k )(s) H=h(t) ds ã < ., e k >.
(5.11)

Moreover the process B (h) is (S) * -differentiable on R and verifies in (S) * dB (h) dt (t)=W (h) (t)= d dt [Λ(t, h(t))].
(5.12)

In order to prove this theorem, we will need two lemmas.

Lemma 5.5. For H in (0, 1) and

f in S (R),d e fi n eg f : R × (0, 1) → R by g f (t, H) := t 0 M H (f )(x)dx. Then (i) The function g f belongs to C ∞ (R × (0, 1), R), (ii) ∀x ∈ R, M H (f )(x)=α H +∞ 0 u H-1/2 (f ′ (x + u) -f ′ (x -u)) du.
where α H has been defined right after (3.7). In particular, the function

(x, H) → M H (f )(x) is differentiable on R × (0, 1). (iii) Assume that h : R → (0, 1) is differentiable. Then, for any real t 0 d dt [g f (t, h(t))] t=t0 = M h(t0) (f )(t 0 )+h ′ (t 0 ) t0 0 ∂MH ∂H (f )(s) H=h(t0) ds. (5.13) Proof. (i) Define µ f on R × (0, 1) by µ f (t, H) := R (u -t)|u -t| H-3/2 f (u) du,f o rf in S (R)
. Using (3.7) we get, for all (t, H) in R × (0, 1),t h ee q u a l i t y

g f (t, H)=α H [µ f (t, H) -µ f (0,H)].
(5.14)

Ac h a n g eo fv a r i a b l e sy i e l d s

µ f (x, H)=- x -∞ |t -x| H-1/2 f (t)dt + +∞ x |t -x| H-1/2 f (t) dt = +∞ 0 u H-1/2 (f (x + u) -f (x -u)) du.
(5.15)

Thanks to (5.14) and to the fact that the map y → α y is C ∞ on (0, 1),i ti ss u ffi c i e n tt os h o wt h a tt h e function µ f belongs to C ∞ (R × (0, 1)).I nv i e wo fa p p l y i n gt h et h e o r e mo fd i ff e r e n t i a t i o nu n d e rt h ei n t e g r a l sign, define j(x, H, u)

:= u H-1/2 (f (x + u) -f (x -u)) for u in R * + .L e tn in N and (α 1 ,α 2 ) in N 2 such that α 1 + α 2 = n.F o ra l m o s te v e r yu in R * + , (x, H) → j(x, H, u) is C n on R × (0, 1) with partial derivatives given by ∂ n j ∂x α1 ∂H α2 (x, H, u) = (ln u) α2 u H-1/2 (f (α1) (x + u) -f (α1) (x -u)). Fix (x 0 ,H 0 ) in R × (0, 1).L e tu ss h o wt h a tµ f is C n in a neighbourhood of (x 0 ,H 0 ).C h o o s e(a, b) such that a<x 0 <band H 1 ,H 2 such that 0 <H 1 <H<H 2 < 1.W eh a v e ∂ n j ∂x α1 ∂H α2 (x, H, u) |u| H1-1/2 | ln u| α2 1 {0<u<1} sup (x,u)∈[a,b]×[0,1] |f (α1) (x ± u)| + |u|| ln u| α2 |f (α1) (x ± u)| 1 {1 |u|} , (5.16)
where

f (α1) (x ± u) := f (α1) (x + u) -f (α1) (x -u).
AT a y l o re x p a n s i o ns h o w st h a tt h e r ee x i s t sar e a lD such that, for all

(u, x) in R × (a, b), |u| 4 |f (α1) (u ± x)| D.
A sac o n s e q u e n c e ,t h e r ee x i s t sar e a lc o n s t a n tC such that, for almost every u in R * + and every

(x, H) in [a, b] × [H 1 ,H 2 ], ∂ n j ∂x α 1 ∂H α 2 (x, H, u) C î |u| H1-1/2 | ln u| α2 1 {0<u<1} + | ln u| α2 1 |u| 3 1 {1 u} ó .
(5.17)

Since the right hand side of the previous inequality belongs to L 1 (R),t h et h e o r e mo fd i ff e r e n t i a t i o nu n d e r the integral sign can be applied to conclude that the function

µ f is of class C n in [a, b] × [H 1 ,H 2 ],f o ra l l
integer n and all f in S (R).T h i se n t a i l s(i).

(ii) (5.14) and (5.15) yield

M H (f )(x)=α H ∂ ∂x [µ f (x, H)] = α H +∞ 0 u H-1/2 (f ′ (x + u) -f ′ (x -u))du, (5.18)
which establishes (ii) and the fact that

(x, H) → M H (f )(x) belongs to C ∞ (R × (0, 1)).
(iii) For a differentiable function h,w eh a v e ,f o re v e r yr e a lt 0 ,

d dt [g f (t, h(t))] t=t0 = ∂g f ∂t (t 0 ,h(t 0 )) + h ′ (t 0 ) ∂g f ∂H (t 0 ,h(t 0 )).
For every f in S (R),(5.14), (5.18) and (3.15) show that,

∂g f ∂H (t, H)= t 0 ∂ ∂H [M H (f )(x)] dx = t 0 ∂MH ∂H (f )(x) dx. Finally, we get d dt [g f (t, h(t))] t=t0 = M h(t0) (f )(t 0 )+h ′ (t 0 ) t0 0 ∂MH ∂H (f )(s) H=h(t0) ds.
Lemma 5.6. The following inequalities hold:

(i) ∀ [a, b] ⊂ (0, 1), ∃ρ ∈ R: ∀k ∈ N, sup (H,u)∈[a,b]×R ∂MH ∂H (e k )(u) ρ (k +1) 2/3 ln(k +1). (ii) ∀t ∈ R, ∀r ∈ R * + , ∃ D t (r) ∈ R, ∀k ∈ N :s u p u∈[t-r,t+r] d du [g e k (u, h(u))] D t (r)( k +1) 2/3 . Proof. (i) Since S (R) is a subset of Γ H (R),( 3. 12 )en t ai l st h at ÿ ∂MH ∂H (e k ) belongs to L 1 (R) ∩ L 2 (R) for every k in N.F u r t h e r m o r e" e k (y)=( -i) k-1 √ 2πe k (y)
for every integer k in N * and for almost every real y (see lemma 1.1.3 p.5 of [START_REF] Thangavelu | Lectures of Hermite and Laguerre expansions[END_REF]). Thus, for every H ∈ [a, b] and almost every u ∈ R,

∂MH (e k ) ∂H (u)= 1 2π ÿ ÿ ∂MH (e k ) ∂H (-u)=-1 2π R e iuy (β H + ln |y|) √ 2π cH |y| 1/2-H " e k (y) dy = -β H 2π R e iuy ÿ M H (e k )(y) dy -1 cH R e iuy |y| 1/2-H (ln |y|)(-i) k-1 e k (y) dy = -β H M H (e k )(u)+ -(-i) k-1 cH R e iuy |y| 1/2-H (ln |y|) e k (y) dy =:V k (u)
.

(5.19)

Then, using (2.5), we see that there exists a family of real constants denoted (ρ i ) 1 i 11 such that we have, for every couple

(H, k) in [a, b] × N and almost every real u, |V k (u)| ρ 1 î |y| 2 √ k+1 |y| 1/2-H | ln |y|||e k (y)|dy + |y|>2 √ k+1 |y| 1/2-H (ln |y|)|e k (y)|dy ó ρ 2 ï 1 (k+1) 1/12 Ä 2 √ k+1 0 | ln y| y 1/2-H dy :=I (k+1) 1 ä + Ä +∞ 2 √ k+1 y 1/2-H (ln y)e -γy 2 dy :=I (k+1) 2 ä ò . (5.20)
An integration by parts yields

I (k) 1 = ρ 3 ((ρ 4 + ln k)(1 + k 3/4-H/2 )) ρ 5 k 3/4-H/2 ln k. (5.21)
Using the change of variables u = y √ γ,w eg e tI

(k) 2 ρ 6 +∞ 2 √ kγ |u| 1/2-H (ln u)e -u 2 du = ρ 6 J (1/2-H) 2 √ kγ where J (α) δ := +∞ δ |u| α ln ue -u 2 du.
W h e nδ>3e,a ni n t e g r a t i o nb yp a r t ss h o w st h a tJ

(α) δ <δ α-1 e -δ 2 ln δ for 0 <α<1/2,a n dt h a tJ (α) δ δ α +∞ δ e -u 2 ln udufor -1/2 <α<0.H e n c ew eg e t J (1/2-H) 2 √ kγ ρ 7 ® k -1/4-H/2 ln k for 0 <H<1/2 k 1/4-H/2 for 1/2 <H<1 (5.22)
and we finally obtain

I (k) 2 ρ 8 ln k k H/2-1/4
.U s i n g( 5 . 1 9 )t o( 5 . 2 2 ) ,t h ep r e v i o u si n e q u a l i t y ,i t e m3 of theorem 3.7 and the fact that both functions H → β H and H → 1 cH are continuous on [a, b] we get, for every k in N, sup

(H,u)∈[a,b]×R ∂MH ∂H (e k )(u) ρ 9 (k +1) 2/3 + ρ 10 sup H∈[a,b] î ((k +1) 2/3-H/2 +(k +1) 1/4-H/2 ) ln(k +1) ó ρ 11 (k +1) 2/3 ln(k +1).
(ii) Let t ∈ R and r>0 be fixed and define

[a, b] := [ inf u∈[t-r,t+r] h(u), sup u∈[t-r,t+r] h(u)].U s i n g( 5 . 1 3 )w eh a v e , for every k in N, sup u∈[t-r,t+r] d du [g e k (u, h(u))] sup (H,u)∈[a,b]×R |M H (e k )(u)| +(|t| + r)s u p u∈[t-r,t+r] |h ′ (u)| sup (H,u)∈[a,b]×R ∂MH ∂H (e k )(u) .
The result then follows from (i) above and item 3 of theorem 3.7.

Remark 5.7. In the sequel, we will only need the bounds of (i) and (ii) with (k +1) in lieu of (k +1) 2/3 .

We may now pro ceed to the pro of of theorem 5.1. Proof of theorem 5.1. From equality (5.10) defining multifractional white noise and equality (5.13), we have formally

W (h) (t)= +∞ k=0 M h(t) (e k )(t) < ., e k > + h ′ (t) +∞ k=0 t 0 ∂MH (e k ) ∂H (s) H=h(t) ds < ., e k >. (5.23)
In order to establish that W (h) (t) is well defined in (S) * and that equality (5.11) holds, it is sufficient to show that both members on the right hand side of the previous equality are in (S) * .

For

t in R,d e fi n i t i o n( 5 . 2 )o ff r a c t i o n a lw h i t en o i s es h o w st h a t +∞ k=0 M h(t) (e k )(t) < ., e k >= W (h(t)) (t) and thus belongs to (S) * . Let us show that V H (t) := +∞ k=0 t 0 ∂MH (e k ) ∂H (s)ds < ., e k > belongs to (L 2 ).U s i n g( 3 . 1 4 )w em a yw r i t e E[V 2 H (t)] = +∞ k=0 < ∂MH ∂H (e k ), 1 [0,t] > 2 L 2 (R) = +∞ k=0 <e k , ∂MH ∂H (1 [0,t] ) > 2 L 2 (R) = ||1 [0,t] || 2 δH < +∞.
As a consequence, W (h) (t) is the sum of an (S) * process and an (L 2 ) process, and thus belongs to (S) * .W e are left with proving equality (5.12), i.e. that W (h) (t) is indeed the (S) * derivative of B (h) (t) for any real t.

Let r =0and t 0 (the case t<0 follows in a similar way). The equality

W (h) (t)=W (h(t)) (t)+h ′ (t) V h(t) (t)
and remark 5.3 entail that W (h) (t) belongs to (S -p ) as soon as p 2.F o rs u c hap,o n ec o m p u t e s :

J p,r (t) := B (h) (t+r)-B (h) (t) r -W (h) (t) 2 -p = +∞ k=0 ïÅ g e k (t + r, h(t + r)) -g e k (t, h(t)) r ã - d dt [g e k (t, h(t))] ò < ., e k > 2 -p = +∞ k=0 1 (2k +2) 2p ïÅ g e k (t + r, h(t + r)) -g e k (t, h(t)) r ã - d dt [g e k (t, h(t))] ò 2 :=J p,r,k (t)
.

(5.24)

Using lemma 5.6 and the Mean-Value theorem we obtain, for r in (-1/2, 1/2)\{0}:

J p,r,k (t) 4 (2k +2) 2p sup u∈[t-1/2,t+1/2] d du [g e k (u, h(u))] 2 4 ‹ D (k +1) 2 (2k +2) 2p ‹ D (2(k +1)) 2(p-1) ,
where

‹ D := ‹ D t (1/2).S i n c eJ p,r,k (t) -→ r→0
0,e q u a l i t y( We note that multifractional white noise is a sum of two terms: a fractional white noise that b elong to (S -p ) as soon as p 2,a n da" s m o o t h "t e r mw h i c hc o r r e s p o n d st ot h ed e r i v a t i v ei nt h e" H"d i r e c t i o n . T h i s is a direct consequence of the fact that the fractional Brownian field Λ(t, H) is not differentiable in the t direction (in the classical sense) but infinitely smooth in the H direction. Proposition 5.9. For p 2,t h em a pt → ||W (h) (t)|| -p is continuous.

Proof. By definition, ||W

(h) (t)|| 2 -p = +∞ k=0 ( d dt [ge k (t,h(t))]) 2 (2k+2) 2p
.U s i n g t h e e s t i m a t e g i v e n i n l e m m a 5 . 6 ( i i ) ,

we see that

||W (h) (t)|| 2
-p is the sum of a sequence of continuous functions that converges normally on any compact.

Generalized functionals of mBm

In the next section, we will derive various Itô formulas for the integral with respect to mBm. It will be useful to obtain such formula for tempered distributions. In that view, we define generalized functionals of mBm as in [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF].

Theorem-Definition 5.2. Let F be a tempered distribution. For t in R * + ,d e fi n e

F (B (h) (t)) := 1 √ 2πt h(t) +∞ k=0 (k!) -1 t -2kh(t) <F,ξ (t,h(t),k) >I k Ä ( M h(t) (1 [0,t] )) ⊗k ä (5.25)
where the functions ξ t,H,k are defined for

(x, H, k) in R × (0, 1) × N by ξ t,H,k (x) := ( √ 2) -k t kH h k (x/( √ 2t H )) exp{- x 2 2 t 2H } = π 1/4 (k!) 1/2 t kH exp{- x 2 4t 2H }e k (x/( √ 2t H )). (5.26)
Then for all

t in R * + , F (B h (t)) is a Hida distribution, called generalized functional of B (h) (t).
Proof. This is an immediate consequence of [START_REF] Kuo | White Noise Distribution Theory[END_REF] p.61-64 by taking f := M h (1 [0,t] ).

Remark 5.10. As shown in [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF], when

F = f is of function type, F (B h (t)) coincides with f (B h (t)).
The following theorem yields an estimate of ||F (B (h) (t)|| Proof. For H ∈ (0, 1) and p ∈ N,T h e o r e m3.3 p.92 of [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF] ensures that there exists C (H) p such that, ∀t>0,

||F (B H (t))|| 2 -p max{t -4pH ,t 4phH }t -H C (H) p |F | 2 -p . Now if H belongs to [H 1 ,H 2 ],
i ti se a s yt os h o w ,b ye x a m i n i n gc l o s e l yt h ei t e r a t i o no f(23) p.94 in [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF], that one can choose a constant C (H1,H2) independent of H. We hence have

∀t>0, ∀H ∈ [H 1 ,H 2 ], ||F (B H (t))|| 2 -p max{t -4pH ,t 4phH }t -H C (H1,H2) p |F | 2 -p .
(5.28)

For t>0,o n eo n l yn e e d st os e tH = h(t) in (5.28) to get (5.27).

S-Transform of mBm and multifractional white noise

The following theorem makes explicit the S-transforms of mBm, multifractional white noise and generalized functionals of mBm. We denote by γ the heat kernel density on

R + × R i.e γ(t, x) := 1 √ 2πt exp { -x 2 2t } if t =0and 0 if t =0.
Theorem 5.12. Let h : R → (0, 1) be a C 1 function and

(B (h) (t)) t∈R (resp. (W (h) (t)) t∈R )b ea nm B m (resp. multifractional white noise). For η ∈ S (R) and t ∈ R, (i) S[B (h) (t)](η)=<η,M h (1 [0,t] ) > L 2 (R) = g η (t, h(t)
),w h e r eg η has been defined in lemma 5.5.

(ii) S[W (h) (t)](η)= d dt [g η (t, h(t))] = M h(t) (η)(t)+h ′ (t) t 0 ∂MH ∂H (η)(s) H=h(t) ds. (iii) For p ∈ N and F ∈ S -p (R), S[F (B (h) (t))](η)= F, γ t 2h(t) ,.- t 0 M h(t) (η)(u) du .
Furthermore, there exists a constant C p ,i n d e p e n d e n to fF, t and η,s u c ht h a t

|S[F (B (h) (t))](η)| 2 max{t -4ph(t) ,t 4ph(t) }t -h(t) C p |F | 2 -p exp{|A p η| 2 0 }. (5.29)
Proof. (i) Thanks to (5.9), lemma 2.4 and theorem 3.7 we have, for every η in S (R) and t in R,

S(B (h) (t))(η)= +∞ k=0 <M h(t) (1 [0,t] ),e k > L 2 (R) <η,e k > L 2 (R) = <M h(t) (1 [0,t] ),η > L 2 (R) = g η (t, h(t)).
(ii) is a straightforward consequence of lemma 2.7, (5.12) and (i).

(iii) The first equality results from theorem 7.3 p.63 in [START_REF] Kuo | White Noise Distribution Theory[END_REF] with

f = M h(t) (1 [0,t]
) and from (i).E q u a l i t y (5.29) results from (5.27) as in theorem 3.8 p.95 of [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF].

Remark 5.13. Using lemma 2.4 and (5.10) we may also write:

∀(t, η) ∈ R × S (R),S (W (h) (t))(η)= +∞ k=0 d dt [g e k (t, h(t))] <η,e k > L 2 (R) .
(5.30)

The multifractional Wick-Itô integral

We are now able to define the Multifractional Wick-Itô integral, in a way analogous to the definition of the fractional Wick-Itô integral. In the sequel of this work, we will always assume that h is a C 1 function on R with bounded derivative.

Definition 5.1 (The multifractional Wick-Itô integral). Let Y : R → (S) * be a process such that the process

t → Y (t) W (h) (t) is (S)
* -integrable on R.W et h e ns a yt h a tt h ep r o c e s sY is dB (h) -integrable on R or integrable on R with respect to mBm B (h) .T h ei n t e g r a lo fY with respect to B (h) is defined by

R Y (s) dB (h) (s) := R Y (s) W (h) (s)ds.
(5.31)

For a Borel set I of R,d e fi n e I Y (s)dB (h) (s) := R 1 I (s) Y (s)dB (h) (s).
When the function h is constant, the multifractional Wick-Itô integral coincides with the fractional Itô integral defined in [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF], [START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF], [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF] and [START_REF] Bender | An S-transform approach to integration with respect to a fractional Brownian motion[END_REF]. In particular, when the function h is identically 1/2,(5.31)isnothing but the classical Itô integral with respect to Brownian motion, provided of course Y is Itô-integrable. The multifractional Wick-Itô integral verifies the following properties:

Proposition 5.14. (i) Let (a, b) in R 2 , a<b.T h e n b a dB (h) (u)=B (h) (b) -B (h) (a)
almost surely. (ii) Let X : I → (S * ) be a dB (h) -integrable process over I,aB o r e ls u b s e to fR.A s s u m e I X(s)dB (h)

(s) belongs to (L 2 ).T h e nE[ I X(s)dB (h) (s)] = 0. Proof. (i) From (ii) of theorem 5.12, t → S(1 [a,b] (t) W (h) (t))(η) is measurable on R for any η in S (R).
Moreover, for any integer

p 0 2,w eh a v e|S(1 [a,b] (t) W (h) (t))(η)| ||W (h) (t)|| -p0 e 1 2 |η| 2 p 0 ,t h a n k st ol e m m a 2.4. By proposition 5.9, t → ||W (h) (t)|| -p0 is continuous thus integrable on [a, b].T h e o r e m2 . 8t h e ne n t a i l s that t → 1 [a,b] (t) W (h) (t) is (S * )-integrable over R.I ti se a s i l ys e e nt h a tt h eS -t r a n s f o r m so f b a dB (h) (u) and B (h) (b) -B (h) (a) coincide.
The result then follows from the injectivity of the S-transform.

(ii) The equality S( I X(s)dB (h) (s))(0) = I S(X(s))(0)S(W (h) (s))(0)ds =0is clear since S(W (h) (s))(0) = d ds [g 0 (s, h(s))] = 0.N o w ,w h e n I X(s)dB (h) (s) belongs to (L 2 ), E[ I X(s)dB (h) (s)] = S( I X(s)dB (h) (s))(0).
Theorem 5.15. Let I be a compact subset of R and X : t → X(t) be a process from I to (L 2 ) such that t → S(X(t))(η) is measurable on I for all η in S (R) and t → ||X(t)|| 0 belongs to L 1 (I).T h e n X is dB (h) -integrable on I and there exist a natural integer q and a constant C I such that,

I X(t) dB (h) (t) -q C I I ||X(t)|| 0 dt.
(5.32)

Proof. For η ∈ S (R),t h em e a s u r a b i l i t yo nI of t → S(X(t) W (h) (t))(η) is clear since S(X(t) W (h) (t))(η)=S(X(t))(η) d dt [g η (t, h(t))].
B yl e m m a2 . 5 ,w eh a v e ,f o ra n yi n t e g e rq 2,

|S(X(t)

W (h) (t))(η)| ||X(t)|| 0 ||W (h) (t)|| -q e |η| 2 q
for every t in I.S i n c et → ||W (h) (t)|| -q is continuous by proposition 5.9 and t → ||X(t)|| 0 belongs to L 1 (I) by assumption, the result follows from theorem 2.8. We refer to theorem 13.5 of [START_REF] Kuo | White Noise Distribution Theory[END_REF] for the upper bound.

Remark 5.16. One can show, using appendix X.A,t h a ti n e q u a l i t y(5.32) is true for every integer q 2.

It is of interest to have also a criterion of integrability for generalized functionals of mBm. In that view, we set up the following notation: for p ∈ N, 0 <a<b,w ec o n s i d e ram a pF :[a, b] → S -p (R) (hence F (t) is a tempered distribution for all t). We then define

F (t, B (h) (t)) := F (t)(B (h) (t))
.R e c a l lt h ef o l l o w i n gt h e o r e m (see [START_REF] Chilov | Les distributions,v o l u m e2[END_REF], lemma 1 and 2 p.73-74):

Theorem 5.17. Let I be an interval of R, t → F (t) be a map from I into S -p (R), t → ϕ(t, .) be a map from

I into S (
) (t))](η)| max{t -2ph(t) ,t 2h(t) }t -h(t)/2 » C (H1,H2) p max s∈[a,b] |F (s)| -p exp{ 1 2 |A p η| 2 0 } (( 1 a ) 2pH2 + b 2pH2 )(( 1 a ) H 2 2 + b H 2 2 ) » C (H1,H2) p max s∈[a,b] |F (s)| -p exp{ 1 2 |A p η| 2 0 }, (5.33)
where

H 1 := min s∈[a,b]
F (t, B (h) (t)) is (S) * -integrable over [a, b]. For dB (h) -integrability, we first note that S[F (t, B (h) (t)) W (h) (t)](η)=S[F (t, B (h) (t))](η) d dt [g η (t,
|S[F (t, B (h) (t)) W (h) (t)](η)| ||F (t, B (h) (t))|| -p0 ||W (h) (t)|| -p0 exp {|η| 2 p0 } K exp{|η| 2 p0 }, (5.34)
where

K := sup t∈[a,b] ||W (h) (t)|| -p0 (( 1 a ) 2p0H2 + b 2p0H2 )(( 1 a ) H 2 2 + b H 2 2 ) » C (H1,H2) p0 max s∈[a,b] |F (s)| -p0 .
Theorem 2.8 applies again and shows that t → F (t,

B (h) (t)) W (h) (t) is integrable over [a, b].
Remark 5.19. Recall that a function f : R → R is said to be of polynomial growth if there is an integer m in N and a constant C such that for all

x ∈ R, |f (x)| C(1 + |x| m ).T h ep r e v i o u st h e o r e me n t a i l si n particular that both quantities b a f (B (h) (t)) dt and b a f (B (h) (t)) dB (h) (t) exist in (S) * if f is a function of polynomial growth. Example 5.20 (Computation of T 0 B (h) (t) dB (h) (t)). Let T>0 fixed. Then I := T 0 B (h) (t) dB (h) (t)= T 0 W (h) (t) B (h) (t) dt = T 0 dB (h) (t) dt B (h) (t) dt. (5.35)
Let us prove that the last quantity is equal to

1 2 B (h) (T ) 2 := 1 2 (B (h) (T ) B (h) (T )) = 1 2 (B (h) (T ) 2 -T 2h(T ) ) (see remark 2.6).
It is sufficient to compute the S-transforms of both sides of the equality. For η in S (R),

S( T 0 B (h) (t)dB (h) (t))(η)= T 0 S(B (h) (t))(η)S(W (h) (t))(η)dt = T 0 g η (t, h(t)) d dt [g η (t, h(t))]dt = 1 2 (S(B (h) (T ))(η)) 2 = 1 2 S Ä B (h) (T ) B (h) (T ) ä (η)=S( 1 2 (B (h) (T ) 2 -T 2h(T ) ))(η).
(5.36)

To end this section, we present a simple but classical stochastic differential equation in the frame of mBm.

Example 5.21 (The multifractional Wick exponential). Fol lowing [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF] formula (4.8) and [START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF] example 3.6, let us consider the multifractional stochastic differential equation

® dX(t)=α(t)X(t)dt + β(t)X(t)dB (h) (t) X(0) ∈ (S * ), (5.37)
where t belongs to R + and where α : R → R and β : R → R are two deterministic continuous functions.

(3.10) is a shorthand notation for

X(t)=X(0) + t 0 α(s) X(s) ds + t 0 β(s) X(s) dB (h) (s), (5.38)
where the previous equality holds in (S) * .R e w r i t et h ep r e v i o u se q u a t i o ni nt e r m so fd e r i v a t i v e si n(S) * as:

   dX dt (t)=α(t) X(t)+β(t) X(t) W (h) (t)=(α(t)+β(t)W (h) (t)) X(t) X(0) ∈ (S * ).
(5.39)

We thus are looking for an (S

Z(t) := X(0) exp Å t 0 α(s)ds + t 0 β(s)dB (h) (s) ã ,t ∈ R + , (5.40)
where exp has been defined in section 2.4.

Theorem 5.22. The process Z defined by (3.12) is the unique solution in (S * ) of (3.11).

Proof. This is a straightforward application of theorem 3.1.2 in [START_REF] Holden | Stochastic Partial Differential Equations, A Modeling, White Noise Functional Approach[END_REF].

Remark 5.23. [START_REF] Holden | Stochastic Partial Differential Equations, A Modeling, White Noise Functional Approach[END_REF] uses the Hermite transform in order to establish the theorem. However it is possible to start from (5.38), take S-transforms of both sides and solve the resulting ordinary stochastic differential equation. Besides, equation (3.10) may be solved with other assumptions on α, β.W er e f e rt oaf o r t h c o m i n g paper for more on stochastic differential equations driven by mBm.

Remark 5.24. In particular when X(0) is deterministic, equal to x, α() ≡ α and β() ≡ β are constant functions, the solution X of (5.38) reads

X(t)=x exp {βB (h) (t)+αt -1 2 β 2 t 2h(t) },t ∈ R + , (5.41)
which is analogous to formula (3.31) given in [START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF] in the case of the fractional Brownian motion.

Multifractional Wick-Itô integral of deterministic elements versus Wiener integral with respect to mBm

In section 4.2, we have defined a Wiener integral with respect to mBm. It is natural to check whether this definition is consistent with the multifractional Wick-Ito integral when the integrand is deterministic. More precisely, we wish to verify that R f (s) W (h) (s)ds = J h (f ) for all functions f such that both members of the previous equality exist and that the left-hand side member is in (L 2 ).I nt h a tv i e ww efi r s tp r o v et h e following theorem. Theorem 5.25. Let f : R → R be a deterministic function which belongs to L 1 loc (R).L e tZ := (Z(t) t∈R ) be the process defined on R by Z(t) := t 0 f (s)dB (h) (s).T h e nZ is an (S * )-process which verifies the following equality in (S * )

t 0 f (s)dB (h) (s)= +∞ k=0 t 0 f (s) d ds [g e k (s, h(s))]ds < ., e k >. (5.42)
Moreover Z is a (centered) Gaussian process if and only if

+∞ k=0 Ä t 0 f (s) d ds [g e k (s, h(s))]ds ä 2 < +∞,f o r all t. In this case, Z(t)= t 0 f (s)dB (h) (s) N 0, +∞ k=0 t 0 f (s) d ds [g e k (s, h(s))]ds 2 , ∀t ∈ R. (5.43)
In particular, the process Z is Gaussian when the function f belongs to C 1 (R, R) and is such that

sup t∈R |f ′ (t)| < +∞.
Proof. We treat only the case t ∈ R * + .T h eo t h e rc a s ef o l l o w ss i m i l a r l y .L e tf be in L 1 loc (R).I no r d e rt o show (5.42) let us establish a), b) and c) below. a) Let us show that s → f (s) W (h) (s) is (S) * -integrable over [0,t].F o re v e r yη ∈ S (R) and s in [0,t],w e get, using lemma 2.4:

|S(f (s) W (h) (s))(η)| = |f (s)| S(W (h) (s))(η) =:L(s) |f (s)| ||W (h) (s)|| -p0 e 1 2 |η| 2 p 0
for s in [0,t] and for p 0 2.S i n c eL is the product of a continuous function and a function of L 1 loc (R), a) is ac o n s e q u e n c eo ft h e o r e m2 . 8 . b)

Ψ f := +∞ k=0 Ä t 0 f (s) d ds [g e k (s, h(s))]ds ä < .
, e k > belongs to (S -p0 ) as soon as p 0 2. Lemma 5.6 entails that there exists a real D such that, for every p 0 2,w eh a v e

||Ψ f || 2 -p0 = +∞ k=0 t 0 f (s) d ds [ge k (s,h(s))] ds (2k+2) p 0 2 ||f || 2 L 1 ([0,t]) +∞ k=0 sup s∈[0,t] | d ds [ge k (s,h(s))]| 2 (2k+2) 2p 0 D 2 ||f || 2 L 1 ([0,t]) +∞ k=0 (k+1) 2 (2k+2) 2p 0 < + ∞. c) t 0 f (s)dB (h) (s)= +∞ k=0 Ä t 0 f (s) d ds [g e k (s, h(s))]ds ä < ., e k > in (S * ). Denote Φ f := t 0 f (s)dB (h) (s)= t 0 Ä +∞ k=0 f (s) d ds [g e k (s, h(s))]< ., e k >
ä ds and define the (S * )-process τ :

[0,t] → (S * ) by τ (s) := +∞ k=0 f (s) d ds [g e k (s, h(s))] < ., e k >.M o r e o v e r ,f o rN in N * ,d e fi n eo n[0,t], τ N : s → τ N (s) := N k=0 f (s) d ds [g e k (s, h(s))]< ., e k >.O b v i o u s l y w e h a v e , i n (S * ), Φ f = t 0 τ (s)ds, Ψ f = lim N →+∞ t 0 τ N (s) ds.I t t h e n r e m a i n s t o s h o w t h a t Φ f = lim N →+∞ t 0 τ N (s) ds in (S * )
.L e t u s u s e , f o r t h i s purpose, theorem X.2. Let p 0 be an integer greater than or equal to 2.I t i s e a s i l y s e e n t h a t τ n and τ are weakly measurable on [0,t] for every n in N (see definition X.1) and that, τ n (s) and τ (s) belongs to (S -p0 ) for every n in N and s in

[0,t].M o r e o v e r ,b o t hf u n c t i o n ss → ||τ n (s)|| -p0 and s → ||τ (s)|| -p0 belong to L 1 ([0,t],du) since ||τ n (s)|| -p0 ||τ (s)|| -p0 |f (s)| D » +∞ k=0 (2k +2) -2(p0-1)
for a certain D given by lemma 5.6 (ii).W eh e n c eh a v es h o w nt h a tb o t hf u n c t i o n sτ n (.) and τ (.) are Bochner integrable on [0,t]. Besides, for every (n, m) in N 2 with n m,w eh a v e

| d ds [g e k (s, h(s))]| 2 (2k +2) 2p0 ê 1/2 MD +∞ k=m+1 1 (2k +2) 2(p0-1) 1/2 ------------→ (n,m)→(+∞,+∞) 0,
where

M := ||f || L 1 ([0,t])
and D is again given by (ii) of lemma 5.6. Theorem X.2 then applies and allows to write that lim

N →+∞ t 0 τ N (s) ds = t 0 τ (s) ds in (S * ).W eh e n c eh a v es h o w nt h a tΨ f = lim N →+∞ t 0 τ N (s) ds = t 0 τ (s)ds =Φ f in (S * ).T h i se n d st h ep r o o fo fc) and establishes formula (5.42). If +∞ k=0 Ä t 0 f (s) d ds [g e k (s, h(s))]ds ä 2 < +∞,forallt,thenZ(t)
is the (L 2 )-limit of a sequence of independent Gaussian variables. Formula (5.43) is then obvious. When f is of class C 1 and such that sup

t∈R |f ′ (t)| < +∞,
an integration by parts yields that

+∞ k=0 t 0 f (s) d ds [g e k (s, h(s))]ds 2 < +∞.
It is easy to check that definitions4 . 1a n d5 . 1c o i n c i d eo nt h es p a c eE(R).I n d e e df o rf :=

n k=1 α k 1 [0,t k ] in E(R)
,r e m a r k4 . 3a n de q u a l i t y( 5 . 1 2 )e n t a i lt h a tJ

h (f )= n k=1 α k B (h) t k almost surely. According to (i) of proposition 5.14, we have the equality R f (s)dB (h) (s)ds = n k=1 α k R 1 [0,t k ] (s)W (h) (s)ds = n k=1 α k B (h) t k almost surely. This implies in particular that || R f (s)dB (h) (s)ds|| (L 2 ) = ||f || h for all f in E(R) since we have ||J h (f )|| (L 2 ) = ||f || h
for such f .S i n c eW i e n e ri n t e g r a l sw i t hr e s pect to standard Brownian motion are the elements of the set

{ R f (s) dB(s),f∈ L 2 (R)} = { R f (s) dB(s),f∈E(R)} (L 2 ) ,i
:= { R f (s) dB (h) (s),f∈E(R)} (L 2 )
.W ec a l lW i e n e r integral with respect to B (h) the elements of Θ h .

Remark 5.26. (i) Obviously, Θ h = { +∞ k=0 Ä R f (s) d ds [g e k (s, h(s))]ds ä < ., e k > : f ∈E(R)} (L 2 )
.T h a n k s to definition 4.1,t h e o r e m5. [START_REF] Corlay | Multifractional volatility models[END_REF] and the fact that

J h (f )= R f (s) dB (h) (s) on E(R),w eh a v e Θ h = {J h (u):u ∈ E(R) <,> h } (5.44)
In other words, the set of Wiener integrals in the sense of definition 4.1 and 5.2 coincide.

(ii) When h is a constant function equal to H we find that

Θ h =Θ H = {< ., M H (f ) > : f ∈ L 2 H (R)} since E(R) <,> h = E(R) <,> H = L 2 H (R).
T h i si se x a c t l yw h a ti se x p e c t e di nv i e wo f(4.1).

In fact we can be a little more precise in the case of fBm. Let supp(K) denote the set of measurable functions f : R → R with compact support.

Proposition 5.27. Let H ∈ (0, 1).T h e n : * .I fw ea s s u m et h a tΦ f belongs to (L 2 ),t h e nt h e equality is valid in (L 2 ).B e s i d e s ,s i n c ef belongs to L 1 loc (R) ∩ L 2 H (R) we have, according to theorem 3.7,

(i) Let f : R → R be in L 1 loc (R) ∩ L 2 H (R).T h e n R f (s) dB (H) (s) belongs to (L 2 ) if and only if R f (s) dB (H) (s)=J H (f ). (ii) L 1 loc (R) ∩ L 2 H (R) ∩ supp(K) ⊂{f : R → R : R f (s) dB (H) (s) ∈ (L 2 )}. (iii) For µ-almost every f in L 1 loc (R) ∩ supp(K) ∩{f : R → R : R f (s) dB (H) (s) ∈ (L 2 )}, f is in L 2 H (R) and verifies R f (s) dB (H) (s)=J H (f ). Proof. (i) Let f ∈ L 1 loc (R) ∩ L 2 H (R) and define Φ f := R f (s) dB (H) (s).B
J H (f )=< ., M H (f ) >= +∞ k=0 <M H (f ),e k > L 2 (R) < ., e k >= +∞ k=0 <f,M H (e k ) >< ., e k > in (L 2 ).T h e converse part is obvious since R f (s) dB (H) (s)=J H (f ) entails that R f (s) dB (H) (s) ∈ (L 2 ). (ii) Since f is in L 1 loc (R) ∩ supp(K) theorem 5.25 (by replacing 1 [0,t] by 1 supp(f ) ,w h e r esupp(f ) de- notes the support of f )e n t a i l st h a t R f (s)dB (H) (s)= +∞ k=0 R f (s)M H (e k )(s)ds < ., e k > = +∞ k=0 < f, M H (e k ) >< ., e k > in (S)
* .B e s i d e s ,s i n c ef belongs to L 2 H (R), J H (f ) exists and is equal to

+∞ k=0 < f, M H (e k ) >< ., e k > in (L 2 ). (iii) Fix f in L 1 loc (R) ∩ supp(K) ∩{f : R → R : R f (s) dB (H) (s) ∈ (L 2 )} and define Ω as subset of ω in Ω such that (5.7) is true for all g in {e k : k ∈ N}.C l e a r l y Ω belongs to G.A ss o o na sf is in Ω,w ec a nw r i t e R f (s) dB (H) (s)= +∞ k=0 <M H (f ),e k >< ., e k > in (L 2 ).T h i se n t a i l st h a tM H (f ) belongs to L 2 (R) and then, by bijectivity of M H ,t h a tf belongs to L 2 H (R).
Remark 5.28. This proposition shows in particular, that for µ-almost every g in supp(K):

g ∈ L 2 H (R) ⇔ R g(s) dB (H) (s) ∈ (L 2 ).
Moreover, in this case, J H (g)

(L 2 ) = R g(s) dB (H) (s).
6I t ô F o r m u l a s 6.1 Itô Formula for generalized functionals of mBm on an interval [a, b] with 0 <a<b

Let us fix some notations. For a tempered distribution G and a positive integer n,l e tG (n) denote the n th distributional derivative of G.W ea l s ow r i t eG ′ := G (1) .H e n c e ,b yd e fi n i t i o n ,t h ee q u a l i t y<G when it exists in S -p (R),f o rac e r t a i ni n t e g e rp.W h e ni te x i s t s , ∂F ∂t (t 0 ) is a tempered distribution, which is said to be the derivative of the distribution F (t) with respect to t at point t = t 0 .I n line with section 5.3, we then define, for t 0 in [a, b] and a positive integer n,t h ef o l l o w i n gq u a n t i t i e s :

′ ,ϕ >= -< G, ϕ ′ > holds for all ϕ in S (R).F o ram a pt → F (t) from [a, b] to S -p (R) we will note ∂ n F ∂x n (t) the quantity (F (t)) (n) ,
∂ n F ∂x n (t 0 ,B h (t 0 )) := (F (t 0 )) (n) (B h (t 0 )) and ∂F ∂t (t 0 ,B h (t 0 )) := Å ∂F ∂t (t 0 ) ã (B h (t 0 )).
F (b, B (h) (b)) -F (a, B (h) (a)) = b a ∂F ∂t (s, B (h) (s)) ds + b a ∂F ∂x (s, B (h) (s)) dB (h) (s) + 1 2 b a Å d ds [R h (s, s)] ã ∂ 2 F ∂x 2 (s, B (h) (s)) ds. (6.1) Remark 6.2. Recall that for all t in [a, b], d dt [R h (t, t)] = 2 t 2h(t)-1 (h ′ (t) t ln t + h(t)).
Proof. We follow closely [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF] p.97-98 for this proof. First notice that the three integrals on the right side of (6.1) exist since all integrands verify the assumptions of theorem 5.18. According to lemma 2.7 it is then sufficient to show equality of the S-transforms of both sides of (6.1). It is easy to see that, for every

η ∈ S (R),t h ef u n c t i o nt → γ Ä t 2h(t) ,.- t 0 M h(t) (η)( u) du ä is differentiable from (0,b] into S (R)
.U s i n g theorem 5.12 and theorem 5.17 we may write, for t in [0,b]:

d dt S(F (t, B (h) (t))(η)= d dt F (t),γ t 2h(t) ,.- t 0 M h(t) (η)(u) du = ∂F ∂t (t),γ t 2h(t) ,.- t 0 M h(t) (η)(u) du +2t 2h(t)-1 (h ′ (t) t ln t + h(t)) F (t), ∂γ ∂t t 2h(t) ,.- t 0 M h(t) (η)(u) du - d dt [g η (t, h(t))] F (t), ∂γ ∂x t 2h(t) ,.- t 0 M h(t) (η)( u) du =: I 1 + I 2 + I 3 . Now, I 1 = S Å ∂F ∂t (t)(B (h) (t)) ã (η)=S Å ∂F ∂t (t, B (h) (t))
ã (η) using theorem 5.12 (iii).B e s i d e s ,s i n c eγ fulfills the equality ∂γ ∂t = 1 2 ∂ 2 γ ∂x 2 ,i ti sc l e a rt h a t

I 2 = t 2h(t)-1 (h ′ (t) t ln t + h(t)) F (t), ∂ 2 γ ∂x 2 t 2h(t) ,.- t 0 M h(t) (η)(u) du = 1 2 d dt [R h (t, t)] S Å ∂ 2 F ∂x 2 (t, B (h) (t)) ã (η).
Using (ii) of theorem 5.12, we get

I 3 = S(W (h) (t))(η) ∂F ∂x (t),γ t 2h(t) ,.- t 0 M h(t) (η)(u) du .F i n a l l y , we obtain d dt S(F (t, B (h) (t))(η)=S Å ∂F ∂t (t, B (h) (t)) ã (η)+S Å W (h) (t) ∂F ∂x (t, B (h) (t)) ã (η) + 1 2 d dt [R h (t, t)] S Å ∂ 2 F ∂x 2 (t, B (h) (t)) ã (η).
In the proof of theorem 6.6 we will need the particular case where the function F (.) is constant, equal to a tempered distribution that we denote F .I nt h i sc a s ew eh a v et h ef o l l o w i n g Corollary 6.3. Let 0 <a<band F be a tempered distribution. Then the following equality holds in (S) * :

F (B (h) (b)) -F (B (h) (a)) = b a F ′ (B (h) (s)) dB (h) (s)+ 1 2 b a Å d ds [R h (s, s)] ã F ′′ (B (h) (s)) ds.
Remark 6.4. Of course when the function h is constant on R, we get the Itô formula for fractional Brownian motion given in [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF].

6.2 Itô Formula in (L 2 )

In this subsection, we give two further versions of Itô formula. The first one holds for functions with polynomial growth but weak differentiability assumptions, whereas the second one deals with C 1,2 functions with sub-exponential growth.

Itô Formula for certain generalized functionals of mBm on an interval Theorem 6.1 does not extend immediately to the case a =0because the generalized functional is not defined in this situation, since M h (1 [0,t] ) converges to 0 a.s and in L 2 (R) when t tends to 0 (see theorem-definition 5.2). As in [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF], we now extend the formula to deal with this difficulty. We will need the following lemma which is a particular case of lemma 6.8 below. L f (g(t),x)=f (x 0 ),f o ra l lr e a lx 0 .

Theorem 6.6. Let F : R → R be continuous at 0 and of polynomial growth. Assume that the first distributional derivative of F is of function type (defined at the beginning of section 3.1).T h e nt h ef o l l o w i n ge q u a l i t y holds in (L 2 ):

F (B (h) (b)) -F (0) = b 0 F ′ (B (h) (s)) dB (h) (s)+ 1 2 b 0 Å d ds [R h (s, s)] ã F ′′ (B (h) (s)) ds. (6.2)
Proof. We follow again closely [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF].

Step 1: lim

t→0+ F (B (h) (t)) = F (0) in (S *
).I no r d e rt oe s t a b l i s ht h i sf a c t ,l e tu su s et h e o r e m8.6 of [START_REF] Kuo | White Noise Distribution Theory[END_REF]. Since F is of polynomial growth, we may write, thanks to formula (29) of [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF], that there exist two reals C and M and a positive integer

m such that E[F (B (h) (t)) 2 ] C 2 (1 + (2m)! 2 m m! |t| 2mh(t) ) M 2 ,
|S[F (B (h) (t))](η)| = |E[F (B (h) (t)) : e <.,η> :]| ||F (B (h) (t))|| 0 || : e <.
lim t→0+ S[F (B (h) (t))](η) = lim t→0+ R F (y) 1 √ 2πt 2h(t) exp ® - 1 2t 2h(t) t 0 M h(t) (η)(u) du -y 2 ´dy = lim t→0+ R F (y) γ t 2h(t) , t 0 M h(t) (η)(u) du -y dy = lim t→0+ L F t 2h(t) , t 0 M h(t) (η)(u) du = F (0). (6.4)
Step 2: lim

a→0+ b a F ′ (B (h) (t)) dB (h) (t)= b 0 F ′ (B (h) (t)) dB (h) (t) in (S) * . Define [H 1 ,H 2 ] := [ min t∈[0,b] h(t), max t∈[0,b] h(t)
] and let us prove the two following facts (i) There exists a constant D 1 which depends only of

F such that ||F ′ (B (h) )(t)|| -1 D 1 max ¶ 1 t H 1 , 1 t H 2 © for all t in (0,b]. Let us first notice that, for all (x, b, t) in R×R * + ×(0,b],wehaveexp {-x 2 /4t 2h(t) } e -x 2 /4 +ε(b)e x p{-x 2 /4b 2H2 } where ε(b)=1if b 1 and ε(b)=0if b<1.
Note moreover that the function x → F ′ (x)(e -x 2 /4 + ε(b)e x p{-x 2 /4b 2H2 }) belongs to L 1 (R) since F ′ is of function type and belongs to S ′ (R).S i n c et h eo p e r a t o rA -1 has a norm operator equal to 1/2 (see [START_REF] Kuo | White Noise Distribution Theory[END_REF] p.17)a n du s i n gt h ee q u a l i t y|M h(t) (1 [0,t] )| 2 0 = t 2h(t) ,w eg e tt h ef o l l o w i n gu p p e rb o u n d ,v a l i df o ra l lk in N,

| <F ′ ,ξ (t,h(t),k) > | 2 = R F ′ (x) π 1/4 (k!) 1/2 t kh(t) exp{-x 2 4t 2h(t) }e k (x/( √ 2t h(t) )) dx 2 π 1/2 sup u∈R |e 2 k (u)| Ä R |F ′ (x)| exp{-x 2 4t 2h(t) } dx ä 2 t 2kh(t) k! π 1/2 sup sup u∈R |e 2 k (u)| : k ∈ N Å R |F ′ (x)| (e -x 2 /4 + ε(b)e x p{-x 2 /4b 2H2 }) dx ã 2 =:D0 t 2kh(t) k!.
Using (ii) of remark 2.3 and again the fact that the operator A -1 has a norm operator equal to 1/2 (see (2) p.17 of [START_REF] Kuo | White Noise Distribution Theory[END_REF] ) we can write, for all real t in (0,b],t h a t

||F ′ (B (h) (t))|| 2 -1 = 1 √ 2πt h(t) +∞ k=0 (k!) -1 t -2kh(t) <F ′ ,ξ (t,h(t),k) >I k Ä ( M h(t) (1 [0,t] )) ⊗k ä 2 -1 = 1 2πt 2h(t) +∞ k=0 (k!) -1 t -4kh(t) | <F ′ ,ξ (t,h(t),k) > | 2 |(A -1 ) ⊗k Ä ( M h(t) (1 [0,t] )) ⊗k ä | 2 0 =|A -1 (Mh(t)(1[0,t]))| 2k 0 D 0 2πt 2h(t) +∞ k=0 (k!) -1 t -4kh(t) t 2kh(t) k! Å 1 2 ã k t 2kh(t) D 0 π max ß 1 t 2H1 , 1 t 2H2 ™ . (6.5) (ii) b 0 F ′ (B (h) (t)) dB (h) (t) exists in (S) * and is equal to lim a→0+ b a F ′ (B (h) (t)) dB (h) (t) in the sense of (S) * .
In order to establish the existence of

b 0 F ′ (B (h) (t)) dB (h) (t) in (
S(F ′ (B (h) (t)) W (h) (t))(η) ||F ′ (B (h) (t))|| -1 ||W (h) (t)|| -2 e |η| 2 2 " K max { 1 t H 1 , 1 t H 2 } =:L(t) e |η| 2 2 , (6.6)
where we have defined "

K := D0 π sup t∈[0,b] ||W (h) (t)|| -2 .T h ef u n c t i o nt → S(F ′ (B (h) (t)) W (h) (t))(η) is mea- surable on [0,b] since S(F ′ (B (h) (t)) W (h) (t))(η)=S(F ′ (B (h) (t)
))(η)S(W (h) (t))(η) using theorems 5.12 and 5.17. Moreover, since L belongs to

L 1 ([0,b]),theorem2.8appliesandsho wsthat b 0 F ′ (B (h) (t)) dB (h) (t) is in (S)
* .I t t h e n j u s t r e m a i n s t o u s e t h e o r e m 8.6 in [START_REF] Kuo | White Noise Distribution Theory[END_REF] to show the convergence, in the sense of

(S) * ,o f b a F ′ (B (h) (t)) dB (h) (t) to b 0 F ′ (B (h) (t)) dB (h) (t)
as a tends to 0 + .L e t (a n ) n∈N be a decreasing sequence of real numbers which tends to 0 when n tends to +∞ and [START_REF] Kuo | White Noise Distribution Theory[END_REF] applies and shows that lim

Ψ n := b 0 F ′ (B (h) (t)) dB (h) (t) - b an F ′ (B (h) (t)) dB (h) (t).F o re v e r yη ∈ S (R) and every n ∈ N, S(Ψ n )(η)= b 0 1 [0,an] (t) S(F ′ (B (h) (t)) W (h) (t))(η) dt.U s i n g( 6 .
a→0+ b a F ′ (B (h) (t)) dB (h) (t)= b 0 F ′ (B (h) (t)) dB (h) (t) in (S) * .
Step 3: Proof of (6.2) For any real a such that 0 <a<b,w eh a v e ,t h a n k st oc o r o l l a r y6 . 3 ,

F (B (h) (b)) -F (B (h) (a)) - b a F ′ (B (h) (s)) dB (h) (s)= 1 2 b a Å d ds [R h (s, s)] ã F ′′ (B (h) (s)) ds.
Steps 1 and 2 ensure that the left hand side has a limit in (S) * when a tends to 0.U s i n g t h e o r e m 2.8, it is easy to see that b 0

Ä d ds [R h (s, s)] ä F ′′ (B (h) (s))
Ä d ds [R h (s, s)] ä F ′′ (B (h) (s)) ds is equal to b 0 Ä d ds [R h (s, s)] ä F ′′ (B (h) (s)) ds in (S) * .S i n c ew eh a v ep r o v e dt h a t ,f o ra l lt in [0,b], F (B (h) (t)) belongs
to (L 2 ),t h es a m eh o l d sf o rt h er i g h th a n ds i d eo f(6.2) and then this equality holds also in (L 2 ).

Remark 6.7. As in the case of fBm (see [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF]), the fact that both sides of the equality (6.2) are in (L 2 ) does not imply that every single element of the right hand side is in (L 2 ). This will be true if, for instance,

F ′′ (B (h) (t)) belongs to (L 2 ) and b 0 d ds [R h (s, s)] ||F ′′ (B (h) (s))|| 0 ds < +∞.
Itô Formula in (L 2 ) for C 1,2 functions with sub-exponential growth

Let us begin with the following lemma: Lemma 6.8. Let T>0 and f : [0,T] × R → R be a continuous function such that there exists a couple

(C T ,λ T ) of R × R * + such that max t∈[0,T ]
|f (t, y)| C T e λT y 2 for all real y.D e fi n ea>λ T , I a := (0, 1 4a ) and J f : R × R + × I a → R by J f (x, t, u) := R f (t, y)γ(u, x -y) dy.T h e nJ f is well defined and moreover lim (x,t,u)→(x0,0 + ,0 + ) J(x, t, u)=f (0,x 0 ).

Proof. This is an immediate consequence of theorems 1 p.88 and 2 p.89 of [START_REF] Widder | Positive temperatures on an infinite rod[END_REF].

Let us now give an Itô formula for functions with subexponential growth. Theorem 6.9. Let T>0 and h : R → (0, 1) be a C 1 function such that h ′ is bounded on R.L e tf be a C 1,2 ([0,T] × R, R) function. Furthermore, assume that there are constants C 0 and λ<

1 4m a x t∈[0,T ] t 2h(t) such that for all (t, x) in [0,T] × R, max t∈[0,T ] ß |f (t, x)| , ∂f ∂t (t, x) , ∂f ∂x (t, x) , ∂ 2 f ∂x 2 (t, x) ™ Ce λx 2 . (6.7)
Then, for all t in [0,T],t h ef o ll o w i n ge q u a l i t yh o l d si n(L 2 ):

f (T,B (h) (T )) = f (0, 0) + T 0 ∂f ∂t (t, B (h) (t)) dt + T 0 ∂f ∂x (t, B (h) (t)) dB (h) (t) + 1 2 T 0 Å d dt [R h (t, t)] ã ∂ 2 f ∂x 2 (t, B (h) (t)) dt. (6.8)
Proof. Our proof is similar to the one of theorem 5.3 in [START_REF] Bender | An S-transform approach to integration with respect to a fractional Brownian motion[END_REF]. Let T>0 and t ∈ [0,T].F o r m u l a( 6 . 8 )m a y be rewritten as

T 0 ∂f ∂x (t, B (h) (t)) dB (h) (t)=f (T,B (h) (T )) -f (0, 0) - T 0 ∂f ∂t (t, B (h) (t)) dt - 1 2 T 0 Å d dt [R h (t, t)] ã ∂ 2 f ∂x 2 (t, B (h) (t)) dt. (6.9)
In order to show that all members on the right hand side of (6.9) are in (L 2 ),l e tu su s et h e o r e m2 . 9 . I fG belongs to

¶ f, ∂f ∂t , ∂f ∂x , ∂ 2 f ∂x 2
© we may write, thanks to (6.7),

E G(t, B (h) (t)) 2 C 2 R exp ß - 1 2 Å 1 t 2h(t) -4λ ã v 2 ™ dv √ 2πt 2h(t) =:M 2 C 2 (1 -4λ max t∈[0,T ] t 2h(t) ) -1/2 . (6.10) Since B (h) (t)=B H (t) |H=h(t)
a.s,i ti se a s yt os e e( i nv i e wo f [START_REF] Bender | An S-transform approach to integration with respect to a fractional Brownian motion[END_REF]p . 9 7 8 )t h a tB (h) (t) is a Gaussian variable with mean equal to

t 0 M h(t) (η)(u) du = g η (t, h(t)
) and variance equal to t 2h(t) under the probability Q η which has been defined in (2.11). Hence, for every t in (0,T] and η in S (R),

S(G(t, B (h) (t))(η)=E Qη [G(t, B (h) (t))] = R G( t, u + g η (t, h(t)) ) γ(t 2h(t) ,u) du = R G t, u t h(t) + t 0 M h(t) (η)(x) dx 1 √ 2π e -u
Å d dt [R h (t, t)] ã ∂ 2 f ∂x 2 (t, B (h) (t)) 0 dt 2M T 0 t 2h(t)-1 |h ′ (t) t ln t + h(t)| dt < +∞.
Thus, theorem 2.9 applies and shows that all members on the right side of (6.9) are in (L 2 ).

Let us now show that t → ∂f ∂x (t,

B (h) (t)) W (h) (t) is (S) * -integrable over [0,T].
Reasoning as in the estimate (6.6), we note that there exists an integer q 2 such that ∂f ∂x (t, B (h) (t)) W (h) (t) belongs to (S -q ) for every t in [0,T].M o r e o v e r ,f o re v e r yη in S (R) and every t in (0,T] we have, using lemma 2.5,

S Å ∂f ∂x (t, B (h) (t)) W (h) (t) ã (η) ∂f ∂x (t, B (h) (t)) 0 ||W (h) (t)|| -2 exp{|η| 2 2 } Ç sup t∈[0,T ] ||W (h) (t)|| -2 å ∂f ∂x (t, B (h) (t)) 0 exp{|η| 2 2 }, Since t → ∂f ∂x (t, B (h) (t)) 0 belongs to L 1 ([0,T]
),t h e o r e m2 . 8a p p l i e so n[0,T] to the effect that T 0 ∂f ∂x (t, B (h) (t)) dB (h) (t) belongs to (S) * .I tt h e nj u s tr e m a i n st os h o wt h ef o l l o w i n ge q u a l i t yf o ra l lt in [0,T] and η in S (R):

S T 0 ∂f ∂x (t, B (h) (t)) dB (h) (t) (η)=S f (T,B (h) (T )) -f (0, 0) - T 0 ∂f ∂t (t, B (h) (t)) dt (η) -S 1 2 T 0 Å d dt [R h (t, t)] ã ∂ 2 f ∂x 2 (t, B (h) (t)) dt (η). (6.12)
Using (6.11) with G = f and applying the theorem of differentiation under the integral sign, we get

d dt î S(f (t, B (h) (t))(η) ó = R d dt î f ( t, u + g η (t, h(t)) ) γ(t 2h(t) ,u) ó du = R γ(t 2h(t) ,u) d dt [f ( t, u + g η (t, h(t)) )] du =:U1(t) + R f ( t, u + g η (t, h(t)) ) d dt î γ(t 2h(t) ,u) ó du =:U2(t) . Now, U 1 (t)= R γ(t 2h(t) ,u) ∂f ∂t (t, u + g η (t, h(t))) du + R γ(t 2h(t) ,u) ∂f ∂x (t, u + g η (t, h(t))) d dt g η (t, h(t))du = S Å ∂f ∂t Ä t, B (h) (t) ä ã (η)+S Å ∂f ∂x Ä t, B (h) (t)) ä ã (η) S Ä W (h) (t) ä (η).
Besides, using the equality ∂γ ∂t = 1 2 ∂ 2 γ ∂x 2 and an integration by parts, we get

U 2 (t)= 1 2 d dt [t 2h(t) ] R ∂ 2 f ∂x 2 (t, u + g η (t, h(t))) γ(t 2h(t) ,u) du = 1 2 d dt [t 2h(t) ] S ∂ 2 f ∂x 2 Ä t, B (h) (t) ä (η).
Hence we obtain, for any ε>0,u p o ni n t e g r a t i n gt → U 1 (t)+U 2 (t) between ε and T ,

S(f (t, B (h) (t))(η) -S(f (ε, B (h) (ε))(η)= T ε S Å ∂f ∂t Ä t, B (h) (t) ä ã (η) dt + T ε S Å ∂f ∂x Ä t, B (h) (t)) W (h) (t) ã (η) dt + 1 2 T ε d dt [t 2h(t) ] S ∂ 2 f ∂x 2 Ä t, B (h) (t) ä (η) dt. (6.13)
Let us now show that lim

ε→0+ S Ä f (ε, B (h) (ε) ä (η)=f (0, 0) = S Ä f (0,B (h) (0) ä (η).
F o re v e r yε>0,( 6 . 1 1 )c a n be rewritten as S(f (ε,

B (h) (ε))(η)= R f (ε, y) γ(ε 2h(ε) ,g η (ε, h(ε)) -y) dy.
For a fixed T>0,l e tλ T ,C T be such that (6.7) is fulfilled. There exists b>0 such that ε 2h(ε) belongs to I a (defined in lemma 6.8) as soon as 0 <ε<b .H e n c ew em a yw r i t e ,f o ra n yε in (0,b), S(f (ε,

B (h) (ε))(η)= J f (g η (ε, h(ε)),ε,ε 2h(ε)
).S i n c e lim ε→0+ ε 2h(ε) and lim ε→0+ g η (ε, h(ε)) are equal to 0,lemma6.8applieswithx 0 =0, and yields lim

ε→0+ S Ä f (ε, B (h) (ε) ä (η)=f (0, 0).
Let us now establish (6.12). Thanks to the fact that both the functions

t → S[G(t, B (h) (t))](η) and t → S[G(t, B (h) (t)) W (h) (t)](η)
are continuous on [0,T] and using the dominated convergence we can take the limit when ε tends to 0 on the right hand side of (6.13) and finally get

S[f (T,B (h) (T )) -f (0, 0)](η)=S T 0 ∂f ∂t Ä t, B (h) (t) ä dt (η)+S T 0 ∂f ∂x Ä t, B (h) (t) ä dB (h) (t) (η) + S 1 2 T 0 d dt [t 2h(t) ] ∂ 2 f ∂x 2 Ä t, B (h) (t) ä (η) dt.
Remark 6.10. We observe that if we take expectations on both sides of Itô's formula (6.7),w eg e te x a c t l y formula (1) of theorem 2.1 of [START_REF] Hirsch | From an Itô type calculus for Gaussian processes to integrals of log-normal processes increasing in the convex order[END_REF], which is a general weak Itô formula for Gaussian processes, in the particular case where the Gaussian process is chosen to be an mBm.

7T a n a k a f o r m u l a a n d e x a m p l e s

In this section we first give a Tanaka formula as a corollary to theorem 6.6 with F : x →| x -a|.W et h e n consider the case of two particular h functions that give noteworthy results.

Tanaka formula

Theorem 7.1 (Tanaka formula for mBm). Let h : R → (0, 1) be of class C 1 , a ∈ R and T>0.T h e following equality holds in (L 2 ):

|B (h) (T ) -a| = |a| + T 0 sign Ä B (h) (t) -a ä dB (h) (t)+ T 0 d dt [R h (t, t)] δ {a} (B (h) (t)) dt, (7.1)
where the function sign is defined on R by sign(x)

:= 1 R * + (x) -1 R-(x).
Proof. This is a direct application of theorem 6.6 with F : x →|x -a|.

Remark 7.2. That the previous equality holds true in (L 2 ) does not imply of course that both integrals above are in (L 2 ). This last result will be established in a forthcoming paper.

Itô formula for functions

h such that d dt [R h (t, t)] = 0 If h verifies d dt [R h (t, t
)] = 0,t h e nt h es e c o n do r d e rt e r m ∂ 2 f ∂x 2 (t, B (h) (t)) disappears in Itô formula. The formula is then formally the same one as in ordinary calculus. In this case, (7.1) reads: In order to obtain an mBm defined on a compact interval, we may choose a compact subset of (-∞, -e λ ) ∪ (e λ , +∞) when λ>0 and a compact subset of (-e λ , 0) ∪ (0,e λ ) when λ<0.

|B (h) (T ) -a| = |a| + T 0 sign(B (h) (t) -a) dB (h) (t).

Note moreover that lim

t→+∞ h i (t) = lim t→0 h i (t)=0for i =1, 2.

Itô formula for functions

h such that d dt [R h (t, t)] = 1
The situation where d dt [R h (t, t)] = 1 is interesting since then Itô formula is formally the same as in the case of standard Brownian motion. As a consequence, Tanaka formula takes the familiar form :

|B (h) (T ) -a| = |a| + T 0 sign(B (h) (t) -a) dB (h) (t)+ T 0 δ {a} (B (h) (t)) dt.
Thus, instead of a "weighted" local time as in (7.1), we get here an explicit expression for the local time of mBm for a family of h functions that we describe now. The solutions (h c ) c∈R of the differential equation are given by h c :( c, +∞)\{-1, 0, 1}→R and h c (t) := 1 2 ln(t-c) ln |t| .R e c a l lt h a th c is required to range in (0, 1).D e n o t e ,f o rc ∈ R,

I c := { t ∈ (c, +∞)\{-1, 0, 1} : 0 <h c (t) < 1}.F o rc in (-∞, 1/4),l e tt 1 := t 1 (c) := 1- √ 1-4c 2 and t 2 := t 2 (c) := 1+ √ 1-4c 2
.T h e nI c is explicitly given as follows:

∀c ∈ (-∞, -2],I c =(1+c, t 1 ) ∪ (t 2 , +∞), ∀c ∈ (-2, -1],I c =(t 1 , 1+c) ∪ (t 2 , +∞), ∀c ∈ (-1, 0),I c =(t 1 , 0) ∪ (0, 1+c) ∪ (t 2 , +∞), ∀c ∈ [0, 1/4),I c =(t 1 ,t 2 ) ∪ (1 + c, +∞), ∀c 1/4,I c =(1+c, +∞).

8C o n c l u s i o n a n d f u t u r e w o r k

In this paper we have used a white noise approach to define a stochastic integral with respect to multifractional Brownian motion which generalizes the one for fBm based on the same approach. This stochastic calculus allows to solve some particular stochastic differential equations. We are currently investigating several extensions of this work. In order to apply this calculus to financial mathematics or to physics, it is necessary to study further the theory of stochastic differential equations driven by a mBm. This is the topic of a forthcoming paper. The Tanaka formula we have obtained suggests that one can get several integral representations of local time with respect to mBm. Finally, since mBm is a Gaussian process, it seems also natural to investigate the links between the construction of stochastic integral withr e s p e c tt om B mt h a tw eg a v ea n dt h eo n ep r o v i d e db y Malliavin calculus. An extension in higher dimension is also desirable.

9A c k n o w l e d g m e n t s

The authors are thankful to Marc Yor for many helpful remarks and comments. They are also grateful to Erick Herbin for suggesting this research topic and to Sylvain Corlay for the simulations of mBm.

XA p p e n d i x X.A Bochner integral

All the following notions about the integral in the Bochner sense come from [START_REF] Hille | Functional Analysis and Semi-Groups,v o l u m e3 1 . A m e r i c a nM a t h e m a t i c a l Society[END_REF] p.72, 80 and 82 and from [START_REF] Kuo | White Noise Distribution Theory[END_REF] p.247. Definition X.1 (Bochner integral [START_REF] Kuo | White Noise Distribution Theory[END_REF] p.247). Let I be a subset of R endowed with the Lebesgue measure. One says that Φ:I → (S) * is Bochner integrable on I if it satisfies the two following conditions:

1. Φ is weakly measurable on I i.e u →< < Φ(u),ϕ > > is measurable on I for every ϕ in (S).

2. ∃p ∈ N such that Φ(u) ∈ (S -p ) for almost every u ∈ I and u → ||Φ(u)|| -p belongs to L 1 (I).

The Bochner-integral of Φ on I is denoted I Φ(s) ds .

Properties X.1. If Φ:I → (S)
* is Bochner-integrable on I then 1. there exists an integer p such that I Φ(s) ds -p I ||Φ(s)|| -p ds. 2. Φ is also Pettis-integrable on I and both integrals coincide on I.

Remark X.1. The previous property shows that there is no risk of confusion by using the same notation for both the Bochner integral and the Pettis integral. Furthermore, if there exists an (S) * -process, denoted Ψ,w h i c hv e r i fi e s(X.1),t h e nΨ(s)=Φ ( s) for almost every s in I.F i n a l l yt h ef o l l o w i n ge q u a l i t yh o l d s :

lim n→+∞ I Φ n (s) ds = I Φ(s) ds in (S * ).

X.B Proof of proposition 4.1

Let B (h) be a normalized mBm on R with covariance function noted R h .I ti sw e l lk n o w nt h a to n ec a nd e fi n e on the linear space span R {R h (t, .):t ∈ R} an inner product, denoted <, > R h ,b y<R h (t, .),R h (s, .) > R h := R h (t, s) (see [START_REF] Janson | Gaussian Hilbert spaces[END_REF] p.120 ff.). Define Ξ h the closure of span R {R h (t, .):t ∈ R} for the norm || || R h .T h e space (Ξ h , || || R h ) is called the Cameron-Martin space (or Reproducing Kernel Hilbert Space (R.K.H.S.)) associated to the the Gaussian process B (h) .L e t E(R) denote the quotient space obtained by identifying all functions of E(R) which are equal almost everywhere. On E(R) × E(R) define a bilinear form, noted

<, > h ,b y< 1 [0,t] , 1 [0,s] > h := R h (t, s).
T h e n <, > h is an inner product provided the linear map κ h :

E(R) → Ξ h defined by κ h (1 [0,t] ) := R h (t, .), t ∈ R,i si n j e c t i v e . D e fi n eI h := vect R {B (h) (t):t ∈ R} (L 2 )
the first Wiener chaos of B (h) .I ti saw e l l -k n o w np r o p e r t yo fR . K .

H . S .t h a tt h em a pτ

h :( Ξ h , || || R h ) → ( span R {B (h) (t):t ∈ R} (L 2 )
, || || (L 2 ) ),d e fi n e df o ra l lr e a lt by τ h (R h (t, .)) = B (h) (t) is an isometry. As a result, κ h is injective if and only if τ h • κ h is injective. The next proposition states that this is indeed the case for any continuous function h: Proposition X.3. Let h be a continuous function defined on R and ranging in (0, 1).T h ef a m i l y(B (h) (t)) t∈R * is linearly independent on R,i . ef o re v e r yp o s i t i v ei n t e g e rn, (

β 1 ,β 2 ••• ,β n ) in R n and (t 1 ,t 2 ••• ,t n ) in (R * )
n ,s u c ht h a tt i = t j for i = j,t h ee q u a l i t y n j=1

β j B (h) (t j )=0 a.s, (X.2) implies β 1 = β 2 = ••• = β n =0.
The proof of this proposition requires the following lemma, the proof of which is easy and left to the reader.

Lemma X.4. Define, for t ∈ R,t h ef u n c t i o nA t : R → C by A t (ξ) := e itξ -1 iξ if ξ =0and A t (0) := t.
Then, for all t, A t is C ∞ on R and verifies, for every n ∈ N, A

t (0) = t (it) n n+1 ,w h e r eA (n) t (n)
denotes the n th derivative of A t .

Proof. of proposition X.3. Let us use a proof by contradiction. By decreasing n if necessary we may always assume that

(β 1 ,β 2 ••• ,β n ) belongs to (R * ) n .B e s i d e s ,s i n c eB (h) (t)=B (H) (t) | H=h(t)
and thanks to lemma 4.2 (i),e q u a l i t y( X . 2 )a l s or e a d s< .,

n j=1 β j M h(tj) (1 [0,tj] ) >=0 a.s.B y t a k i n g F o u r i e r transforms, we get n j=1 β j ¤ M h(tj) (1 [0,tj ] )=0 a.e..U s i n g( 3 . 1 )t h i sy i e l d s n j=1 α j |ξ| 1/2-h(tj ) ' 1 [0,tj] (ξ)=0, ∀ξ ∈ R * , (X.3)
where we have defined, for j in {1; 2; •••; n}, α j := β j (c h(tj ) ) -1 .B yr e -a r r a n g i n gi fn e c e s s a r yt h e(t i ) i ,w e may assume without loss of generality that h(t 1 ) h(t 2 ) ••• h(t n ).L e tcard(A) denote the cardinal of the set A. We distinguish three cases.

First case: card({h(t 1 ); h(t 2 );

•••; h(t n )})=1 . Since h(t 1 )=h(t 2 )=••• = h(t n )=:
H,w ege t ,b ym u l t i p l y i n ge q u al i t y( X . 3)b y|ξ| H-1/2 and taking inverse Fourier transform, n j=1 α j 1 [0,tj] =0 almost everywhere on R.T h i se n t a i l st h a t{α 1 ; α 2 ; •••; α n } and then

{β 1 ; β 2 ; •••; β n } is equal to {0}.
Second case: h(t 1 ) >h(t 2 ).U s i n gt h a t ' 1 [0,t] (ξ)=A t (ξ),( X . 3 )r e a d s :

α 1 ( e it 1 ξ -1 iξ )=- n j=2 α j |ξ| h(t1)-h(tj) ( e it j ξ -1 iξ ), ∀ξ ∈ R * . (X.4)
By lemma X.4 and taking the limit when ξ tends to 0 in (X.4), we get α 1 =0which constitutes a contradiction.

Third case: h(t 1 )=h(t 2 ).

There exists an integer r in {2; 3;

•••; n -1}, (k 1 ,k 2 , ••• ,k r ) in (N * ) r with 2 k 1 <k 2 < ••• <k r = n,s u c h that h(t 1 )=h(t 2 )=••• = h(t k1 )= : H 1 h(t k1+1 )=h(t k1+2 )=••• = h(t k2 )= : H 2 . . . . . . h(t kr-1+1 )=h(t kr-1+2 )=••• = h(t kr )= :H r , (X.5)
where

1 >H 1 >H 2 > ••• >H r > 0.N o t
:= {1; 2; •••; k 1 }, I 2 := {k 1 +1;k 1 +2;•••; k 2 }, ••• ,I r := {k r-1 +1;k r-1 + 2; •••; k r }.
U s i n gl e m m aX . 4 ,e q u a l i t y( X . 3 )c a nb er e w r i t t e na s r l=1 j∈I l

α j |ξ| 1/2-H l A tj (ξ)=0, ∀ξ ∈ R * . (X.6)
Lemma X.5. For every p in N * , j∈I1

α j t p j =0. (L p )
Let us admit this lemma for the moment. The equalities (L p )forp in {1; 2; •••; k 1 } yield the following linear system:

á t 1 t 2 ••• t k1-1 t k1 t 2 1 t 2 2 ••• t 2 k1-1 t 2 k1 . . . . . . ••• . . . . . . t k1 1 t k1 2 ••• t k1 k1-1 t k1 k1 ë =:D á α 1 α 2 . . . α k1 ë = á 0 0 . . . 0 ë . (X.7)
The determinant of this system is a Vandermonde determinant which is non zero since all the t i are distinct from each other. As the result, the only solution is

α 1 = α 2 = •••α k1 =0which constitutes a contradiction
and proves the proposition.

We now present a sketch of pro of of lemma X.5.

Proof of lemma X.5. By multiplying both sides of equality (X.6) by |ξ| H1-1/2 and then taking the limit when ξ tends to 0,w eg e t ,u s i n gl e m m aX . 4 , j∈I1 α j t j =0 ,w h i c hi se q u a l i t y(L 1 ).N o w ,fi xp in N * . Starting from equality (X.6) we first -m u l t i p l yb o t hs i d e so fe q u a l i t y( X . 6 )b y|ξ| H1-1/2 and call (X.6 bis) the resulting equality. For any ξ in R * ,w et h e nt a k et h ep th derivative of both sides of equality (X.6 bis) at point ξ.W ec a l l(E 1 ) the equality thus obtained. It reads:

r l=1 j∈I l α j [|ξ| H1-H l A tj (ξ)] (p) =0, ∀ξ ∈ R * , (E 1 )
where [g(ξ)] (p) denotes the p th derivative of the p-times differentiable map ξ → g(ξ). Now, starting from (E 1 ), we recursively perform the following operations successively for l =2,...,r: -m u l t i p l yb o t hs i d e so fe q u a l i t y(E l-1 ) by |ξ| H l -H l-1 +p and call (E l-1 bis) the resulting equality.

-tak ethep th derivative of both sides of equality (E l-1 bis) at every point ξ in R * and call (E l ) the resulting equality.

Equality (E r ) then reads:

r l=1 j∈I l α j ×   ••• [|ξ| H1-H l .A tj (ξ)] (p) .|ξ| H2-H1+p (p)
.|ξ|

H3-H2+p

(p)

•••|ξ|

Hr-1-Hr-2+p

.|ξ|

Hr -Hr-1+p   (p) =:[••• ] (p) l,j (ξ) =0, ∀ξ ∈ R * , (E r ) Lemma X.4 yields that lim ξ→0 α j A (p) tj (ξ)= i p p+1 α j t p+1 j
.W ew a n tt ol e tξ tend to 0 in the previous equality.

However, for (l, j) in {1; 2;

•••; r}×I l , lim ξ→0 [•••] (p)
l,j (ξ)=+ ∞.N e v e r t h e l e s s , i t i s e a s y t o s h o w t h a t , f o r every (l, j) in {1; 2;

•••; r}×I l , lim ξ→0 [••• ] (p) l,j (ξ) A (p) t j
(ξ) |ξ| Hr -H l exists in C * and is independent of j.D e fi n e c l := lim ξ→0

[••• ] (p) l,j (ξ) A (p) t k l
(ξ) |ξ| Hr -H l , l =1,...,r.D e n o t e ,f o r(l, j) in {1; 2; •••; r}×I l , U l,j : R → C,t h ec o n t i n u o u sm a po n

R such that [•••] (p) l,j (ξ)=c l |ξ| Hr -H l A (p) tj (ξ)(1+ U l,j (ξ)).E q u a l i t y(E r ) then reads r l=1 c l j∈I l α j |ξ| Hr -H l A (p) tj (ξ)( 1+ U l,j (ξ)) = 0, ∀ξ ∈ R * . (X.8)
Upon multiplying both sides of the previous equality by |ξ| H1-Hr we get, for

ξ in R * c 1 j∈I1 α j A (p) tj (ξ)(1+ U 1,j (ξ)) = - r l=2 c l j∈I l α j |ξ| H1-H l A (p) tj (ξ)(1+ U l,j (ξ)). (X.9)
Since H 1 >H l for l in {2; •••; r},t a k i n gt h el i m i tw h e nξ tends to 0 in (X.9) and using lemma X.4 yields

c 1 i p (p +1) -1 j∈I1 α j t p+1 j =0,

w h i c hi sn o t h i n gb u t(L p+1 ).T h i se n d st h ep r o o f .

Remark X.6. Another way to establish that R H (., .) defines an inner product on E(R) for H in (0, 1) is to use (3.3) and (3.9).

INTRODUCTION AND BACKGROUND

57

these relations. It turns out that the following condition is sufficient to prove all the results we will need in this section. (E[Y ] denotes the expectation of a random variable Y in L 1 (Ω, F ,P)):

(H 1 ):∀[a, b] ⊂ R, ∀[c, d] ⊂ (0, 1), ∃(Λ,δ) ∈ (R * + ) 2 , such that E[(B(t, H) -B(t, H ′ )) 2 ] Λ |H -H ′ | δ , for every (t, H, H ′ ) in [a, b] × [c, d] 2 .
Note that assumption (H 1 ) is equivalent to the following one:

(H):∀[a, b] × [c, d] ⊂ R × (0, 1), ∃(Λ,δ) ∈ (R * + ) 2 ,s.t. E[(B(t, H) -B(s, H ′ )) 2 ] Λ Ä |t -s| 2c + |H -H ′ | δ ä , for every (t, s, H, H ′ ) ∈ [a, b] 2 × [c, d] 2 .
Indeed, since B is a fractional field, for every

(t, s, H) in [a, b] 2 × [c, d], E[(B(t, H) -B(s, H)) 2 ]=|t -s| 2H . Then if (H 1 ) is true we have, for all (t, s, H, H ′ )[ a, b] 2 × [c, d] 2 , E[(B(t, H) -B(s, H ′ )) 2 ] 2Λ (1 + |b -a| 2(d-c) ) Ä |t -s| 2c + |H -H ′ | δ ä .
Thus, we will refer either to assumption (H 1 ) or (H) in the sequel.

Remark

) ∈ [a, b] 2 × [c, d] 2 , E[(B(t, H) -B(s, H ′ )) 2p ]=κ p Ä E[(B(t, H) -B(s, H ′ )) 2 ]) ä p κ p Λ p Ä |t -s| 2c + |H -H ′ | δ ä p κ p (2 Λ) p max{|t -s|; |H -H ′ |} p(δ∧2c) ,
where Λ := Λ (1 + b -a) δ∨2c-δ∧2c , x ∧ y := min{x; y} and x ∨ y := max{x; y}. Kolmogorov's criterion implies that the field B has a d-Hölder continuous version for any d in (0, 2p p(δ∧2c)-2 ). In the sequel we will always work with this version.

An mBm is defined as follows:

Definition 1.2 (Multifractional Brownian motion). Let h : R → (0, 1) be a deterministic continuous function. A multifractional Brownian motion with functional parameter h is the Gaussian process B h := (B h t ) t∈R defined by B h t := B(t, h(t)) for every t in R.

Aw o r do nn o t a t i o n : B H

. or B h(t)

.

will always denote an fBm with Hurst index H or h(t),w h i l eB h . will stand for an mBm. Note that

B h t := B(t, h(t)) = B h(t) t
,f o re v e r yr e a lt. Furthermore, we say that h is the regularity function of the mBm. We will say that the fractional field (B(t, H)) (t,H)∈R×(0,1) is normalized when, for all H in (0, 1),t h ef r a c t i o n a lB r o w n i a nm o t i o n(B H t ) t∈R , resulting from the fractional field (B(t, H)) (t,H)∈R×(0,1) ,i san o r m a l i z e df B m . I nt h i sc a s ew ew i l la l s os a y that the mBm B h := (B h t ) t∈R is normalized. It is straightforward to check that an mBm in the sense of [75, def.1.1] is also an mBm with our definition. [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF] provides the correlation structure of various mBms. In general, it is clear that the correlation structures of two mBms derived from different fractional Brownian fields will differ whenever the "inter-line" correlations of the fields differ. Fractional fields (B(t, H)) (t,H)∈R×(0,1) leading to previously considered mBms include:

B 1 (t, H) := 1 c H R e itu -1 |u| H+1/2 W 1 (du) , B 2 (t, H) := R Ä |t -u| H-1/2 -|u| H-1/2 ä W 2 (du), B 3 (t, H) := R Ä (t -u) H-1/2 + -(-u) H-1/2 + ä W 3 (du) , B 4 (t, H) := T 0 {0 u<t T } (t, u) K H (t, u) W 4 (du),
where c H :=

Å 2cos(πH)Γ(2-2H) H(1-2H) ã 1 2 , α H := Ä 2H (1-2H)β(1-2H,H+1/2) ä 1/2 and K H (t, u) :=    α H ï Ä t u ä H-1/2 (t -u) H-1/2 -(H -1/2)u 1/2-H t u (v -u) H-1/2 v H-1/2 dv ò if 0 <H<1/2, Ä H(2H-1) β(2-2H,H-1/2) ä 1/2 u 1/2-H t u (v -u) H-3/2 v H-1/2 dv if 1/2 <H<1,
and where W i denotes an independently scattered standard Gaussian measure on R, i ∈{ 2; 3; 4},a n d W 1 denotes the complex-valued Gaussian measure which can be associated in a unique way to any independently scattered standard Gaussian measure W 1 on R (see [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF] p.203-204 and [73] p.325-326 for more details on the meaning of R f (u) W(du) and R f (u) W(du) for a real or complex-valued function f in L 2 (R,du)). B 1 (t, H) and B 2 (t, H) lead to the so-called harmonisable mBm, first considered in [7]. B 2 (t, H) yields the moving average mBm defined in [START_REF] Peltier | Multifractional brownian motion[END_REF]. All these are particular cases of mBms in the sense of [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF]. B 4 (t, H) corresponds to the Volterra multifractional Gaussian process studied in [START_REF] Boufoussi | Local time and Tanaka formula for a Volterra-type multifractional Gaussian process[END_REF].

This paper is organized as follows. The section 2 is devoted to a new definition of mBm from fractional field and then to a result of approximation of a mBm by a sum of sequence of fBm. After giving an idea of what should be a reasonable integral w.r.t. mBm we give, in section 3,ac r i t e r i o nt h a ta l l o w st ok n o ww h e na sequence of integrals with respect to several fBms converge. We then give a definition of integral with respect to mBm as a sum of two terms, depending both on, the method of integration and the fractional field from which the mBm is defined.In section 4 we apply the definition given in section 3 to the Skorohod integral w.r.t. fBm obtained, using Malliavin calculus, in [3] and show that our integral with respect to mBm is the one obtained by [START_REF] Boufoussi | Local time and Tanaka formula for a Volterra-type multifractional Gaussian process[END_REF]. In section 5 we first define an integral w.r.t. mBm (from fractional field B 1 ). We then compare it to the one which has been defined in [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF] and show that, when they exist, they both coincide.

2A p p r o x i m a t i o n o f m u l t i f r a c t i o n a l B r o w n i a n m o t i o n

Since an mBm is just a continuous path traced on a fractional Brownian field, a natural question is to enquire whether it may be approximated by patching adequately chosen fBms, and in which sense. Heuristically, for a<b,w edivide[a, b) into "small" intervals [t i ,t i+1 ),an drep laceoneac hoft h e s eB h by the fBm B Hi where H i = h(t i ).I ts e e m sr e a s o n a b l et oe x p e c tthat the resulting process i B Hi t [ti,ti+1) (t) will "converge", in as e n s et ob em a d ep r e c i s e ,t oB h when the size of the intervals [t i ,t i+1 ) goes to 0. This idea is further supported by the fact that the moving average, harmonisable and Volterra mBms are all "tangents" to fBms in the following sense. This idea is further supported by the fact that the moving average, harmonisable and Volterra mBms are all "tangents" to fBms in the following sense: for every real u,

B h u+rt -B h u r h(u) ; t ∈ [a, b] law ----→ r→0 + {B h(u) t ; t ∈ [a, b]}
Our aim in this section is to make this line of thought rigorous.

Approximation of mBm by piecewise fBms

In the sequel, we fix a fractional Brownian field B and a continuous function h,t h u sa nm B m . W ea i mt o prove that this mBm can be approximated on every compact interval [a, b] by patching together fractional Brownian motions defined on a sequence of partitions of [a, b].I nt h a tv i e w ,d e fi n ea ni n c r e a s i n gs e q u e n c e (q n ) n∈N of integers such that q 0 := 1, 2 n q n 2 2 n for every n in N * .F o rac o m p a c ti n t e r v a l[a, b] of R and n in N,l e tx (n)

:= {x (n) k ; k ∈ [[0,q n ]] } where x (n) k := a + k (b-a)
qn for every k in [[0,q n ]] (for integers p and q with p<q, [[ p, q]] denotes the set {p; p +1;•••; q}). Now, if we define, for every n in N,t h ep a r t i t i o n

A n := {[x (n) k ,x (n) k+1 ); k ∈ [[0,q n -1]]}∪{x (n)
qn },i ti sc l e a rt h a tA := (A n ) n∈N is a decreasing nested sequence of subdivisions of [a, b] (i.e A n+1 ⊂A n ,f o re v e r yn in N). For every t in [a, b] and n in N there exists a unique integer p in [[0,q n -1]] such that x

B hn t := B(t, h n (t)) = qn-1 k=0 [x (n) k ,x (n) k+1 ) (t) B(t, h(x (n) k )) + {b} (t) B(b, h(b)). (2.1)
Note that, despite the notation, the process B hn is not an mBm, as h n is not continuous in general. We believe however there is no risk of confusion in using this notation. B hn is almost surely càdlàg and discontinuous at times x

(n) k , k in [[0,q n ]
] . Theorem 2.1 (Approximation theorem). Let B be a fractional Brownian field, h : R → (0, 1) be a continuous deterministic function and B h be the associated mBm. Let [a, b] be a compact interval of R, A be a sequence of partitions as defined above, and consider the sequence of processes defined in (2.1).T h e n :

1. If B is such that the map C (defined in (i) of remark 1.1) is continuous on ([a, b]) 2 × h([a, b]) 2 then the sequence of processes (B hn ) n∈N converges in L 2 (Ω) to B h ,i.e ∀t ∈ [a, b], lim n→+∞ E Ä B hn t -B h t ä 2 =0.
2. If B satisfies assumption (H) and if h is β-Hölder continuous for some positive real β,thenthese quenc e of processes (B hn ) n∈N * converges 2 . The cases where t = b or s = t are consequences of point 1. above. We now assume that a s<t<b. For every integer n in N,w eg e t

(i) in law to B h ,i . e {B hn t ; t ∈ [a, b]} law -----→ n→+∞ {B h t ; t ∈ [a, b]}. (ii) almost surely to B h ,i . e P Å {∀t ∈ [a, b], lim n→+∞ B hn t = B h t } ã =1. Proof. 1. Let t ∈ [a, b].F o ra n yn in N,a n du s i n gt h en o t a t i o n so f(i) in remark 1.1, E Ä B hn t -B h t ä 2 = C(t, t, h(x (n) t ),h(x (n) t )) -2 C(t, t, h(x (n) t ),h(t)) + C(t,
E î B hn t B hn s ó = (k,j)∈[[ 0 ,qn-1]] 2 [x (n) k ,x (n) k+1 ) (t) [x (n) j ,x (n) j+1 ) (s) E [B(t, h n (t))B(s, h n (s))] . We hence get E î B hn t B hn s ó = E î B(t, h(x (n) t ))B(s, h(x (n) s ))
ó for all large enough integers n,( i.e such that

x (n) s s<x (n) t t)
. The continuity of h, (i) of remark 1.1, and the fact that lim n→∞ (x

(n) t ,x (n) s )=(t, s),e n t a i lt h a t lim n→∞ E î B hn t B hn s ó = E î B h t B h s ó .
b) Tightness of the sequence of probability measures (P • B hn ) n∈N .

We are in the particular case where a sequence of càdlàg pro cesses converges to a continuous one. Theorem page 92 of [START_REF] Pollard | Convergence of stochastic processes[END_REF] applies to this situation: it is sufficient to show that, for every positive reals ε and τ ,t h e r e exist an integer m and a grid

{t i } i∈[[ 0 ,m]] such that a = t 0 <t 1 < ••• <t m = b
® A α (1 + q n )e x p -ατ 2 4|b -a| 2H2 q 2H2 n ;4 (1+q n )e x p -τ 2 2 7 D q δβ n ; b -a q n ; GD 1/2 q n δβ 2 ´< 1 ∧ τ 8 ∧ ε 2 .
(2.4) Set m := m(τ, ε)=q N and t

i := x (m) i for i in [[0,m]] .N o t
:= P Ä ¶ max 0 i m sup t∈[ti,ti+1) |B hn t -B hn ti | >τ ©ä .T h e n J τ,m n (1 + m)m a x 0 i m P Ä ¶ sup t∈[ti,ti+1) |B(t, h n (t)) -B(t i ,h n (t i ))| >τ ©ä (1 + m) Ä max 0 i m L τ,i n +m a x 0 i m Q τ,i n ä , (2.5
|B(t i ,h n (t)) -B(t i ,h n (t i ))| >τ/2 © å . -U p p e rb o u n df o r(1 + m)m a x 0 i m L τ,i n : The couple (i, n) being fixed in [[0,m]] × N,theprocess(B(s, h n (t))) s∈[ti ,ti+1] is

a fractional Brownian motion of Hurst index h n (t).U s i n gi n c r e m e n t -s t a t i o n a r i t ya n ds e l f -s i m i l a r i t yo ff r a c t i o n a lB r o w n i a nm o t i o ny i e l d s :

L τ,i n = P Ç sup t∈[ti,ti+1) |B(t -t i ,h n (t))| >τ/2 å = P Ç sup u∈[0,ti+1-ti) |t i+1 -t i | hn(u+ti) B Å u t i+1 -t i ,h n (u + t i ) ã >τ/2 å P Ç sup v∈[0,1] |B(v, h n (t i + v(t i+1 -t i )))| > τ 2|t i+1 -t i | H2 å .
(2.6)

Using Markov identity and Fernique's theorem,

L τ,i n P Ç sup (v,H)∈F |B(v, H)| > τ 2|t i+1 -t i | H2 å = P Ç exp ¶ α sup (v,H)∈F |B(v, H)| 2 © > exp ß ατ 2 4|t i+1 -t i | 2H2 ™ å E ñ exp ® α sup (t,H)∈F B(t, H) 2 ´ô exp -ατ 2 4|t i+1 -t i | 2H2 = A α exp ® -ατ 2 q 2H2 N 4|b -a| 2H2 < ε 2(1 + q N ) = ε 2(1 + m) .
We have shown that

∀i ∈ [[0,q N ]] , ∀n N, (1 + m)m a x 0 i m L τ,i n < ε 2 . (2.7) -U p p e rb o u n df o r(1 + m)m a x 0 i m Q τ,i n : Fix a couple (i, n) in [[0,m]] × N.R e c a l lt h a th n (t) belongs to h([t i ,t i+1 ]) =: K (i) for every t in [t i ,t i+1 ).W e hence have, Q τ,i n P Ä ¶ sup H∈K (i) |B(t i ,H) -B(t i ,h(t i ))| >τ/2 ©ä 2 P Ä ¶ sup H∈K (i) =:Xi(H) B(t i ,H) -B(t i ,h(t i )) >τ/4 ©ä . (2.8)
Observe that the right hand side of the previous inequality does not depend on n any more.

Our aim is to apply [2, (2.6) p.43]. In that view, we first prove the following estimate:

Lemma 2.2. For al l i in [[0,q N ]] , µ i := E ï sup H∈K (i) X i (H) ò < GD 1/2 q δβ/2 N < τ 8 .
Proof of lemma 2.2: Fix i in [[0,q N ]] . We recall some notions from [START_REF] Talagrand | n o g r a p h si nM a t h e m a t i c s .S p r i n g e r -V e r l a g[END_REF]. A sequence

C (i) := (C (i) n ) n∈N of partitions of K (i) is called admissible if it is increasing and such that card(C (i) n )
2 2 n ,f o re v e r yn in N.L e td i denote the pseudo-distance associated to the Gaussian process (X i (H))

H∈K (i) ,( i.e d i (H, H ′ ) := (E[(X i (H) -X i (H ′ )) 2 ]) 1/2 ,f o re v e r y(H, H ′ ) in K (i) ×K (i) ).
For (H, p) in K (i) ×N,letC d i (H, H ′ ).[ 7 6 ,T h e o r e m2.1.1]e n t a i l st h a t :

µ i Lγ 2 (K (i) ,d i ), (2.9)
where γ 2 (K (i) ,d i ) := inf sup

H∈K (i) p 0 2 p/2 ∆ i (C (i)
p (H)) and where the infimum is taken over all admissible sequences of partitions of K (i) .L e tH (i)

1 and

H (i) 2 be such that K (i) =: [H (i) 1 ,H (i) 2
].C o n s i d e rt h es e q u e n c e of partitions C (i) := (C (i) n ) n∈N of K (i) defined, for every integer n,b ys e t t i n gC

(i) n := {[y (n) k ,y (n) k+1 ); k ∈ [[0,q n -1]]}∪{y (n) qn },w h e r ey (n) 0 = H (i) 1 and y (n) k+1 -y (n) k = H (i) 2 -H (i) 1 qn ,f o r(n, k) in N × [[0,q n -1]]. It is clear that C (i) is a decreasing nested sequence of partitions of K (i) and is hence admissible. For (H 0 ,p) in K (i) × N,d e n o t e[y (p) k0 ,y (p) k0+1 ) the unique element of C (i)
p which contains H 0 .T h e n ,u s i n g(H),

µ i L p 0 2 p/2 sup H0∈K (i) sup (H,H ′ )∈[y (p) k 0 ,y (p) k 0 +1 ) 2 (E[(X i (H) -X i (H ′ )) 2 ]) 1/2 L p 0 2 p/2 sup k∈[[ 0 ,qp-1]] sup (H,H ′ )∈[y (p) k ,y (p) k+1 ) 2 (E[(X i (H) -X i (H ′ )) 2 ]) 1/2 L Λ 1/2 p 0 2 p/2 sup k∈[[ 0 ,qp-1]] |y (p) k+1 -y (p) k | δ/2 = G Λ 1/2 (H (i) 2 -H (i) 1 ) δ/2
.

By Hölder continuity of h, H

(i) 2 -H (i) 1 η |t i+1 -t i | β = η |b-a| β q β N
.U s i n g( 2 . 4 ) ,w efi n a l l yg e tµ i < GD

Q τ,i n 4exp ß -(τ/4 -µ i ) 2 2σ 2 i ™ ,
where σ 2 i := sup

H∈K (i) E î X i (H) 2 ó
.B yd e fi n i t i o no f(X i (H)) H∈K (i) and assumption (H):

σ 2 i Λs u p H∈K (i) |H -h(t i )| δ Λ η δ |t i+1 -t i | βδ =Λη δ |b -a| βδ q βδ N = D q βδ N . This yields that Q τ,i n 4e x p ¶ -τ 2 q δβ N 2 7 D
© and finally: (ii) Almost sure convergence

∀i ∈ [[0,q N ]] , ∀n N, (1 + m)m a x 0 i m Q τ,i n < ε 2 . ( 2
Denote Ω the measurable subset of Ω,v e r i f y i n gP ( Ω) = 1, such that for every ω in Ω, (t,

H) → B(t, H)(ω) is continuous on [a, b] × [H 1 ,H 2 ].
T h e n ,f o re v e r yω in Ω,w eg e t :

|B hn t (ω ′ ) -B h t (ω ′ )| = |B(t, h n (t))(ω ′ ) -B(t, h(t))(ω ′ )| = B(t, h Ä x (n) t ä )(ω ′ ) -B(t, h(t))(ω ′ )|-→ n→+∞ 0.
This ends the proof.

The four fractional fields [START_REF] Torben | The distribution of realized stock return volatility[END_REF]] defined at the end of section 1 verify assumption (H),a ss h o w nb y the next proposition:

(B i (, )) i∈[[ 1 ,
Proposition 2.3. The fractional Brownian fields

B i := (B i (t, H)) (t,H)∈R×(0,1) , i ∈ [[1, 4]],f u l fi llA s s u m p t i o n (H 1 ).
Proof: The case of B 1 and B 2 :

(3.10) of [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF] yields that B 2 (t, H)

(law) = 2cos((H+1/2) π 2 ) √ 2π
Γ(H +1/2) c H B 1 (t, H).I ti st h u ss u ffi c i e n tt oe s t a b l i s h

(H 1 ) for B 1 .L e t[a, b] × [c, d] be fixed in R × (0, 1).F o ra l l(t, H, H ′ ) in [a, b] × [c, d] 2 , I H,H ′ t := E î (B(t, H) -B(t, H ′ )) 2 ó = R e itξ -1 c H |ξ| H+1/2 - e itξ -1 c H ′ |ξ| H ′ +1/2 2 dξ = R e itξ -1 ξ 2 1 cH |ξ| 1/2-H -1 c H ′ |ξ| 1/2-H ′ 2 dξ. (2.11) For ξ in R * ,t h em a pf ξ :[c, d] → R + , defined by f ξ (H) := 1 cH |ξ| 1/2-H is C 1 since H → c H is C ∞ on (0, 1). Thus there exists a positive real D such that ∀(ξ, H) ∈ R * × [c, d], |f ′ ξ (H)| D |ξ| 1/2-H (1 + | ln(|ξ|)|) D Ä |ξ| 1/2-c + |ξ| 1/2-d ä (1 + | ln(|ξ|)|).
Thanks to the mean-value theorem, (2.5) then reads

I H,H ′ t D 2 |H -H ′ | 2 R |e itξ -1| 2 |ξ| 2 Ä |ξ| 1/2-c + |ξ| 1/2-d ä 2 (1 + | ln(|ξ|)|) 2 dξ D 2 |H -H ′ | 2 2 3 |ξ|>1 (1 + ln |ξ|) 2 |ξ| 1+2c dξ +(2t) 2 |ξ| 1 |ξ| 1-2d (1 + | ln(|ξ|)|) 2 dξ (2 3 + T 2 ) D 2 |ξ|>1 (1 + ln |ξ|) 2 |ξ| 1+2c dξ + |ξ| 1 |ξ| 1-2d (1 + | ln(|ξ|)|) 2 dξ |H -H ′ | 2 .
Since the two integrals in the last line are finite, assumption (H 1 ) is verified with δ =2. The case of B 3 is settled in [68, proof of theorem 4, (2.10)]a n dt h eo n eo fB 4 in [18, Proposition 3, ( 5)].

Particular cases of Kernel integral representations of the fractional field B

An integral representation of fBm is used, mostly, for analytical computations. It is thus relevant to recast assumption (H) in terms of the kernel used in these representations. Two situations are present in the literature: an integral over a compact interval, and an integral over R.W e d e a l w i t h b o t h i n t h e n e x t subsections.

On a compact set [0,T]

In this case (see, e.g [3]), the fractional field (B(t, H)) (t,H)∈[0,T ]×(0,1) is defined by

B(t, H) := T 0 K(t, u, H) W(du),
where W is a Gaussian measure, K is defined on [0,T] 2 × (0, 1) and verifies u → K(t, u, H) belongs to L 2 ([0,T],du),f o ra l l(t, H) in [0,T] × (0, 1).T h i sc a s ei sf o ri n s t a n c et h eo n eo fB 4 .A so n ec a ne a s i l ys e e , the following condition (C K ) entails (H).

(C K ):∀(c, d) with 0 <c<d<1,H → K(t, u, H) is Hölder continuous on [c, d], uniformly in (t, u) in [0,T] 2 , i.e ∃(M, δ) ∈ (R * + ) 2 , ∀(t, u) in [0,T] 2 , |K(t, u, H) -K(t, u, H ′ )| M |H -H ′ | δ . Condition (C K ) is fulfilled by the kernel K defining B 4 (see [18, Proposition 3, (5)]).
On R ArepresentationwithanintegraloverR is used for instance in [START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF][START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF]. The fractional field (B(t, H)) (t,H)∈R×(0,1) is then defined by B(t, H) := R M (t, u, H) W(du) where M is defined on R 2 × (0, 1),a n dv e r i fi e s u → K(t, u, H) belongs to L 2 (R,du),f o re v e r y(t, H) in R × (0, 1).T h i s i s t h e c a s e f o r t h e fi e l d s B 1 , B 2 and B 3 .C o n d i t i o n(C M ) entails (H):

(C M ):∀[a, b] ⊂ R, ∀[c, d] ⊂ (0, 1), ∃δ ∈ R * + , ∀t ∈ [a, b], ∃Φ t ∈ L 2 (R,du), verifying sup t∈[a,b] R |Φ t (u)| 2 du < +∞, s.t. ∀(u, H, H ′ ) ∈ R × [c, d] 2 , |M (t, u, H) -M (t, u, H ′ )| Φ t (u) |H -H ′ | δ .
Condition (C M ) is fulfilled by the kernel M defining B 1 and B 2 (see [18, Proposition 3, (5)]).

3S t o c h a s t i c i n t e g r a l s w . r . t . m B m a s l i m i t s o f i n t e g r a l s w . r .

t . f B m

Our aim in this section is to show that one may define an integral with respect to mBm by using integrals with respect to fBms as approximations.

We consider as ab ove a fractional field (B(t, H)) (t,H)∈R×(0,1) ,b u tw ea s s u m ei na d d i t i o nt h a tt h efi e l di sC 1 in H on (0, 1) in the L 2 (Ω) sense, i.e we assume that the map H → B(t, H),f r o m(0, 1) to L 2 (Ω),i sC 1 for every real t.W ew i l ld e n o t e ∂B ∂H (t, H ′ ) the L 2 (Ω)-derivative at point H ′ of the map H → B(t, H).T h e field ( ∂B(t,H) ∂H ) (t,H)∈R×(0,1) is of course Gaussian. We will need that the derivative field satisfies the same assumption (H 1 ) as B(t, H).M o r ep r e c i s e l y ,f r o mn o wo n ,w ea s s u m et h a tB(t, H) satisfies Assumption (H 2 ) we define now,

(H 2 ):∀[a, b] × [c, d] ⊂ R × (0, 1),H → B(t, H) is C 1 in the L 2 (Ω) sense from (0, 1) to L 2 (Ω) for every t in [a, b] and ∃(∆,α,λ) ∈ (R * + ) 3 s.t. E Ä ∂B ∂H (t, H) -∂B ∂H (s, H ′ ) ä 2 ∆ Ä |t -s| α + |H -H ′ | λ ä ,f o re v e r y (t, s, H, H ′ ) in [a, b] 2 × [c, d] 2 .
Proposition 3.1. The fractional Brownian fields

B i := (B i (t, H)) (t,H)∈R×(0,1) , i ∈ [[1, 4]],verifyAssumpti on (H 2 ).
Proof: The proof of this proposition in the case of B 1 and B 2 may be found in Appendix VI. The ones for B 3 and B 4 are easily obtained using results from [START_REF] Peltier | Multifractional brownian motion[END_REF] and [START_REF] Boufoussi | Local time and Tanaka formula for a Volterra-type multifractional Gaussian process[END_REF] and are left to the reader.

Let us consider from now on a C 1 deterministic function h : R → (0, 1),af r a c t i o n a lfi e l dB which fulfills assumptions (H 1 ) and (H 2 ),a n dt h ea s s o c i a t e dm B mB h t := B(t, h(t)).

We now explain in a heuristic way how to define an integral with resp ect to mBm using approximating fBms. Write the total differential of B(t, H):

dB(t, H)= ∂B ∂t (t, H) dt + ∂B ∂H (t, H) dH.
Of course, this is only formal as t → B(t, H) is not differentiable in L 2 -sense nor almost surely with respect to t.I ti s ,h o w e v e r ,i nt h ew h i t en o i s es e n s e ,b u tw ea r en o ti n t e r e s t e di nt h e s ea s p e c t sa tt h i ss t a g e .W i t h ad i ff e r e n t i a b l ef u n c t i o nh in place of H,t h i s( a g a i nf o r m a l l y )y i e l d s

dB(t, h(t)) = ∂B ∂t (t, h(t)) dt + h ′ (t) ∂B ∂H (t, h(t)) dt. (3.1)
The second term on the right-hand side of (3.1) is defined for almost every ω and every real t by assumption. Moreover, it is almost surely continuous as fonction of t and thus Riemann integrable on compact intervals. On the other hand, the first term of (3.1) has no meaning a priori since mBm is not differentiable with respect to t.H o w e v e r ,s i n c es t o c h a s t i ci n t e g r a l sw i t hr e s pect to fBm do exist, we are able to give a sense to t → ∂B ∂t (t, H) for every fixed H in (0, 1).C o n t i n u i n gw i t ho u rh e u r i s t i cr e a s o n i n g ,w et h e na p p r o x i m a t e ∂B ∂t (t, h(t)) by lim

n→+∞ qn-1 k=0 [x (n) k ,x (n)
k+1 ) (t) ∂B ∂t (t, h n (t)).T h i sf o r m a l l yy i e l d s :

dB(t, h(t)) ≈ lim n→+∞ qn-1 k=0 [x (n) k ,x (n) k+1 ) (t) ∂B ∂t (t, h n (t)) dt + h ′ (t) ∂B ∂H (t, h(t)) dt. (3.2)
Assuming we may exchange integrals and limits, we would thus like to define, for suitable processes Y ,

1 0 Y t dB(t, h(t)) = lim n→+∞ qn-1 k=0 x (n) k+1 x (n) k Y t dB h(x (n) k ) t + 1 0 Y t h ′ (t) ∂B ∂H (t, h(t)) dt, (3.3)
where the first term of (3.3) is a sum of integrals with respect to several fBms and the second term is, for almost every ω,aR i e m a n ni n t e g r a l . The interest of this approach is that one may use any of the numerous definitions of stochastic integration with respect to fBm, and automatically obtain a corresponding integral with respect to mBm. It is worthwhile to note that, with this approach, an integral with respect to mBm is a sum of two terms: the first one seems to depend only on the chosen method for integrating with respect to fBm, (e.g. aw h i t en o i s eo rp a t h w i s e integral), while the second is, in most cases, a Riemann integral which appears to depend only on the field used to define the chosen mBm, i.e essentially on its correlation structure. This second term will imply that the integral with respect to the moving average mBm, for instance, is different from the one with respect to the harmonisable mBm. However, as the example of simple processes in the next subsection will show, the second term does also depend on the integration method with respect to fBm.

Let us now make the above idea precise. We first fix some notations. (M) denotes a fixed method of integration with respect to fBm (e.g Skohorod, white noise, pathwise, •••). For the sake of notational simplicity, we will consider integrals over the interval [0, 1].F o r H in (0, 1),d e n o t e

1 0 Y t d (M) B H t the integral of Y := (Y t ) t∈[0,1] on [0, 1]
with respect to the fBm B H ,i ns e n s eo fm e t h o d(M),a s s u m i n gi t does exist. With the same notations as in subsection 2.1, we consider a sequence (q n ) n∈N ,af a m i l yx (n)

:= {x (n) k ; k ∈ [[0,q n ]] } (which is defined on [0, 1] by x (n) k := k qn for k in [[0,q n ]] )a n dt h ef a m i l yo fp a r t i t i o n A n := {[x (n) k ,x (n) k+1 ); k ∈ [[0,q n -1]]}∪{x (n) qn } of [0, 1].T h ef o l l
o w i n gn o t a t i o n sw i l lb eu s e f u l : Definition 3.1 (i n t e g r a lw i t hr e s p e c tt ol u m p e df B m s ) . Let Y := (Y t ) t∈[0,1] be a process on [0, 1] which is integrable with respect to all fBms of index H in h([0, 1]) in the sense of method (M).W ed e n o t et h ei n t e g r a l with respect to lumped fBms in sense of method (M) by:

1 0 Y t d (M) B hn t := qn-1 k=0 1 0 [x (n) k ,x (n) k+1 ) (t) Y t d (M) B h(x (n) k ) t ,n ∈ N. (3.4)
Note that the nature of 1 0 Y t d (M) B hn t depends on the method (M) of integration. For example, We will write

1 0 Y t d (M) B H
1 0 Y t d (M) B h
t for the integral of Y on [0, 1] with respect to mBm in the sense of method (M) (which is yet to be defined). When we do not want to specify a particular method (M) but instead wish to refer to all methods at the same time, we will write

1 0 Y t dB hn t and 1 0 Y t dB h t instead of 1 0 Y t d (M) B hn t and 1 0 Y t d (M) B h t .
In order to gain a better understanding of our approach, we treat in the following subsection the cases of simple deterministic and then random integrands.

Example: simple integrands

Deterministic simple integrands

Any reasonable definition of an integral with respect to mBm must be linear. Thus it suffices to consider the case of Y =1 .O b v i o u s l y ,w es h o u l dfi n dt h a t

1 0 1 dB h t = B h 1 .

L e tu st h e nc o m p u t et h el i m i to ft h e sequence (

1 0 1 dB hn t ) n∈N . In that view, it is convenient to use the S-transform. For a function η in the Schwartz space S (R) we will denote S(Φ)(η) the S-transform of the L 2 -random variable Φ at point η (the reader who is not familiar with the S-transform may refer to section 5 of this paper where this notion is recalled in the more general setting of Hida spaces). For a field B and (t, H, η) in R × (0, 1) × S (R),w es h a l lw r i t eg η (t, H) for the S-transform at point η of the random variable B(t, H).I no t h e rw o r d s ,g η (t, H) := S(B(t, H))(η).

Proposition 3.2. Assume that the map

h is C 2 ,thatthemapH → g η (t, H) is C 2 for every (t, η) in R×S (R)
and that (t, H) → H) is continuous on R × (0, 1) for i in {1; 2}.T h es e q u e n c e(

∂ i gη ∂H i (t,
1 0 1 dB hn t ) n∈N then converges in L 2 (Ω) to B(1,h(1)) - 1 0 h ′ (t) ∂B ∂H (t, h(t)) dt,
where the second term is a pathwise integral.

Proposition 3.2 implies that, for regular enough fields B and h functions, formula (3.3) does indeed yield

1 0 1 dB h t = B h 1 . Proof: Let us first show that 1 0 h ′ (t) ∂B ∂H (t, h(t)) dt belongs to L 2 (Ω)
.C a u c h y -S c h w a r zi n e q u a l i t ya n da s s u m p t i o n (H 2 ) entail:

E ñ 1 0 h ′ (t) ∂B ∂H (t, h(t) dt 2 ô 1 0 |h ′ (t)| 2 dt :=µ 1 0 E Å ∂B ∂H (t, h(t)) ã 2 dt µ∆ 1 0 (|t| α + |h(t)| λ ) dt < +∞.
Let I n := 1 0 1 dB hn t .T h a n k st o( 3 . 4 )w eh a v e :

I n = qn-1 k=0 1 0 [x (n) k ,x (n) k+1 ) (t)1dB h(x (n) k ) t = qn-1 k=0 Å B h(x (n) k ) x (n) k+1 -B h(x (n) k ) x (n) k ã = B h x (n) qn -1 1 - qn-1 k=1 Å B h(x (n) k ) x (n) k -B h(x (n) k-1 ) x (n) k ã =:Jn (3.5)
It is thus sufficient to show that the sequence (J n ) n∈N converges in L 2 (Ω) to 1 0 h ′ (t) ∂B ∂H (t, h(t) dt.T h a n k s to [9, theorem 2.3], this is equivalent to showing that the sequence ((S(J n )(η)) n∈N (respectively

(E J 2 n ) n∈N ) converges to S( 1 0 h ′ (t) ∂B ∂H (t, h(t) dt)(η) (respectively to E Ä 1 0 h ′ (t) ∂B ∂H (t, h(t) dt ä 2 )f o ra l lη in S (R).
Now, for every integer n and η in S (R),t a k i n gS-transform of J n at point η and using two times the finite increment theorem, there exist (r

(n) k ,w (n) k ) in [x (n) k-1 ,x (n) k ] 2 such that S(J n )(η)= qn-1 k=1 S Å B h(x (n) k ) x (n) k ã (η) -S Å B h(x (n) k-1 ) x (n) k ã (η)= qn-1 k=1 g η (x (n) k ,h(x (n) k )) -g η (x (n) k ,h(x (n) k-1 )) = qn-1 k=1 ∂g η ∂H Ä x (n) k ,h(r (n) k ) äÄ h(x (n) k ) -h(x (n) k-1 ) ä = qn-1 k=1 ∂g η ∂H Ä x (n) k ,h(r (n) k ) ä h ′ (w (n) k )(x (n) k -x (n) k-1 ) = 1 q n qn-1 k=1 Å ∂g η ∂H Ä x (n) k ,h(r (n) k ) ä h ′ (w (n) k ) - ∂g η ∂H Ä x (n) k ,h(x (n) k ) ä h ′ (x (n) k ) ã =:Kn + 1 q n qn-1 k=1 ∂g η ∂H Ä x (n) k ,h(x (n) k ) ä h ′ (x (n) k ) =:Ln . (3.6)
By definition of Riemann integral,

lim n→+∞ L n = 1 0 h ′ (t) ∂g η ∂H (t, h(t)) dt = 1 0 h ′ (t) S Å ∂B ∂H (t, h(t)) ã (η) dt = S 1 0 h ′ (t) ∂B ∂H (t, h(t)) dt (η).
(3.7)

Besides, the assumptions made on the map g η and the mean value theorem entail that there exists a couple

(t (n) k ,v (n) k ) in [x (n) k ,x (n) k+1 ] 2 such that |K n | 1 q n qn-1 k=1 Å |h(r (n) k ) -h(x (n) k )|| h ′ (w (n) k )| ∂ 2 g η ∂H 2 (x (n) k ,h(t (n) k ) + |h ′′ (v (n) k )|| w (n) k -x (n) k | ∂g η ∂H Ä x (n) k ,h(x (n) k ) ä ã .
We hence have,

|K n | A 1 1 q n qn-1 k=1 Ä η |r (n) k -x (n) k | β + |w (n) k -x (n) k | ä A 1 (1 + η) 1 q β n + 1 q n -→ n→+∞ 0, (3.8)
where

A 1 := sup u∈[0,1] |h ′ (u)| sup (u,v)∈[0,1]×h([0,1]) ∂ 2 gη ∂H 2 (u, v) +s u p u∈[0,1] |h ′′ (u)| sup (u,v)∈[0,1]×h([0,1]) ∂gη ∂H (u, v) .
Thanks to (3.7) and (3.8) we get

lim n→+∞ S(J n )(η)=S Ä 1 0 h ′ (t) ∂B ∂H (t, h(t)) dt ä (η) for every η in S (R).I t then remains to show that lim n→+∞ E[J 2 n ]=E[ Ä 1 0 h ′ (t) ∂B ∂H (t, h(t)) dt ä 2 ].U s i n gF u b i n i ' st h e o r e m , Q :=E ñ 1 0 h ′ (t) ∂B ∂H (t, h(t)) dt 2 ô = 1 0 1 0 h ′ (t) h ′ (s) =:Θ((t,s,h(t)),h(s))) E ï ∂B ∂H (t, h(t)) ∂B ∂H (s, h(s)) ò ds dt.
For the same reason as in (i) of remark 1.1, assumption (H 2 ) entails that the function Θ is continuous on R 2 × (0, 1) 2 .T h u s ,

Q := lim n→+∞ 1 q 2 n 1 k,j qn-1 h ′ (x (n) j ) h ′ (x (n) k ) E î ∂B ∂H (x (n) j ,h(x (n) j )) ∂B ∂H (x (n) k ,h(x (n) k )) ó . (3.9)
Besides for any n,t h a n k st o( 3 . 6 ) ,

J n = 1 q n qn-1 k=1 Å ∂B ∂H Ä x (n) k ,h(r (n) k ) ä h ′ (w (n) k ) - ∂B ∂H Ä x (n) k ,h(x (n) k ) ä h ′ (x (n) k ) ã =:J (1) n + 1 q n qn-1 k=1 ∂B ∂H Ä x (n) k ,h(x (n) k ) ä h ′ (x (n) k ) =:J (2) n . (3.10)
In view of (3.9), it is clear that (J

n ) n∈N tends to Q in L 2 ( (2)

Ω).L e tu sn o ws h o wt h a t(J

(1) n ) n∈N tends to 0 in L 2 (Ω).I naw a ys i m i l a rt ot h ec o m p u t a t i o n sl e a d i n gt o( 3 . 8 ) ,w eg e t

J (1) n = 1 q n qn-1 k=1 Å h ′′ (v (n) k )( w (n) k -x (n) k ) ∂B ∂H Ä x (n) k ,h(x (n) k ) ä ã + 1 q n qn-1 k=1 h ′ (w (n) k ) Å ∂B ∂H Ä x (n) k ,h(r (n) k ) ä - ∂B ∂H Ä x (n) k ,h(x (n) k ) ä ã =: M n + N n . (3.11)
Using (3.8), we get on the one hand

E î M 2 n ó = 1 q 2 n 1 k,j qn-1 h ′′ (v (n) k ) h ′′ (v (n) j )( w (n) k -x (n) k )( w (n) j -x (n) j )Θ Ä x (n) k ,x (n) j ,h(x (n) k ),h(x (n) j ) ä A 2 q 2 n 1 k,j qn-1 (w (n) k -x (n) k )( w (n) j -x (n) j ) A 2 q 2 n -→ n→+∞ 0, (3.12)
where we set A 2 := sup 2) applies to the four fields considered in the introduction, since they all satisfy the required assumptions: in the case of B 1 ,i th a sb e e np r o v e di n [ 5 3 ,l e m m a5 . 5 ] ,t h a t(t, H) → g η (t, H) belongs to C ∞ (R × (0, 1)) for every η in S (R). Moreover, the result is similar when B = B 3 ,s i n c e [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF] example 2 shows that there exist two C ∞ maps H → α H and H → υ H from (0, 1) to R such that

u∈[0,1] |h ′′ (u)| 2 sup (u,v)∈[0,1] 2 |Θ(u, v, h(u),h(v)) |.O
ó A 3 q 2 n 1 k,j qn-1 ∂B ∂H Ä x (n) k ,h(r (n) k ) ä -∂B ∂H Ä x (n) k ,h(x (n) k ) ä L 2 (Ω) ∂B ∂H Ä x (n) j ,h(r (n) j ) ä -∂B ∂H Ä x (n) j ,h(x (n) j ) ä L 2 (Ω) A 3 ∆ q 2 n 1 k,j qn -1 Ä |h(r (n) j ) -h(x (n) j )|| h(r (n) k ) -h(x (n) k )| ä λ/2 A 3 ∆(1 + η λ ) q 2 n Ñ 1 k qn-1 |r (n) k -x (n) k | βλ/2 é 2 A 3 ∆(1 + η λ ) q 2 n Ñ 1 k qn-1 1 q βλ/2 n é 2 A 3 ∆(1 + η λ ) q βλ n -→ n→+∞ 0. ( 3
B 3 (t, H)=α H B 1 (t, H)+υ H R e itξ -1 iξ|ξ| H-1/2 W 1 (dξ) =:ZH (t) .S i n c eS(Z H (t))(η)= R η(u) e itu -1
iu|u| H-1/2 du for every η in S (R) the conclusion follows. The case of B 4 is similar and left to the reader.

Simple processes

Given the result of Proposition 3.2 it seems natural to set

1 0 Y t d (M) B h t := lim n→∞ 1 0 Y t d (M) B hn t +Φ (M) h (Y ), (3.14)
where the process Y is integrable with respect to B hn in sense of method (M),f o re v e r yn in N,a n dw h e r e we assume moreover that:

-lim n→∞

1 0 Y t d (M) B hn t
exist in a sense to be made precise -the functional Φ h is defined on the set containing process Y with value in L 2 (Ω) and fulfills

Φ (M) h (Y )= 1 0 h ′ (t) Y t ∂B ∂H (t, h(t)) dt for every constant process (Y t ) t∈[0,1] and Φ (M) h (Y )=0as soon as h is constant.
The following examples gives indication of what sould be the functional Φ h in general cases.

Examples: 1. Let (M 1 ) denote the White Noise Theory

The reader which is not familiar with integral with respect to fBm in the sense of White noise Theory (also called fractional Wick-Itô integral) may refer to [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF] and [START_REF] Bender | An S-transform approach to integration with respect to a fractional Brownian motion[END_REF]. Let m be a fixed integer in N.D e fi n e ,f o re v e r yi in [[0,q m ]] , t i := x (m) i and let Y be the process defined by

Y t := qm-1 i=0 Y i 1 (ti,ti+1] (t)
where, for every i, Y i is a centred Gaussian random variable which is F (B h )mesurable where F (B h ) denotes the σ-field generated by the random variables in the first Wiener chaos of the mBm B h . Since the sequence (A n ) n∈N of partitions of [0, 1] is nested, we know that, for every integer n m and every ineteger i in [[0,q m ]] ,t h e r ee x i s t sa nu n i q u ei n t e g e rk

(n) i in [[0,q n ]] such that t i = x (n) k (n) i . We hence have [t i ,t i+1 ]= k (n) i k k (n) i+1 -1 [x (n) k ,x
Y i d (M1) B h(x (n) k ) t = qm-1 i=0 k (n) i+1 -1 k=k (n) i x (n) k+1 x (n) k Y i d (M1) B h(x (n) k ) t = qm-1 i=0 k (n) i+1 -1 k=k (n) i x (n) k+1 x (n) k Y i W h(x (n) k ) t dt = qm-1 i=0 k (n) i+1 -1 k=k (n) i Y i Å x (n) k+1 x (n) k W h(x (n) k ) t dt ã = qm-1 i=0 Y i Ö k (n) i+1 -1 k=k (n) i Å B h(x (n) k ) x (n) k+1 -B h(x (n) k ) x (n) k ã è . (3.15)
Using exactly the same method we used to show that

(J n ) n∈N converged in L 2 (Ω) to 1 0 h ′ (t) ∂B ∂H (t, h(t)) dt in the proof of Proposition 3.2 we get the convergence in L 2 (Ω) of the sequence ( k (n) i+1 -1 k=k (n) i Å B h(x (n) k ) x (n) k+1 - B h(x (n) k ) x (n) k ã ) n∈N to B h ti+1 -B h ti - ti+1 ti h ′ (t) ∂B ∂H (t, h(t)) dt,f o re v e r yi in [[0,q m -1]].
T h ec o n t i n u i t yo fW i c k product then allow us to write

lim n→∞ S n = qm-1 i=0 Y i (B h ti+1 -B h ti ) - qm-1 i=0 Y i ti+1 ti h ′ (t) ∂B ∂H (t, h(t)) dt, (3.16)
where the previous limit holds in L 2 (Ω).S i n c e w e h a v e , f o r e v e r y i, Y i ti+1 ti

h ′ (t) ∂B ∂H (t, h(t)) dt = ti+1 ti h ′ (t) Y i ∂B ∂H (t, h(t)) dt = ti+1 ti h ′ (t) Y t ∂B ∂H (t, h(t)) dt and ti+1 ti 1 d (M1) B h t = B h ti+1 -B h ti , equality (3.16) reads, qm-1 i=0 Y i ti+1 ti 1 d (M1) B h t = lim n→∞ 1 0 Y t d (M1) B hn t + 1 0 h ′ (t) Y t ∂B ∂H (t, h(t)) dt, (3.17)
where the previous equality and limit hold in L 2 (Ω).R e m e m b e r i n gt h a t ,i nt h ec a s eo fi n t e g r a lw i t hr e s p e c t to fBm in the White Noise sense we get the equality

Y i ti+1 ti 1 d B H t = ti+1 ti Y i d B H t = ti+1 ti Y t d B H t ,
for every H in (0, 1),a n da s s u m i n gw eh a v ea na n a l o g o u se q u a l i t y ,i nt h ec a s eo ft h ei n t e g r a lw . r . t . m B m , we would then get

Y i ti+1 ti 1 d B h t = ti+1 ti Y i d B h t = ti+1 ti Y t d B h t ,
for the White noise integral we want to define. Thus we will hence have

qm-1 i=0 Y i ti+1 ti 1 d (M1) B h t = ti+1 ti Y i d (M1) B h t = 1 0 Y t d (M1) B h
t .E q u a l i t y (3.17) will hence read

1 0 Y t d (M1) B h t = lim n→∞ 1 0 Y t d (M1) B hn t + 1 0 h ′ (t) Y t ∂B ∂H (t, h(t)) dt,
where the limit and the equality hold in L 2 (Ω).T h ep r e v i o u se q u a l i t ya n dt h ef a c tt h a tn a t u r a ls p a c e so f white noise theory are the spaces (S -p ),f o re v e r yp in N (see section 5 below), suggest to define the integral of any (S -p )-valued process Y := (Y t ) t∈[0,1] with respect to mBm, in sense of (M 1 ),b ys e t t i n g ,

1 0 Y t d (M1) B h t := lim n→∞ 1 0 Y t d (M1) B hn t + 1 0 h ′ (t) Y t ∂B ∂H (t, h(t)) dt, (3.18)
where the limit, the last integral, assuming they both exist, and the equality hold in (S -q ) for some integer q in N.T h i se n t a i l si np a r t i c u l a rt h a tt h ef u n c t i o n a lΦ (M2) h is (S -q )-valued for some q in N and is defined, assuming it exists, by

Φ (M1) h (Y ) := 1 0 h ′ (t) Y t ∂B ∂H (t, h(t)
) dt,w h e r et h el a s ti n e t g r a li st a k e ni ns e n s eo f Bochner. In fact it will be shown in section 5 that this definition of integral w.r.t. mBm in sense of white Noise theory is relevant since it coincide with the definition of the multifractional Wick-Itô integral given in [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF] (see also definition 5.3 below).

Let (M 2 ) denote the Malliavin Calculus

Thanks to [64, proposition 8]and [9,corollary3.5]weknowthat Y i 1 (ti,ti+1] (t),w eg e t :

1 0 Y t d (M1) B H t = 1 0 Y t d (M2) B H t as
lim n→∞ T n = qm-1 i=0 Y i (B h ti+1 -B h ti ) - qm-1 i=0 Y i ti+1 ti h ′ (t) ∂B ∂H (t, h(t)) dt, (3.19)
where the previous limit holds in L 2 (Ω). Now, in the case of Skorohod integral with respect to any Gaussian process X := (X t ) t∈[0,1] ,a si th a sb e e n defined in [47,

Y i (B h ti+1 -B h ti )= qm-1 i=0 Y i ti+1 ti 1 d (M2) B h t = qm-1 i=0 ti+1 ti Y i d (M2) B h t = qm-1 i=0 ti+1 ti Y t d (M2) B h t = 1 0 Y t d (M2) B h t .
The previous equality and (3.19) then suggest to define the integral of Y on [0, 1] with respect to (M 2 ) by setting:

1 0 Y t d (M2) B h t := lim n→∞ 1 0 Y t d (M2) B hn t + 1 0 h ′ (t) Y t ∂B ∂H (t, h(t)) dt, (3.20)
where the previous equality and limit hold in L 2 (Ω).T h i se n t a i l si np a r t i c u l a rt h a tt h ef u n c t i o n a lΦ

(M2) h is L 2 (Ω)-valued and is defined, assuming it exists, by Φ (M2) h (Y ) := 1 0 h ′ (t) Y t ∂B ∂H (t, h(t)) dt,w h e ret h el ast integral is pathwise (i.e Φ (M2) h (Y )(ω)
is defined, for almost every ω in Ω,a saR i e m a n ni n t e g r a l ) .

Remark 3.4. A notable advantage of these two definitions is to provide, by construction, the equality

1 0 Y t d (M1) B h t = 1 0 Y t d (M2) B h t as soon as Y is integrable w.r.t. mBm in sense of M 2 .
T h i sr e s u l t generalizes the case of integral w.r.t. the fractional Brownian motion.

3.

We can also see that, when (M 3 ) denotes the Zähle integrale or the Rough path theory (see [START_REF] Coutin | An Introduction to (Stochastic) Calculus with Respect to Fractional Brownian Motion[END_REF][START_REF] Zähle | On the link between fractional and stochastic calculus[END_REF] for more details), we will want to set

1 0 Y t d (M2) B h t := lim n→∞ 1 0 Y t d (M2) B hn t + 1 0 h ′ (t) Y t ∂B ∂H (t, h(t)) dt. (3.21)

Integral with respect to mBm in sense of method (M)

In regard of the previous examples it is now possible to give a rigorous definition of integral with respect to mBm in sense of method (M).M o r ep r e c i s e l y ,l e t(E, || || E ) and (F, || || F ) are two normed linear spaces, endowed with their Borelian σ-field noted B(E) and B(F ),a n dY := (Y t ) t∈[0,1] be an E-valued process (i.e Y t belongs to E for every real t in [0, 1] and t → Y t is measurable from (0, 1) to (E, B(E)) ). The underlying method (M) and the space F being fixed once and for all, define for every α ∈ (0, 1), It turns out that there exists a simple suuficient condition that guarantees the existence of the limit lim n→∞ 1 0 Y t d (M) B hn t ,w h e nY belongs to Λ E . Define, for n ∈ N,t h em a p

H α E := Y ∈ E [0,1] : [0,1] Y t d (M) B α t exists
L n :Λ E → F Y → [0,1] Y t d (M) B hn t . (3.23)
The following theorem provides a necessary condition under which (L n (Y )) n∈N converges in F .

Theorem 3.5. Let (a n ) n∈N be an increasing sequence of positive integers such that 2 n

0 k n-1 a k 2 2 n
for every n in N and such that lim n→+∞ (n(a n -1)) (

0 k n-1 a k ) -1 =0
.C h o o s et h es e q u e n c e(q n ) n∈N used in (3.4) such that q 0 =1and q n+1 = a n q n for all n in N.A s s u m et h a tt h ef u n c t i o nI :Λ E × (0, 1) → F defined by

∀ Y ∈ Λ E , ∀α ∈ (0, 1), I(Y, α) := [0,1] Y t d (M) B α t ,
is θ-Hölder continuous with respect to α uniformly in Y for some positive real θ>χ ,i . e . t h e r ee x i s t sK>0 such that

∀Y ∈ Λ E , ∀(α, α ′ ) ∈ (0, 1) 2 , sup Y Λ E 1 I(Y, α) -I(Y, α ′ ) F K |α -α ′ | θ . (3.24)
Then the sequence of functions (L n ) n∈N defined in (3.23) converges pointwise to a function L :Λ E → F .

Remark 3.6. As the following proof will show, it is of crucial importance that the sequence (q n ) n∈N fulfills the assumption q n+1 = a n q n for every integer n.T h ec o n d i t i o n ,

0 k n-1
a k 2 2 n for every n in N,w e asked results from section 2 where we have had to assume that 2 n q n 2 2 n for every integer n.S i n c ew e want to use in the same time the result of section 2 and this theorem it is normal to have this supplementary condition.

Proof: The general case being similar we will establish, for sake of simplicity, the proof in the case of the constant sequence (a n ) n∈N equal to 2 which obviously fulfills the growing condition. For all n ∈ N and all Y ∈ Λ E we remark, thanks to (3.1), that L n (Y ) can be decomposed as

L n (Y )= 2 n -1 k=0 Ç [ 2k 2 n+1 , 2k+1 2 n+1 ) Y t d (M) B h( k 2 n ) t + [ 2k+1 2 n+1 , 2k+2 2 n+1 ) Y t d (M) B h( k 2 n ) t å . Now, L n+1 (Y )= 2 n -1 k=0 Ç [ 2k 2 n+1 , 2k+1 2 n+1 ) Y t d (M) B h( 2k 2 n+1 ) t + [ 2k+1 2 n+1 , 2k+2 2 n+1 ) Y t d (M) B h( 2k+1 2
L n (Y ) -L n+1 (Y ) F = 2 n -1 k=0 Ç I Ä Y. î 2k+1 2 n+1 , k+1 2 n ä ,h( k 2 n ) ä -I Ä Y. î 2k+1 2 n+1 , k+1 2 n ä ,h( 2k+1 2 n+1 ) ä å F 2 n -1 k=0 Ç I Ä Y. î 2k+1 2 n+1 , k+1 2 n ä ,h( k 2 n ) ä -I Ä Y. î 2k+1 2 n+1 , k+1 2 n ä ,h( 2k+1 2 n+1 ) ä å F K 2 n -1 k=0 Y. [(2k+1).2 -(n+1) ,(k+1).2 -n ) ΛE h((2k +1).2 -(n+1) ) -h(k.2 -n ) θ K 2 -θ(n+1) sup t∈[0,1] |h ′ (t)| θ 2 n -1 k=0 Y. [(2k+1).2 -(n+1) ,(k+1).2 -n ) ΛE KM2 -θ(n+1) sup t∈[0,1] |h ′ (t)| θ 2 nχ Y ΛE .
It follows that the series n∈N (L n+1 (Y ) -L n (Y )) converges absolutely for any fixed Y ∈ Λ E ,a n dc o n s equently (L n (Y )) n∈N converges to a limit L(Y ) as n goes to infinity.

Definition of integral with respect to mBm in sense of (M)

The previous criterion allow us to define now rigorously the integral with respect to mBm in sense of method (M).L e t u s fi r s t p r e c i s e a n o t a t i o n .W e w i l l s a y t h a t a n E-valued process (Z t ) t∈[0,1] is integrable on I ⊂ [0, 1],i ns e n s eo fF ,i f I Z t dt exists:

-for almost every ω in sense of Riemann if F ⊂ L 2 (Ω), -in sense of Bochner if L 2 (Ω) ⊂ F , and belongs to F (the reader who is no familiar wiht the integral in Bochner sense may refer to section 5 below and references therein). Now, let ϕ (M) : E×E → F be the map defined by ϕ (M) (U, V ) := U V if (M) denotes the White noise theory or the Mallivin calculus and by ϕ (M) (U, V ) := UV otherwise. For every process

Y := (Y t ) t∈[0,1] ∈ E [0,1] , such that the map t → h ′ (t) ϕ (M) (Y t , ∂B ∂H (t, h(t)) is integrable in sense of F ,d e fi n e Φ (M) h (Y ) := 1 0 h ′ (t) ϕ (M) (Y t , ∂B ∂H (t, h(t)) dt.
We get the following definition.

Y := (Y t ) t∈[0,1] be an element of Λ E such that the map t → h ′ (t) ϕ (M) (Y t , ∂B ∂H (t, h(t))
is integrable in sense of F .W ec a l li n t e g r a lo f Y with respect to mBm in sense of (M),a n dn o t e

1 0 Y t d (M) B h t ,t h e quantity 1 0 Y t d (M) B h t := lim n→∞ 1 0 Y t d (M) B hn t +Φ (M) h (Y ), (3.25)
where the limit and the equality hold in F .

Remark 3.7. (i)

Contrary to what we might expect in regard of proposition 3.2, the second term on the right-hand side of (3.25) does not only depend on the choice of the fractional field B but also on the method (M) of integration with respect to fBm that has been chosen. Note that the same is also true of the first term on the right-hand side of (3.25).

(ii) The main advantage of expression (3.25) is that any known construction of a stochastic integral with respect to fBm (e.g. pathwise, Malliavin calculus, white noise, rough path, ...) gives rise to a corresponding stochastic integral with respect to mBm.

(iii) Once again, note that E is not necessary a space of random variables (e.g E := (S -p ) for some positive integer p;s e es e c t i o n5b e l o w )a n dt h a tE could be different of F as it will be the case in section 5

4S k o h o r o d i n t e g r a l w i t h r e s p e c t t o m u l t i f r a c t i o n a l B r o w n i a n m otion

In this section, we apply Theorem 3.5 to define a Skohorod-type integral with respect to mBm. The reference method of integration with respect to fBm here is the one based on Malliavin calculus, as exposed in [3]. We assume throughout this section that H>1/2 and that h ranges in (1/2, 1).F o rd e fi n i t e n e s s ,w ea l s os e t B = B 4 in this section.

Our notations are as follows (for a presentation of Malliavin calculus, see e.g. [6,[START_REF] Nualart | The Malliavin Calculus and related Topics[END_REF]). Let:

S = ¶ R := f (W (h 1 ),W(h 2 ),...,W(h n )) ,f ∈ C ∞ b (R n ),h i ∈ L 2 ([0,T]),i =1,...,n © where W (h i ) := [0,T ] h i (s) dW s with W := (W s ) s∈[0,T ] aB r o w n i a nm o t i o n ,C ∞ b (R n
) is the set of functions which are bounded as well as all their derivatives. For an element of S,o n ed e fi n e st h ed e r i v a t i v eo p e r a t o r D as:

DR = n i=1 ∂ i f (W (h 1 ),W(h 2 ),...,W(h n )) h i .
D extends to the domain D which is the completion of S with respect to the norm:

R 1,2 = Ä E(R 2 )+E(||DR|| 2 L 2 ([0,T ] ) ä 1 2 .
We denote by δ the adjoint of D,a n db yD o m (δ) its domain. More precisely, Dom(δ) is the set of u ∈ L 2 (Ω, [0,T]) such that:

|E(<DR,u>)| c u E(R 2 )
for all R ∈S (we use < ., . > to denote the scalar product on L 2 ([0,T]), and δ is defined on Dom(δ) by the relation:

E(Rδ(u)) = E(<DR,u>).
The operator δ is a closed linear operator on Dom(δ).I tc o i n c i d e sw i t ht h eS k o h o r o di n t e g r a l .

Let us now recall briefly the approach of [3] for the construction of a stochastic integral w.r.t. a class of Gaussian processes. Assume thec o n t i n u o u sG a u s s i a np r o c e s sX may be written:

X t = t 0 K(t, s) dW s , (4.1)
where the kernel K(t, s) is defined for 0 <s<t<T and verifies

sup t∈[0,T ] t 0 K(t, s) 2 ds < ∞. (4.2)
Define the operator K * on the set of step functions on [0,T]:

(K * ϕ)(s) := ϕ(s)K(s + ,s)+ T s ϕ(t)K(dt, s) (4.3)
where K(s + ,s)=K(T,s) -K ((s, T ],s).T h e nt h es t o c h a s t i ci n t e g r a lw . r . t .X is defined for processes in Dom(δ X )( [ 3 ] ,f o r m u l a( 1 2 ) ) : Dom(δ X ) := (K * ) -1 (Dom(δ)).

For a pro cess v in Dom(δ X ), one sets:

δ X (v) := T 0 v(s)δX(s) := T 0 (K * v)(s)δW (s).
In the case of fBm, one has:

B H t = t 0 K H (t, s) dW s ,
where

K H (t, s)=d H (t -s) H-1 2 + c H Å 1 2 -H ã t s (u -s) H-3 2 1 - s u 1 2 -H du (4.4)
and

d H = 2HΓ( 3 2 -H) Γ( 1 2 + H)Γ(2 -2H) 1 2
.

We will index our op erators and sets with H, i.e. we will write:

Dom(δ H )=(K * H ) -1 (Dom(δ))
for the domain of the Skohorodi n t egralwithrespecttoB H and

δ H (v) := T 0 v(s)δB H (s) := T 0 (K * H v)(s)δW (s)
for the integral. In other words, δ H (v)=δ(K * H v). As the methodology of [3] works in a general framework, one may wonder whether it is possible to apply it directly to mBm. The prerequisite is to exhibit a kernel K such that (4.1) holds when X is an mBm. This is indeed the case for B 4 ,a n dt h ew o r k[ 1 8 ]d e v e l o p st h i sa p p r o a c h . I nc o n t r a s t ,i td o e sn o ts e e mt ob ea n easy task to find such a kernel for the moving average and harmonizable mBms, as they have very different correlation structures (see [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF]). As a matter of fact, we conjecture that such a kernel does not exist for the harmonizable mBm, based on the following fact: for all t 1 ,t 2 in R and H 1 ,H 2 in (0, 1),

E [B 1 (t 1 ,H 1 )B 1 (t 2 ,H 2 )] = E [B 1 (t 1 ,H 2 )B 1 (t 2 ,H 1 )] .
This will investigated in a forthcoming work. We now seek to apply Theorem 3.5 in order to define a Skohoro d integral w.r.t. mBm through approximating Skohorod integrals w.r.t. fBms. In that view, we set

F = L 2 (Ω) and I(Y, α)=δ α (Y ). It is straightforward to check that, for every (t, s) in [0,T] 2 ,t h ef u n c t i o nH → K H (t, s) is C 1 .W ed e n o t e its derivative by G H , i.e. G H1 (t, s) is the derivative of the function H → K H (t, s) evaluated at H 1 .W e associate to G H an operator G *
H in a way similar to (4.3). Note that G * H is the derivative of the function H → K * H .O n ee a s i l yv e r i fi e st h a tG H fulfils (4.2), so that one may define as above δ G (.) := δ(G * H .) for a suitable class of processes. Let:

D := H∈h([0,1]) Dom(δ H ), and
F := H∈h([0,1]) F H ,
where

F H := (G * H ) -1 (Dom(δ)). Set Λ := D∩F, equipped with the norm v Λ =sup H∈h([0,1]) E Ä T 0 (K * H v)(s) 2 ds ä +sup H∈h([0,1]) E Ä T 0 (G * H v)(s) 2 ds ä ,w h i c h satisfies condition (3.22) with χ =0 .B yd e fi n i t i o n ,δ H (v) and T 0 (G * H v)(s)δW (s) both exist for all H in h([0, 1]) and all v in Λ.F i x (v, s, H, H ′ ) in Λ × [0,T] × h([0, 1]) 2 with H<H ′ .C o n s i d e r t h e f u n c t i o n ϕ :Ω× [H, H ′ ] → R defined by: ϕ(ω, H 1 ) := (K * H1 v(ω))(s) -(K * H v(ω))(s) -(H 1 -H) (K * H ′ v(ω))(s) -(K * H v(ω))(s) H ′ -H .
For every ω in Ω, ϕ(ω, .) is C 1 and ϕ(ω, H)=ϕ(ω, H ′ )=0 .A sac o n s e q u e n c e ,t h e r ee x i s t sH

′′ ω in [H, H ′ ] such that ∂ϕ ∂H (ω, H ′′ ω )=0.T h u s ,t h es e tA ω := {H 1 ∈ [H, H ′ ]: ∂ϕ ∂H (ω, H 1 )=0} is a non-empty closed subset of [H, H ′ ].
I th a sam i n i m u m ,t h a tw ed e n o t eH 0 (ω).T h em a pω → H 0 (ω) is measurable (i.e. H 0 is a random variable), and so is the map (v, s, H, H ′ ,ω) → H 0 (v, s, H, H ′ ,ω). We wish to estimate I(v, H) -I(v, H ′ ) F for v in Λ.A s w e h a v e j u s t s e e n , t h e r e e x i s t s a m e a s u r a b l e function H 0 = H 0 (H, H ′ ,v,s,ω) such that:

u(s) := (K * H ′ v)(s) -(K * H v)(s) =( H ′ -H)(G * H0 v)(s).
Thus:

I(v, H ′ ) -I(v, H)=δ(u) =( H ′ -H) T 0 (G * H0 v)(s)δW (s), and
I(v, H ′ ) -I(v, H) L 2 (Ω) |H -H ′ | v Λ
i.e. (3.24) holds with θ =1, E = F := L2 (Ω) and Λ E := Λ.

In order to define our integral with (3.25), we need to check that h ′ (t)Y (t) ∂B ∂H (t, h(t)) is integrable. Define

Z t := t 0 ∂B ∂H (s, h(s)) ds = t 0 L(t, u) dB u with L(t, u) := t u G h(s) (s, u) ds.
T h u sZ is a Volterra process. It follows from Proposition 7 in [START_REF] Nualart | A white noise approach to fractional Brownian motion[END_REF] 2 that any process in L 2 (Ω × [0, 1]) is Wick integrable w.r.t. Z.T h i si m p l i e si np a r t i c u l a rt h a t

1 0 h ′ (t)Y (t) ∂B ∂H (t, h(t)) dt exists for Y in Λ.I na d d i t i o n ,

a d a p t i n gt h ea r g u m e n t si nP r o p o s i t i o n8o f[ 6 4 ] ,o n em a ys h o w that

1 0 h ′ (t)Y (t) ∂B ∂H (t, h(t)) dt = 1 0 h ′ (t)Y (t)
δZ t is a Skohorod integral and thus belongs to L 2 (Ω).W e are then able to set the following definition and theorem:

Theorem-Definition 4.1. Let Y ∈ Λ.T h e nt h eS k o h o r o di n t e g r a lo fY with respect to mBm is well-defined and given by:

1 0 Y t d (M2) B h t := 1 0 Y t δB h t := lim n→∞ 1 0 Y t δB hn t + 1 0 h ′ (t) Y (t) ∂B ∂H (t, h(t)) dt. (4.5)
where the equality holds in L 2 (Ω).

Remark 4.1. One may verify that the integral defined above coincides with the one studied in [START_REF] Boufoussi | Local time and Tanaka formula for a Volterra-type multifractional Gaussian process[END_REF] when they are both defined. Comparing their domains would be an interesting task.

5 White noise approach to stochastic integration w.r.

t. mBm

One can easily see that the function R h , defined on R 2 ,b y

R h (t, s)= c 2 ht,s c h(t) c h(s) ï 1 2 Ä |t| 2ht,s + |s| 2ht,s -|t -s| 2ht,s ä ò , (5.1)
is the covariance function of mBm build from the fractional field B 1 ,w h i c hh a sbe e ng i v e na tt h ebe g i n n i n g of section 1.

In [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF] an integral with respect to mBm (built from fractional field B 1 )hasbeendev elopedandstudied. Our aim in this section is firstly to define an integral with respect to mBm using definition 3.2 (for M = M 1 , where (M 1 ) still denote the white noise theory ) and, secondly, to compare the integral w.r.t. mBm hence defined with the one defined in [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF]. For this purpose, we have to choose and fix B = B 1 what we do from now and untill the end of this section 3 .B e f o r ec o m p a r i n gt h e s et w oi n t e g r a l sw . r . t .m B m ,w efi r s tn e e dt o particularize the construction of the stochastic integral given in definition 3.2 to B 1 . Moreover, since this section is devoted to integral w.r.t mBm in sense of White noise theory, we recall, in the next subsection, the basic ideas and tools of white noise theory.

Reminders about White Noise theory and Bochner integral

White Noise theory

Define the measurable space (Ω, F ) by setting Ω := S ′ (R) and F := B(S ′ (R)),w h e r eb eB denotes the σ-algebra of Borel sets. There exists an unique probability measure µ on (Ω, F ) such that, for every f in L 2 (R),t h em a p< ., f >:Ω→ R defined by < ., f > (ω)= <ω,f >(where <ω,f>is by definition ω(f ), i.e the action of the distribution ω on the function f )i sac e n t e r e dG a u s s i a nr a n d o mv a r i a b l ew i t hv a r i a n c e equal to ||f ||

2 L 2 (R) under µ.F o re v e r yn in N,d e fi n ee n (x) := (-1) n π -1/4 (2 n n!) -1/2 e x 2 /2 d n dx n (e -x 2 ) the nth Hermite function. Let (|| p ) p∈Z be a family norms defined by |f | 2 p := +∞ k=0 (2k +2) 2p <f,e k > 2 L 2 (R) ,f o r a l l (p, f ) in Z × L 2 (

R).T h eo p e r a t o rA defined on S (R) by

A := -d 2 dx 2 +
x 2 +1admits the sequence (e n ) n∈N as eigenfunctions and the sequence (2n +2) n∈N as eigenvalues. From now on we will denote as is customary (L 2 ) the space L 2 (Ω, G,µ) where G is the σ-field generated by (< ., f >) f ∈L 2 (R) .F o re v e r yr a n d o mv a r i a b l eΦ of (L 2 ) there exists, according tot h eW i e n e r -I t ôt h e o r e m , au n i q u es e q u e n c e(f n ) n∈N of functions f n in L 2 (R n ) such that Φ can be decomposed as Φ= I n (A ⊗n f n ),w h ereA ⊗n denotes the nth tensor power of the operator A (see [START_REF] Janson | Gaussian Hilbert spaces[END_REF] appendix Ef o rm o r ed e t a i l sa b o u tt e n s o rp r o d u c t so fo p e r a t o r s ) . T h eo p e r a t o rΓ(A) is densely defined on (L 2 ) is called the second quantization operator of A.I ti si n v e r t i b l ea n d i t si n v e r s e Γ(A)

-1 is bounded. Let us denote ||ϕ|| 2 0 := ||ϕ|| 2 (L 2 ) for any random variable ϕ in (L 2 ) and, for n in N,l e tDom(Γ(A) n ) be the domain of the nth iteration of Γ(A).D e fi n et h ef a m i l yo fn o r m s(|| || p ) p∈Z by:

||Φ|| p := ||Γ(A) p Φ|| 0 = ||Γ(A) p Φ|| (L 2 ) , ∀p ∈ Z, ∀Φ ∈ (L 2 ) ∩ Dom(Γ(A) p ).
For any p in N,define(S p ) := {Φ ∈ (L 2 ): Γ(A) p Φ exists and belongs to (L 2 )} and define (S -p ) as being the completion of the space (L 2 ) with respect to the norm || || -p .A si n[ 5 2 ] ,w el e t(S) denote the projective limit of the sequence ((S p )) p∈N and (S) * the inductive limit of the sequence ((S -p )) p∈N .T h es p a c e(S) is called the space of stochastic test functions and (S) * the space of Hida distributions. One can show that, for any p in N,t h ed u a ls p a c e(S p ) * of S p is (S -p ).T h u sw ew i l lw r i t e(S -p ),i nt h es e q u e l ,t od e n o t et h e space (S p ) * .N o t ea l s ot h a t(S) * is the dual space of (S).W ew i l ln o t e< <, > > the duality bracket between (S) * and (S).I fΦ belongs to (L 2 ) then we have the equality < <Φ,ϕ> > = < Φ,ϕ > (L 2 ) = E[Φ ϕ].S i n c ew e have defined a topology given by a family of norms on the space (S) * it is possible to define a derivative and an integral in (S) * .Af u n c t i o nΦ:R → (S) * is called a stochastic distribution process, or an (S) * -process, or a Hida process for every real t 0 .F o re v e r yr e a lt 0 ,t h eH i d ap r o c e s sΦ is said to be differentiable at t 0 if lim r→0 r -1 (Φ(t 0 + r) -Φ(t 0 )) exists in (S) * .

For every f in L 2 (R) we define the Wick exponential of < ., f >,n o t e d: e <.,f > :,a sb e i n gt h e(L 2 ) random variable equal to e <.,f >-1 2 |f | 2 0 .T h eS-transform of an element Φ of (S * ),n o t e dS(Φ),i sd e fi n e da sb e i n g the function from S (R) to R given by S(Φ)(η) := < <Φ,: e <.,η> :> > for every η in S (R).F i n a l l yf o re v e r y (Φ, Ψ) ∈ (S)

* × (S) * ,t h e r ee x i s t sau n i q u ee l e m e n to f(S) * ,c a l l e dt h eW i c kp r o d u c to fΦ and Ψ and noted Φ Ψ,s u c ht h a tS(Φ Ψ)(η)=S(Φ)(η) S(Ψ)(η);f o re v e r yη in S (R).

Fracti o nal and mul ti fracti o nal W hi te no i s e

As we will see now, the operators M H and ∂MH ∂H are crucial in the study of an integral with respect to mBm in the framework of the white noise theory.

Operators M H and ∂MH ∂H Let H be a fixed real in (0, 1).F o l l o w i n g [ 3 4 ]a n dr e f e r e n c e stherein, define the operator M H ,s p e c i fi e di n the Fourier domain, by ◊ M H (u)(y) :=

√ 2π
cH |y| 1/2-H u(y) for every y in R * .T h i so p e r a t o ri sw e l ld e fi n e do n the homogeneous Sobolev space of order 1/2 -H,d e n o t e dL 2 H (R) and defined by

L 2 H (R) := {u ∈ S ′ (R): u = T f ; f ∈ L 1 loc (R) and ||u|| H < +∞},w h eret h en orm|| || H derives from the inner product <, > H defined on L 2 H (R) by <u,v> H := 1 c 2 H R |ξ| 1-2H " u (ξ)" v (ξ)
dξ and where c H has been given in the definition of the fractional field B 1 . The definition of the operator ∂MH ∂H is quite similar. More precisely, define for every H in (0, 1),t h es p a c e

Γ H (R) := {u ∈ S ′ (R): u = T f ; f ∈ L 1 loc (R) and ||u|| δH (R) < +∞},wherethenorm|| || δH (R) derives from the inner product on Γ H (R) defined by <u,v> δH := 1 c 2 H R (β H + ln |ξ|) 2 |ξ| 1-2H " u (ξ) " v (ξ) dξ.F o l l o w i n g [53], define the operator ∂MH ∂H from (Γ H (R),<,> δH (R) ) to (L 2 (R),<,> L 2 (R) ),i

Fracti o nal and mul ti fracti o nal W hi te no i s e

Since we only consider mBm B h with covariance function R h ,w ec a nw r i t e ,t h a n k st o[ 5 3 ,( 5 . 1 0 ) ]a n du s i n g the two last subsections, Almost surely, for every real t,

B h t = B 1 (t, h(t)) =< ., M h(t) (1 [0,t] ) >. (5.2)
However and for sake of simplicity we will only restrict ourselves to the case where t belongs to [0, 1].W e can also write (5.2) under the form Almost surely, for every real t,

B h t = +∞ k=0 t 0 M h(t) (e k )(s)ds < ., e k >. (5.3)
We are now able to define the derivative in sense of (S) * of the mBm. Define the (S * )-valued process

W h := (W h t ) t∈[0,1] by W h t := +∞ k=0 ï d dt Å t 0 M h(t) (e k )(s) ds ãò < ., e k >. (5.4)
The following theorem states that, for all real t,t h er i g h th a n ds i d eo f( 5 . Theorem-Definition 5.1. [53, Theorem-definition 5.1] The process W h defined by (5.4) is an (S) * -process which verifies,in (S) * ,t h ef o l l o w i n ge q u a l i t y :

W h t = +∞ k=0 M h(t) (e k )(t) < ., e k > + h ′ (t) +∞ k=0 Å t 0 ∂MH ∂H (e k )(s) H=h(t) ds ã < ., e k >. (5.5)
Moreover the process B h is (S) * -differentiable on [0, 1] and verifies dB h dt (t)=W h t in (S) * .

Once again if the function h 0 is constant, identically equal to H,w ew i l ln o t eW H := (W H t ) t∈[0,1] and call fractional white noise the (S)

* -process W h0 .O n ec a nr e w r i t e( 5 . 5 ) ,f o re v e r yt in [0, 1],u n d e rt h ef o r m

W h t = W h(t) t + h ′ (t) ∂B 1 ∂H (t, h(t)), (5.6)
where

W h(t) t is nothing but W H t | H=h(t)
and where the equality holds in (S) * .

Bochner integral

Since the objects we are now handling with are no longer, in general, random variables, the Riemann or Lebesgue integrals are not relevant here. However, taking advantage on the fact that we are working with vector linear spaces, we can use Pettis or Bochner integrals. We know that the space E,d e fi n e da tt h e beginning of section 3.2, will be a space (S -p ) for some integer p.T h ef a c tt h a tw en e e dt ofi n dan o r mo n Λ E suggests the use of Bochner integral. A very good survey of this topic can be found in [52, p.247]. We here just recall the definition.

Definition 5.1 (Bochner integral [START_REF] Kuo | White Noise Distribution Theory[END_REF],p.247). Let I be a subset of [0, 1] endowed with the Lebesgue measure. One says that Φ:I → (S) * is Bochner integrable on I if it satisfies the two following conditions:

1. Φ is weakly measurable on I i.e t →< < Φ t ,ϕ > > is measurable on I,f o re v e r yϕ in (S).

2. There exists p in N such that Φ t ∈ (S -p ) for almost every t in I and t → ||Φ t || -p belongs to L 1 (I,dt).

The Bochner-integral of Φ on I is denoted I Φ t dt.M o r e o v e rw ew i l ls a yt h a tΦ is Bochner-integrable of index p.

Propertie 1. If Φ:I → (S) * is Bochner-integrable on I with index p then we have the following inequality

I Φ t dt -p I ||Φ t || -p dt.
Theorem 5.1 ([52], theorem 13.5). Let Φ := (Φ t ) t∈[0,1] be a (S) * -valued process such that:

(i) t → S(Φ t )(η) is measurable for every η in S (R). (ii) There exist p in N, b in R + and a function L in L 1 ([0, 1],dt) such that, for a.e. t in [0, 1],s u c ht h a t |S(Φ t )(η)| L(t) e b|η| 2 p ,f o re v e r yη in S (R).
Then Φ is Bochner integrable on [0, 1] and 1 0 Φ(s) ds ∈ (S -q ) for any q>psuch that 2be 2 D(q -p) < 1 where e denotes the base of the natural logarithm and where D(r) is defined by setting D(r

) := 1 2 2r +∞ n=1 1 n 2r
for every r of (1/2, +∞).

Wick-Itô integral with respect to fBm

The definition of the fractional Wick-Itô integral withrespecttofBmorin tegralw.r.tfBminthewhitenoise sense has been firstly given in [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF] and then extend in [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF] using the Pettis integral. However, in order to use Theorem 3.5, we need that Y s belongs to (S -p ) for almost every real s.I tt h e ns e e m sr e a s o n a b l et oa s s u m e that (Y s ) s∈[0,1] is Bochner integrable on [0, 1].F o rt h i sr e a s o n ,w en o wp a r t i c u l a r i z et h ef r a c t i o n a lW i c k -I t ô integral with respect to fBm of [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF] and [START_REF] Bender | An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter[END_REF] into the framework of the Bochner integral. Definition 5.2 (Wick-Itô integral w.r.t fBm in Bochner sense). Let H be a fixed real in (0, 1), I be a Borel subset of [0, 1], B H := (B H t ) t∈I be a fractional Brownian motion of hurst index H,a n dY := (Y t ) t∈I be a (S)

* -valued process such that (i) There exists p ∈ N such that Y t ∈ (S -p ) for almost every t ∈ I, (ii) the process t → Y t W H t is Bochner integrable on I We then say that the process Y is Bochner-integrable with respect to fBm, or integrable on I with respect to fBm, in the Bochner sense and define this integral by setting:

I Y s d B H s := I Y s W H s ds. (5.7)
The following lemma, states that every Bochner integrable process is integrable with respect to fBm of any Hurst index H,i nt h eB o c h n e rs e n s e .

Lemma 5.2. Let Y := (Y t ) t∈[0,1] be a (S) * -valued process, Bochner integrable of index p 0 ∈ N.T h e nY is integrable on [0, 1],w i t hr e s p e c tt of B mo fa n yH u r s ti n d e xH, in the Bochner sense. Moreover, for any H in (0, 1), [0,1] Y s d B H s belongs to (S -r0 ) for every r 0 p 0 +1 if p 0 2 and for every r 0 p 0 +2 if p 0 ∈{0; 1}.

Proof: Let H ∈ (0, 1), p 0 2 and r 0 p 0 +1 are fixed. For every t in [0, 1],i ti sc l e a rt h a tt

→ Y t W H t is weakly measurable since t → S(Y t W H t )(η) is measurable for every η ∈ S (R).U s i n g[ 5 2
,r e m a r k2 p.92], we get, for almost every

t in [0, 1], ||Y t W H t || -r0 ||Y t || -p0 ||W H t || -p0 < +∞ and hence Y t W H t belongs to (S -r0 ).S i n c et h em a pt → ||W H
t || -r is continuous for every integer r 2 (see [53, proposition 5.9]), we also get:

1 0 ||Y t W H t || -r0 dt Ç sup t∈[0,1] ||W H t || -p0 å 1 0 ||Y t || -p0 dt < +∞.
This shows that t → Y t W H t is Bochner-integrable of index r 0 . Let us now assume that p 0 ∈{ 0; 1}.W ej u s th a v et ov e r i fi yt h a tt h e o r e m5 . 1a p p l i e s .T h ec o n d i t i o n(i) is obviously fulffiled. Moreover, using, [52, p.79]

w eh a v ef o re v e r y(t, η) in [0, 1] × S (R), |S(Y t W H t )(η)| Y t -p0 e 1 2 |η| 2 2 sup t∈[0,1] ||W H t || -2 =: L(t) e 1 2 |η| 2 2 .
Since Y is Bochner integrable of index p 0 ,itisclearthatL belongs to L The following lemma, the proof of which is obvious in regard of hypothesis (H 2 ),w i l lb eu s e f u li nt h ep r o o f of the proposition 5.5 below.

Lemma 5.4. For every integer p in N and every real t in [0, 1],t h em a p(t, H) → ∂B1 ∂H (t, H) is continuous from [0,t] into ((S -p ), || || p ). In particular, for every subset [a, b] of (0, 1) there exists a positive real κ such that:

∀p ∈ N, sup (s,H)∈[0,t]×[a,b] ∂B 1 ∂H (s, H) -p κ.
(5.8)

Stochastic integral with respect to mBm

The purpose of the next proposition is to show that we can apply Theorem 3.5 to the Wick-Itô integral w.r.t fBm and hence define an integral w.r.t to mBm. Let p 0 be a fixed integer in N and s 0 max{p 0 +1;3} be fixed in N.D e fi n eE := (S -p0 ) and F := (S -s0 ).B yd e fi n i t i o nw eh a v et h ee q u a l i t y

H E = (Y t ) t∈[0,1] ∈ (S -p0 ) R : [0,1] Y t d B α t ∈ (S -s0 ), ∀α ∈ h([0, 1]) .
Define also the set

Λ E := ¶ (Y t ) t∈[0,1] ∈ (S -p0 ) R : Y is Bochner integrable of index p 0 on [0, 1] © and the norm ΛE on Λ E by Φ ΛE := 1 0
Φ t -p0 dt.T h ei n c l u s i o nΛ E ⊂H E results from lemma 5.2. Moreover the fact that (Λ E , ΛE ) is complete is a straightforward consequence of [START_REF] Hille | Functional Analysis and Semi-Groups,v o l u m e3 1 . A m e r i c a nM a t h e m a t i c a l Society[END_REF]Theorem 3.7.7 p.82]. The notations being fixed above we get the following Proposition 5.5. The white noise theory method (i.e (M 1 ))f ul fill sc o nd i ti o ns (3.22) and (3.24). Moreover, for every process

Y := (Y t ) t∈[0,1] Bochner-integrable on [0, 1],t h em a pt → h ′ (t) ϕ (M1) (Y t , ∂B1 ∂H (t, h(t)) is integrable in sense of F .
The proof of this proposition stands on the two following lemmas.

:= (Y t ) t∈[0,1] Bochner-integrable on [0, 1],themapt → h ′ (t) Y t ∂B1 ∂H (t, h(t)) is integrable in sense of F .
Proof of lemma 5.6: Since (S -p0 ) ⊂ (S -2 ) if p 0 belongs to {0; 1},w eassummefromno wthatp 0 2 and that

s 0 p 0 +1.L e tA 1 ,A 2 , ••• ,A n are some fixed disjoint borelians of [0, 1].W eg e t n k=1 ||Y 1 A k || ΛE = n k=1 1 0 ||Y 1 A k (s)|| -p0 ds n k=1 1 0 1 A k (s)||Y s || -p0 ds 1 0 n k=1 1 A k (s) ||Y s || -p0 ds ||Y || ΛE . (5.9)
Besides we know, thanks to lemma 5.2, that Y is integrable with respect to fBm, in the Bochner sense, for any Hurst index α in (0, 1) and that I(Y, α)= [0,1] Y t dB α t belongs to (S -s0 ).N o wf o ra n y(α, α ′ ) in (0, 1) 2 , we get, using the same arguments we used in the proof of lemma 5.2,

||I(Y, α) -I(Y, α ′ )|| -s0 = [0,1] Y t (W α t -W α ′ t ) dt -s0 1 0 ||Y t || -p0 ||W α t -W α ′ t || -p0 dt Ç sup t∈[0,1] ||W α t -W α ′ t || -p0 å ||Y || ΛE , and then sup ||Y || Λ E 1 ||I(Y, α) -I(Y, α ′ )|| -s0 sup t∈[0,1] ||W α t -W α ′ t || -p0 . (5.10)
Furthermore, we have

||W α t -W α ′ t || 2 -p0 = +∞ k=0 (Mα(e k )(t)-M α ′ (e k )(t)) 2 (2k+2) 2p 0 .F o re v e r y(t, k) in [0, 1] × N,d e fi n e f t,k :( 0, 1) → R by f t,k (α) := M α (e k )(t).T h a n k st o[ 5 3 ,l e m m a5.5]w ek n o wt h a tf t,k is differeientable on (0, 1),f o re v e r y(t, k) in [0, 1] × N,a n dt h a ti t sd e r i v a t i v e ,n o t e df ′ t,k ,s a t i s fi e sf ′ t,k (α)= ∂ ∂α [M α (e k )(t)] = ∂Mα ∂α (e k )(t)
where the operator ∂Mα ∂α has been defined in [53, section 3.2]. Using point 1 of [53, lemma 5.6] and then the meanvalue theorem we get for every [a, b] ⊂ (0, 1), the existence of a positive real ρ such that for all (t, α, α

′ ,k) ∈ [0, 1] × [a, b] 2 × N, |M α (e k )(t) -M α ′ (e k )(t)| ρ (k +1) 2/3 ln(k +1)|α -α ′ |.
We hence have

||W α t -W α ′ t || 2 -p0 ρ 2 |α -α ′ | 2 +∞ k=0 (k+1) 4/3 ln 2 (k+1) 2 2p 0 (k+1) 2p 0
and finally, using (5.10), we get

sup ||Y || Λ E 1 ||I(Y, α) -I(Y, α ′ )|| -s0 |α -α ′ | γ p0 (5.11)
where

γ p0 := ρ Ä +∞ k=1 ln 2 k k 2(p 0 -2/3)
ä 1/2 which is finite since p 0 2.S i n c eγ p0 is independent of α and α ′ , Theorem 3.5 applies and gives us the existence of lim n→+∞ [0,1] Y t d B hn t in (S -s0 ).

Proof of lemma 5.7: Let p 0 2 and s 0 p 0 +1 are fixed. Using the same arguments as in the proof of lemma 5.2 we easily get the weak measurability of t → h ′ (t) Y t ∂B1 ∂H (t, h(t)) on [0, 1] and, using lemma 5.4, sup

(s,H)∈[0,1]×h([0,1]) ∂B1 ∂H (s, H) -p0 κ,f o re v e r yp 0 .W eh e n c eg e t h ′ (t) Y t ∂B 1 ∂H (t, h(t)) -s0 ||Y s || -p0 Ç sup s∈[0,1] |h ′ (s)| å sup s∈[0,1] ∂B 1 ∂H (s, h(s)) -p0 < +∞.
Thus there exists δ ∈ R * + ,s u c ht h a t

1 0 h ′ (s) Y s ∂B1 ∂H (s, h(s)) -s0 ds δ 1 0 ||Y s || -p0 ds < +∞ which shows that 1 0 h ′ (t) Y t ∂B1 ∂H (t, h(t)
)dt is well defined in sense of Bochner. Corollary 5.8. Let (a n ) n∈N and (q n ) n∈N are two sequences of integers that fulfill the growth condition of Theorem 3.5, then the element

1 0 Y t d (M1) B h t defined by 1 0 Y t d (M1) B h t := lim n→∞ 1 0 Y t d (M1) B hn t + 1 0 h ′ (t) Y t ∂B 1 ∂H (t, h(t)) dt, (5.12)
where the limit and the equality hold in (S -s0 ),i sw e l l -d e fi n e da n db e l o n g st o(S -s0 ),w h e r es 0 has been defined at the beginning of this subsection.

Proof: Obvious in regard of Proposition 5.5, Theorem 3.5 and Definition 3.2.

5.4

Comparison between integrals in sense of (M 1 ) and Wick-Itô integral w.r.

t. mBm

The definition of the multifractional Wick-Itô integral with respect to mBm or integral w.r.t mBm in the white noise sense has been given in [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF]. In order to compare it to the integral w.r.t mBm defined by (5.5), we now particularize the multifractional Wick-Itô integral w.r.t mBm, defined in [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF] in the Pettis integral framework, into the Bochner integral framework.

Definition 5.3 (Wick-Itô integral w.r.t mBm in Bochner sense). Let I be a Borelian connected subset of [0, 1], B h := (B h t ) t∈I be a multifractional Brownian motion and Y := (Y t ) t∈I be a (S) * -valued process such that: (i) There exists p ∈ N such that Y t ∈ (S -p ) for almost every t ∈ I, (ii) the process t → Y (t) W h t is Bochner integrable on I. We then say that the process Y is integrable on I,w i t hr e s p e c tt om B m ,i nt h eB o c h n e rs e n s ea n dd e fi n et h i s integral by:

I Y s d B h s := I Y s W h s ds. (5.13)
Remark 5.9. (i) Of course this definition generalizes the defintion 5.2 since we recover definition 5.2 when the function h is identically constant on I,e q u a lt oH.

(ii) Using the definition of

Ä W h t ä t∈[0,1]
,[ 5 3 ,p r o p o s i t i o n5.9], and the proof of lemma 5.2, it is clear that every (S) * -valued process Y := (Y t ) t∈I ,B o c h n e ri n t e g r a b l eo nI of index p 0 ,i si n t e g r a b l eo nI with respect to mBm, in the Bochner sense. Moreover [0,1] Y t dB h t belongs to (S -r0 ),w h e r er 0 has been defined in lemma 5.2.

We now have two integrals with resp ect to mBm using white noise theory. The first one, which is noted

1 0 Y t d (M1) B h
t has been defined in (5.12) and the second one, which is noted I Y s d B h s ,h a sb e e nd e fi n e d in (5.13).

In order to compare these two integrals with respect to mBm when they both exist, it seems natural, in view of lemma 5.2, proposition 5.5 and of the previous remark, to assume that Y =( Y t ) t∈[0,1] is a Bochner integrable process of index p 0 ∈ N.T h e s p a c e E and the norm || || ΛE will then be taken as stated in Proposition 5.5. The following theorem shows that the integrals Proof: Since Y is a Bochner integrable process of index p 0 ∈ N,w ek n o w ,t h a n k st op r o p o s i t i o n5 . 5a n dt o (ii) of the remark 5.9, that [0,1] Y s d (M1) B h s exists in (S -s0 ) and [0,1] Y s d B h s exist in (S -r0 ),w h e r es 0 has been defined just above proposition 5.5 and r 0 has been defined in lemma 5.2. Moreover, thanks to (5.12) and using (3.4) and (5.7), we can write, in (S -s0 ),

1 0 Y t d (M1) B h t = lim n→∞ 1 0 Y t d (M1) B hn t + 1 0 h ′ (t) Y t ∂B 1 ∂H (t, h(t)) dt = lim n→∞ qn-1 k=0 1 0 [x (n) k ,x (n) k+1 ) (t) Y t d (M1) B h(x (n) k ) t + 1 0 h ′ (t) Y t ∂B 1 ∂H (t, h(t)) dt = lim n→∞ qn-1 k=0 1 0 [x (n) k ,x (n) k+1 ) (t) Y t W h(x (n) k ) t dt + 1 0 h ′ (t) Y t ∂B 1 ∂H (t, h(t)) dt. (5.14)
Besides, thanks to equality (5.13) and using (5.6), we have, in (S -r0 ),

[0,1] Y s d B h s = 1 0 Y t W h(t) t dt + 1 0 h ′ (t) Y t ∂B 1 ∂H (t, h(t)) dt.
(5.15)

Since s 0 r 0 we have (S -r0 ) ⊂ (S -s0 ).T h u sw eh e n c ej u s th a v et os h o wt h a t ,i n(S -s0 ),

L(Y ) := lim n→∞ qn-1 k=0 1 0 [x (n) k ,x (n) k+1 ) (t) Y t W h(x (n) k ) t dt is equal to M (Y ) := 1 0 Y t W h(t) t
(n) k ,x (n) k+1 ) (t) Y t W h(x (n) k ) t dt (η).
Using now (ii) of [53, theorem 5.12] we get,

S(L(Y ))(η) = lim n→+∞ qn-1 k=0 x (n) k+1 x (n) k S(Y t )(η) S(W h(x (n) k ) t )(η) dt = lim n→+∞ [0,1] S(Y t )(η) qn-1 k=0 1 [x (n) k ,x (n) k+1 ) (t) M h x (n) k (η)(t) dt. (5.16)
Furthermore, thanks to [53, lemma 5.5], we know that the map (t, H) → M H (η)(t) is continuous on every compact set of R × (0, 1).D e fi n eK η := sup

(t,H)∈[0,1]×h([0,1])
M H (η)(t).F o re v e r yn in N and t in [0, 1] we have,

S(Y t )(η) qn-1 k=0 1 [x (n) k ,x (n) k+1 ) (t) M h x (n) k (η)(t) sup (t,H)∈[0,1]×h([0,1]) M H (η)(t) e 1 2 |η| 2 p 0 ||Y t || -p0 K η e 1 2 |η| 2 p 0 ||Y t || -p0 .
(5.17)

Since the map t → K η e |η| 2 p 0 ||Y t || -p0 belongs to L 1 (R,dt) and since it is clear that, for almost every t in [0, 1],

lim n→+∞ qn-1 k=0 1 [x (n) k ,x (n) k+1 ) (t) M h x (n) k (η)(t)=M h(t) (η)(t),
the dominated convergence theorem applies and gives us, using (5.17),

S(L(Y ))(η)= [0,1] S(Y t )(η) M h(t) (η)(t) dt = [0,1] S(Y t )(η) S(W h(t) t )(η) dt = S [0,1] Y t W h(t) t dt (η),
The injectivity of the map S :Φ → S(Φ) from (S) * into itself provides the equality

L(Y )= [0,1] Y t W h(t) t
dt, which ends the proof.

Remark 5.11. Thanks to the previous theorem we can use the Itô formulas and the Tanaka formula we obtained in chapter 2 (sections 6 and 7) for multifractional Wick-Itô integral.

VI Appendix

ä 2 = ∂MH ∂H (1 [0,t] -1 [0,s] ) 2 L 2 (R) = 1 c 2 H R (β H + ln |y|) 2 |y| 1-2H 1-e iy(t-s) y 2 2 dy.
where β H := τ .N o t et h a tM>1 and that |y| τ 2 for every y such that |y| M . We hence have:

J := 1 c 2 H |y|>M (β H + ln |y|) 2 |y| 1-2H 1 -e iy(t-s) y 2 2 dy + 1 c 2 H |y| M (β H + ln |y|) 2 |y| 1-2H 1 -e iy(t-s) y 2 2 dy 1 c 2 H |y|>M (β H + ln |y|) 2 |y| 1-2H |t -s| 2 |y| 2τ |y| 2 dy + |t -s| 2 2 2 c 2 H |y| M (β H + ln |y|) 2 |y| 2H-1 dy |t -s| 2 4 c 2 H Ç |y|>M (β H + ln |y|) 2 |y| 1+2(H-τ ) dy + |y| M (β H + ln |y|) 2 |y| 2H-1 dy å =: |t -s| 2 Q(H).
Since ∆ 1 := sup

ä 2 ∆ 2 |H -H ′ | 2 . (VI.2)
Setting ∆ := 2(∆ 1 +∆ 2 ) and using (VI.1) and (VI.2) we finally get:

E Ä ∂B1 ∂H (t, H) -∂B1 ∂H (s, H ′ ) ä 2 2E Ä ∂B1 ∂H (t, H) -∂B1 ∂H (t, H ′ ) ä 2 +2E Ä ∂B1 ∂H (t, H ′ ) -∂B1 ∂H (s, H ′ ) ä 2 ∆ Ä |t -s| 2 + |H -H ′ | 2 ä .

Introduction

Volatility in financial markets is b oth of crucial imp ortance and hard to mo del in an accurate way. It has been long known that a constant volatility as in the Black & Scholes model is not consistent with empirical findings, such as the smile effect. More basically, there is no reason to expect that instantaneous volatility should be constant. Popular models allowing for a varying volatility include ARCH models and their generalizations, stochastic volatility models and local volatility models. Local volatility models, in particular, enable the possibility to mimic in an exact way implied volatility surfaces. However, such models do not take into account another well documented fact: while stocks do not typically exhibit correlations, volatility does display long-range correlations (see, e.g. [START_REF] Torben | The distribution of realized stock return volatility[END_REF]). Stochastic volatility models, in contrast, are able to incorporate this feature, provided an adequate driving noise is used. In [START_REF] Comte | Long memory continuous-time stochastic volatility models[END_REF][START_REF] Comte | Affine fractional stochastic volatility models with application to option pricing[END_REF], this is performed by using fractional integration. More precisely, the model considered in [START_REF] Comte | Long memory continuous-time stochastic volatility models[END_REF] for the dynamic of the price of a risky asset reads as follows:

ß dS t = µ(t, S t )dt + S t σ t dW t , d ln(σ t )=θ (µ -ln(σ t )) dt + γdB H t ,σ > 0, (0.1)
where W is a Brownian motion and B H t is an independent fractional Brownian motion (fBm) under the historical probability. Fractional Brownian motion is an extension of Brownian motion, parametrized by ar e a lH in (0, 1),w h i c hh a sc o n s t a n tl o c a lH ö l d e rr e g u l a r i t ye q u a lt oH and whose increments display long-range dependence for H>1/2.S u c ham o d e li sc o n s i s t e n tw i t ht h es l o wd e c a yi nt h ec o r r e l a t i o n so f volatility observed in practice. It also accounts for two features related to the measured smile effect: the volatility process is less persistent in the short term than a standard diffusion, while it is more persistent in the long run ([23, p. 3]). We verify this fact in the case of Model (0.1) numerically in Section 6. However, by the very nature of this model, the evolution in time of the smile is governed by the single parameter H. In this work, we replace fBm appearing in (0.1) with a more general process called multifractional Brownian motion (mBm). Multifractional Brownian motion is an extension of fBm where the Hurst parameter H is replaced by a function h.T h i se n a b l e st h ep o s s i b i l i t yt oa c c o m m o d a t ef o rn o n -s t a t i o n a r yl o c a lr e g u l a r i t y , and to decouple it from long-range dependence properties. Indeed, there is no reason to believe that the regularity of the volatility should be constant. In addition, graphs of estimated historical volatility (see, e.g. [START_REF] Torben | The distribution of realized stock return volatility[END_REF]) seem to indicate that this process is highly irregular. Modelling this evolution with the help of an fBm

B h u+rt -B h u r h(u) ; t ∈ [a, b] law ----→ r→0 + {B h(u) t ; t ∈ [a, b]}.
These properties show that mBm is a more versatile model that fBm: in particular, it is able to mimic in a more faithful way local properties of financial records, Internet traffic and natural landscapes [START_REF] Bianchi | Pathwise identification of the memory function of multifractional brownian motion with application to finance[END_REF][START_REF] Li | Towards describing multi-fractality of traffic using local Hurst function[END_REF][START_REF] Echelard | Terrain modelling with multifractional Brownian motion and self-regulating processes[END_REF]

nC = {C 1 , ••• ,C N } of E such that Proj = N i=1 γ i 1 Ci .I n
other words, Proj performs the process of mapping the continuous set X(Ω) to the finite set Γ.L e tProj Γ denote a nearest neighbour projection on Γ.C l e a r l y ,

|X -Proj Γ (X)| |X -Proj(X)| so that |X -Proj Γ (X)| p |X -Proj(X)| p .
Hence, in order to minimize the quantization error, it is optimal to use a nearest neighbour projection on the codebook Γ.As o l u t i o no f( 2 . 1 )i sc a l l e da nL p -optimal quantizer of X.A ne l e m e n t a r yp r o p e r t yo fa n L 2 -optimal quantizer is stationarity: We now assume that X is a bi-measurable stochastic process on [0,T] verifying

E[X|Y ]=Y .W
T 0 E |X t | 2 dt < ∞,t
h atw e see as a random variable valued in the Hilbert space H = L 2 ([0,T]).S u p p o s et h a ti t sc o v a r i a n c ef u n c t i o nΓ X is continuous. In [START_REF] Luschgy | Functional quantization of Gaussian processes[END_REF], it is shown that, in the centred Gaussian case, linear subspaces U of H spanned by Nstationary quantizers correspond to principal components of X,i no t h e rw o r d s ,a r es p a n n e db ye i g e n v e c t o r s of the covariance operator of X.T h u s ,t h eq u a d r a t i co p t i m a lq u a n t i z a t i o no fG a u s s i a np r o c e s s e sc o n s i s t si n using its Karhunen-Loève decomposition e X n ,λ X n n 1 . To p erform optimal quantization, the Karhunen-Lo ève expansion is first truncated at a fixed order m and then the R m -valued Gaussian vector constituted of the m first coordinates of the process on its Karhunen-Loève decomposition is quantized. To reach optimal quantization, we have to determine both the optimal rank of truncation d X (N ) (the quantization dimension) and the optimal d X (N )-dimensional Gaussian quantizer corresponding to the first coordinates,

d X (N ) j=1 N Ä 0,λ X j ä
.W eh a v et h ef o l l o w i n gr e p r e s e n t a t i o no ft h eq u a d r a t i c

distortion E N (X) := E N,2 Ä X, |•| L 2 ([0,T ]) ä : E N (X) 2 = j m+1 λ X j + E N m j=1 N Ä 0,λ X j ä 2 .
From a numerical viewp oint, we are thus confronted on the one hand with the finite-dimensional quantization of the Gaussian distribution

m j=1 N Ä 0,λ X j ä
and on the other hand with the numerical evaluation of the first Karhunen-Loève eigenfunctions e X n 1 n d X (N ) .V a r i o u sn u m e r i c a la l g o r i t h m sh a v eb e e nd e v e l o p e dt od e a l with the first point. Let us mention Lloyd's algorithm and the Competitive Learning Vector Quantization (CLVQ). A review of these methods is available in [65]. As far as the evaluation of the first Karhunen-Loève eigenfunctions is concerned, closed-expressions are available for standard Brownian motion, standard Brownian bridge and Ornstein-Uhlenbeck process. Other examples of explicit Karhunen-Loève expansions may be found in [START_REF] Deheuvels | A Karhunen-Loève decomposition of a Gaussian process generated by independent pairs of exponential random variables[END_REF] and [START_REF] Pycke | Explicit Karhunen-Loève expansions related to the Green function of the Laplacian[END_REF]. In the general case, the so-called Nyström method for approximating the solution of the associated integral equation may be used. It reads

T 0 Γ X (•,s)e X k (s)ds = λ X k e X k ,k 1, (2.2)
where both the eigenvalues and the eigenvectors have to be determined. The Nyström method relies on the use of a quadrature scheme to approximate the integral, so that it turns into a matrix eigensystem.

When dealing with the midpoint quadrature rule, and for sufficiently regular kernels Γ X ,t h ee r r o ra d m i t s an asymptotic expansion in the form of the sum of even powers of the step size, for both the eigenvalues and the eigenfunctions. We take advantage of this asymptotic expansion by using Richardson-Romberg extrapolation methods. This method has been benchmarked against the Karhunen-Loève eigensystems of standard Brownian motion, Brownian bridge and Ornstein-Uhlenbeck process in [START_REF] Corlay | The Nyström method for functional quantization with an application to the fractional Brownian motion[END_REF].

Instead of using an optimal quantization for the distribution .I nt h ec a s eo fi n d e p e n d e n tm a r g i n a l s , this yields a stationary quantizer, i.e. aquan tizerY of X which satisfies E[X|Y ]=Y .T h i sp r o p e r t y ,s h a r e d with optimal quantizers, results in a convergence rate of a higher order for the quantization-based cubature scheme, as explained in [START_REF] Pagès | Functional quantization for numerics with an application to option pricing[END_REF]. An advantage of this approach is that one-dimensional Gaussian quantization is a fast procedure.

d X (N ) j=1 N Ä 0,λ X
In [65], deterministic optimization methods (e.g. Newton-Raphson) are shown to converge rapidly to the unique optimal quantizer of the one-dimensional Gaussian distribution. Moreover, a sharply optimized database of quantizers of standard univariate and multivariate Gaussian distributions is available at www.quantize.maths-fi.com [START_REF] Pagès | W ebsitedevotedto optimal quantization[END_REF] for download. One sill has to determine the quantization level for each dimension to obtain optimal product quantization. In this case, the minimization of the distortion becomes: These graphs reflect, to some extent, the features of the quantized process, in particular its correlation and regularity properties.

Ä E prod N (X) ä 2 := min    d j=1 E 2 Nj Ä N Ä 0,λ X j ää + j d+1 λ X j ,N 1 ו••×N d N, d 1    . ( 2
In the case of fBm (Figure 4.1), when H increases, the rate of decay of the Karhunen-Loève eigenvalues also increases (and so does the pathwise Hölder regularity of the underlying process), so that even though we do not change the quantization dimension in this example, the contribution of higher-order Karhunen-Loève eigenvalues decreases. In Figure 4.1, one can see that the curves of the functional quantizer localize around the quantization of the first Karhunen-Loève coordinate when H =0 .75,w h i l et h i si sn o tt h ec a s ew h e n H =0.25.

In addition, the distribution of the curves on the plane is related to the fact that the almost sure Hausdorf dimension of the paths of fBm is 2-H: for small H, we expect the set of curves to be more space-filling than for large H,af e a t u r et h a tc a nb ei n d e e db ev e r i fi e do nt h efi g u r e . I na d d i t i o n ,t h el o n g -t e r mc o r r e l a t i o no f fBm for H>1/2,whichresultsinpathst ypicallyhavingstrongtrends,translatesherein tocurveswhichare roughly monotonous. Conversely, the negative correlations which characterizes the case H<1/2 is reflected in the more oscillatory behaviour of the curves in the left pane.

The case of mBm (Figure 4.2) makes even clearer the relation between the properties of the process and the optimal quantizer. In the right pane of Figure 4.2, the function h decreases linearly from 0.9 to 0.1.O n ec a n see that, for small t,b o t ht h ed i s t r i b u t i o no ft h ec u r v e sa n dt h e i rt r e n dl oo kl i k et h eo n e so ff B mw i t hl a r g e H.A st increases, the curves become more space-filling and oscillatory, in agreement with the fact that, for Another way of interpreting these figures is to recall that mBm is tangent, at each t,t of B mw i t he x p o n e n t H = h(t).T h eb e h a v i o u ro ft h ec u r v e so nF i g u r e4 . 2i sat r a n s l a t i o no ft h i sf a c ti nt h eq u a n t i z a t i o nd o m a i n . Finally, note that the shape of the convex envelopes in each of the four figures roughly matches the time evolution of the variances of the corresponding processes, i.e. t 0.25 ,t 0.75 ,t 0.1+0.8t and t 0.9-0.8t .

Rate of decay of the quantization error for mBm

The rate of decay of the quadratic functional quantization error was first investigated in [START_REF] Luschgy | Functional quantization of Gaussian processes[END_REF]. More precise results were then established in [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF]. These results rely on assumptions on the asymptotic behaviour of the Karhunen-Loève eigenvalues of the considered process. In Subsection 2.2, we recall the main result involving the rate of decay of these eigenvalues, leading to sharp rates of convergence for the quantization of fBm.

Unfortunately, such asymptotics for the Karhunen-Loève eigenvalues are not known at this time in the case of mBm. However, since the regularity of mBm is known, one may use another, less precise, type of results: these yield an upper estimate on the rate of decay of the quantization error [START_REF] Luschgy | Functional quantization rate and mean regularity of processes with an application to Lévy processes[END_REF]. This is explained in Subsection 2.2.

In Let X be a bi-measurable centred Gaussian process on [0,T] with a continuous covariance function Γ X and such that T 0 E[X 2 s ]ds < ∞.D e n o t eb y e X n ,λ X n n 1 its Karhunen-Loève eigensystem.

Theorem 2.1 (Quadratic quantization error asymptotics [START_REF] Luschgy | Sharp asymptotics of the functional quantization problem for Gaussian processes[END_REF]). Assume that λ X n ∼ φ(n) as n →∞ ,w h e r e φ :(s, ∞) → (0, ∞) is a decreasing regularly varying function of index -b<-1 and s>0.S e tψ(x) := 1 xφ(x) . Then

E N (X) ∼ Ç Å b 2 ã b-1 b b -1 å 1/2
ψ(log(N )) -1/2 as N →∞.

Moreover, the optimal product quantization dimension m X (N ) verifies m X (N ) ∼ 2 b log(N ) as N →∞ , and the optimal product quantization error E prod N (X) of level N satisfies

E prod N (X) Ç Å b 2 ã b-1 b b -1 + C(1) å 1/2 ψ(log(N )) -1/2 as N →∞,
where C(1) is a universal positive constant.

Though the optimal product quantization is not asymptotically optimal, it still provides a rate-optimal sequence of quantizers. In the case where b =1 ,as i m i l a rr e s u l ti st r u e ,w i t ht h ea d d i t i o n a lp r o p e r t yt h a t the optimal product quantization does yield an asymptotically optimal quadratic quantization error.

The case of fractional Brownian motion

In [START_REF] Luschgy | Functional quantization of Gaussian processes[END_REF][START_REF] Bronski | Asymptotics of Karhunen-Loeve eigenvalues and tight constants for probability distributions of passive scalar transport[END_REF], it is shown that the Karhunen-Loève eigenvalues of fBm on [0,T] verify

λ B H n ∼ ν H n 2H+1 as n →∞,
where ν H is a positive constant. Thus, fBm satisfies the hypotheses of Theorem 2.1 and

E N Ä B H ä ∼ K H log(N ) H as N →∞for some K H > 0, and E prod N Ä B H ä 1 log(N ) H as N →∞.

Mean regularity and domination of the functional quantization rate

We recall the definition of regular variation at 0: Definition 2.2 (Regularly varying function at zero). Am e a s u r a b l ef u n c t i o nφ :(0,s) → (0, ∞),( s>0)i s regularly varying at 0 with index b 0 if for every t>0, lim Definition 2.3 (The φ-Lipschitz assumption). Let X be a bi-measurable process on [0,T].W es a yt h a t X satisfies the φ-Lipschitz assumption for ρ>0,w h i c hw ed e n o t eb y(L φ,ρ ),i ft h e r ei san o n -d e c r e a s i n g function φ : R + → [0, ∞],c o n t i n u o u sa t0 with φ(0) = 0,s u c ht h a t

(L φ,ρ ) ≡    ∀(s, t) ∈ [0,T] 2 , E [|X t -X s | ρ ] (φ(|t -s|)) ρ , if ρ 1 ∀t ∈ [0,T], ∀h ∈ (0,T], E ñ sup t s (t+h)∧T |X s -X t | ρ ô (φ(h)) ρ if 0 <ρ<1. Remark 2.2. The φ-Lipschitz assumption implies that E î |X| ρ L ρ ([0,T ]) ó < ∞ so that P-almost surely, t → X t lies in L ρ ([0,T]).
Theorem 2.3 (Mean regularity and quantization rate). Let X be a bi-measurable process on [0,T] such that X t ∈ L ρ for every t ∈ [0,T], ρ>0.A s s u m et h a tX satisfies (L φ,ρ ) where φ is regularly varying at 0 with index b.T h e n

∀(r, p) ∈ [0,ρ] 2 , E N,r Ä X, |•| L p ([0,T ]) ä C r,p ß φ(1/ log(N )), if b>0, ψ(1/ log(N )), if b =0, with ψ(x) := x 0 (φ(ξ)) (r∧1) ξ dξ 1 r∧1 ,a s s u m i n gi na d d i t i o nt h a t x 0 (φ(ξ)) (r∧1) ξ dξ < ∞ if b =0.
In particular, if φ(u)=cu b ,b>0,t h e n

E N,r Ä X, |•| L p ([0,T ]) ä = O(log(N ) -b ).
The case of multifractional Brownian motion Theorem 2.4 (L 2 -mean regularity of the multifractional Brownian motion). Let B h be an mBm with functional parameter h satisfying assumption (H).A s s u m et h a th is β-Hölder continuous, β>0 3 .T h e n there exists a positive constant M such that

∀(s, t) ∈ [0,T] 2 , E Ä B h t -B h s ä 2 M |t -s| Ä 2i n f u∈[0,T ] h(u)∧βδ ä , (2.4)
where δ is given in assumption (H).

Proof: We may assume that the fractional field (B(t, H)

) (t,H)∈[0,T ]×[c,d] is normalized. For (t, s) in [0,T] 2 : E Ä B h t -B h s ä 2 2 E î (B(t, h(t)) -B(s, h(t))) 2 ó +2 E î (B(s, h(t)) -B(s, h(s))) 2 ó 2 Ä |t -s| 2h(t) +Λ |h(t) -h(s)| δ ä 2 Ä |t -s| 2H1 (1 + T 2(H2-H1) )+Λ η β |t -s| βδ ä 2( 1 + T 2(H2-H1) )( 1+Λη β ) Ä |t -s| 2H1 + |t -s| βδ ä M |t -s| 2H1∧βδ ,
where

H 1 := inf u∈[0,T ] h(u), H 2 := sup u∈[0,T ]
h(u) and M := 2(1 + T 2(H2-H1) )( 1+T 2H1∨βδ-2H1∧βδ )( 1+Λη β ).

3. i.e. ∃η ∈ R * + , ∀(s, t) ∈ [0,T] 2 , |h(s) -h(t)| η |s -t| β .
Corollary 2.5 (Upper bound on the quantization error for multifractional Brownian motion). With the same notations and assumptions as in theorem 2.4:

E N,r Ä B h , |•| L p ([0,T ]) ä = O Ä log(N ) -(H1∧ βδ 2 )
ä , for every (r, p) in (R + ) 2 .

Proof: Since B h is a Gaussian process, Theorem 2.4 shows that B h fulfils the φ-Lipschitz assumption for every integer ρ of the form ρ := 2p where p is a positive integer and for the continuous function φ ρ defined on R + by φ ρ (0) := 0 and φ ρ (x) := (κ ρ/2 ) 1/2ρ

h := (V h t ) t∈[0,T ] , of the form V h t := Z(t, h(t)) where Z := (Z(t, H)) (t,H)∈R×[H1,H2] is a Gaussian field such that one can find (Λ,γ,δ) in (R * + ) 3 with ∀(s, t, H, H ′ ) ∈ [0,T] 2 × [H 1 ,H 2 ] 2 , E Ä Z(t, H) -Z(s, H ′ ) ä 2 Λ(|t -s| γ + |H -H ′ | δ ),
In this case, for every (r, p) in (R * + ) 2 ,w eg e tE N,r

Ä V h , |•| L p ([0,T ]) ä = O log(N ) -( γ 2 ∧ βδ 2 ) .
Remark 2.7 (Sharp rate). We conjecture that the domination of the rate of decay of the quantization error for the mBm established in Corollary 2.5 is in fact a sharp rate and that we have, for every (r, p)

∈ (R * + ) 2 , E N,r Ä B h , |•| L p ([0,T ]) ä ∼ K h,r,p log(N ) H1∧β as N →∞for some K h,r,p > 0,
and that the quadratic optimal product quantization error E prod N Ä B h ä for the mBm satisfies

E prod N Ä B h ä 1 log(N ) H1∧β as N →∞,
which is consistent with the results of previous section on the fractional Brownian motion.

I H,H ′ t := E î (B(t, H) -B(t, H ′ )) 2 ó = R e itξ -1 c H |ξ| H+1/2 - e itξ -1 c H ′ |ξ| H ′ +1/2 2 du = R e itξ -1 ξ 2 1 cH |ξ| 1/2-H -1 c H ′ |ξ| 1/2-H ′ 2 dξ. (2.5)
For every ξ in R * ,t h em a pf ξ :

[ c, d] → R + , defined by f ξ (H) := 1 cH |ξ| 1/2-H is C 1 since H → c H is C 1 on (0,

1).T h u st h e r ee x i s t sap o s i t i v er e a lD such that

∀(ξ, H) ∈ R * × [c, d], |f ′ ξ (H)| D |ξ| 1/2-H (1 + | ln(|ξ|)|) D Ä |ξ| 1/2-c + |ξ| 1/2-d ä (1 + | ln(|ξ|)|).
Thanks to the mean-value theorem, (2.5) yields

I H,H ′ t D 2 |H -H ′ | 2 R |e itξ -1| 2 |ξ| 2 Ä |ξ| 1/2-c + |ξ| 1/2-d ä 2 (1 + | ln(|ξ|)|) 2 dξ D 2 |H -H ′ | 2 2 3 |ξ|>1 (1 + | ln(|ξ|)|) 2 |ξ| 1+2c dξ +(2t) 2 |ξ| 1 |ξ| 1-2d (1 + | ln(|ξ|)|) 2 dξ (2 3 + T 2 ) D 2 |ξ|>1 (1 + | ln(|ξ|)|) 2 |ξ| 1+2c dξ + |ξ| 1 |ξ| 1-2d (1 + | ln(|ξ|)|) 2 dξ |H -H ′ | 2 .
Since the two integrals in the last line are finite, (H)i sv e r i fi e dw i t hδ =2.

Quantization-based cubature

Basic formula and related inequalities in the case of Lipschitz continuous functionals

The idea of quantization-based cubature methods is to approximate the distribution of the random variable X by the distribution of a quantizer Y of X.A sY is a discrete random variable, we have

P Y = N i=1 p i δ yi . Therefore, if F : E → R is a Borel functional, E[F (Y )] = N i=1 p i F (y i ). (2.6)
Hence, the weighted discrete distribution (y i ,p i ) 1 i N of Y allows us to compute the sum (2.6). We review some error bounds that can be derived when approximating E[F (X))] by (2.6). See [START_REF] Pagès | Functional quantization for numerics with an application to option pricing[END_REF] for more details. 1. If X ∈ L 2 , Y aq u a n t i z e ro fX of size N and F is Lipschitz continuous, then

|E[F (X)] -E[F (Y )]| [F ] Lip X -Y 2 . (2.7) where [F ] Lip is the Lipschitz constant of F . In particular, if (Y N ) N 1 is a sequence of quantizers such that lim N →∞ X -Y N 2 =0 , then the distribution N i=1 p N i δ x N i of Y N weakly converges to the distribution P X of X as N →∞. 2. If Y is a stationary quantizer of X,i.e. Y = E[X|Y ],a ndF is differentiable with a Lipschitz continuous derivative DF ,t h e n |E[F (X)] -E[F (Y )]| [DF ] Lip X -Y 2 2 , (2.8)
where

[DF ] Lip is the Lipschitz constant of DF . If F is twice differentiable and D 2 F is bounded, then we can replace [DF ] Lip by 1 2 D 2 F ∞ . 3. If F is a semi-continuous convex functional 4 and Y is a stationary quantizer of X, E[F (Y )] E[F (X)].
(2.9) This is a simple consequence of Jensen's inequality. Indeed,

E[F (Y )] Stationarity = E[F (E[X|Y ])] Jensen E[E[F (X)|Y ])] = E[F (X)].
The case of exponentials of continuous centred Gaussian processes Let (X s ) s∈[0,T ] be a continuous centred Gaussian process on [0,T].T h e nt h ec o v a r i a n c ef u n c t i o no fX is also continuous. In addition, Fernique'st h e o r e me n t a i l st h a tE î T 0 X 2 s ds ó is finite. We view X as a random variable valued in the separable Banach space (C([0,T], R), • ∞ ).L e t " X be a stationary quantizer of X.

By the mean-value theorem, for all (x, y) ∈ R 2 , |e x -e y | e |x|+|y| |x -y|.C o n s e q u e n t l y ,f o rp 1,u s i n g Hölder's inequality:

E T 0 e Xs -e Xs p ds 1/p E ï T 0 e p|Xs|+p Xs X s -" X s p ds ò 1/p E ï T 0 e p p|Xs| e p p Xs ds ò 1 2p p E ï T 0 X s -" X s pq ds ò 1 2p q ,
where (p, q) ∈ (1, ∞) 2 are conjugate exponents. For >0,w ec h oo s e(p, q) such that pq = p + .T h i sg i v e s q =1+ /p and p =1+p/ .

4. In the infinite-dimensional case, convexity does not imply continuity. In infinite-dimensional Banach spaces, a semicontinuity hypothesis is necessary for Jensen's inequality. See [START_REF] Zapała | Jensen's inequality for conditional expectations in Banach spaces[END_REF] for more details. (2.10)

We shall apply (2.10) with p =2-in section 4: this will allow us to control the L 2-quantization error of the exponential of a centred continuous Gaussian process X by the L 2 quantization of X.

Richardson-Romberg extrapolation

With respect to the quantization error

In the general setting of a non-uniform random variable X,aquadraticoptimalN -quantizer Y N of X and a C 1 functional with Lipschitz continuous derivative, Equation (2.8) does not provide a true asymptotic expansion which would allow one to use a Richardson-Romberg expansion, but it suggests the use a higher-order Taylor expansion of F (X) -F (Y N ) to get one. It follows from Taylor's formula that there exists a vector

ζ ∈ [X, Y N ] such that E[F (X)] = E [F (Y N )] + E [ DF (Y N ),X -Y N ] =E[DF (YN ).E[X-YN |YN ]]=0 by stationarity. + 1 2 E D 2 F (Y N )(X -Y N ) ⊗2 + 1 6 E ζ(X -Y N ) ⊗3 + o E |X -Y N | 3 = E [F (Y N )] + 1 2 E D 2 F (Y N )(X -Y N ) ⊗2 + O Ä E î |X -Y N | 3 óä .
(2.11)

In [START_REF] Graf | Distortion mismatch in the quantization of probability measures[END_REF], it is proved that the asymptotics of the L s quantization error induced by a sequence of L r -optimal quantizers remains rate-optimal in the case of probability distributions on R d ,w i t hs<r+ d for a class of distributions including the Gaussian distribution. This leads to E |X -

Y N | 3 = O E |X -Y N | 2 3 2
.T h i s holds e.g. for Brownian motion. Unfortunately, no sharp equivalence between X -Y N

[F (Y l )]E 2 k -E[F (Y k )]E 2 l E 2 k -E 2 l .
(2.12)

Although this kind of Richardson-Romberg extrapolationhasnotreceivedafulltheoretical justification yet, it does dramatically increase the efficiency of quantization-based cubature formulas.

With respect to the quantization level

When the value of E 2 k is not known, one may rely on an asymptotic expansion with respect to the quantization level.

Remark 2.9 (Romberg extrapolation with respect to the quantization level). In Section 2.2, we have seen that under some assumptions on the eigenvalues of the convergence operator, the rate of convergence of optimal quantizers and K-L optimal product quantizers is (ln(N ) -α ) for some α ∈ (0, 1).R e p l a c i n gt h e distortion E N by its asymptotics 1 ln(N ) α as N →∞in Equation (2.12) suggests to approximate E[F (X)] by the quantity

E[F (Y l )](ln l) 2α -E[F (Y k )](ln k) 2α (ln l) 2α -(ln k) 2α . ( 2

.13)

Multidimensional Richardson-Romberg extrapolation Let X 1 and X 2 be two independent random variables. We wish to estimate the expectation E F X 1 ,X 2 for some regular functional F .I n t h a t v i e w , o n e m a y u s e a c u b a t u r e b a s e d o n a p r o d u c t q u a n t i z a t i o n Ä " X 1 , " X 2 ä of (X 1 ,X 2 ),a n dp e r f o r mam u l t i d i m e n s i o n a lR i c h a r d s o n -R o m b e r ge x t r a p o l a t i o n . T h i sa m o u n t s to performing two Richardson-Romberg extrapolations as described already, one related to the quantization error of X 1 between quantization levels N 1 and M 1 ,andonerelatedtothequan tizationerrorofX 2 between quantization levels N 2 and M 2 .T h i sl e a d st oa p p r o x i m a t i n gE[F (X 1 ,X 2 )] by the quantity

E 2 M1 E 2 M2 F N1,N2 -E 2 N1 E 2 M2 F M1,N2 -E 2 M1 E 2 N2 F N1,M2 + E 2 N1 E 2 N2 F M1,M2 Ä E 2 M1 -E 2 N1 äÄ E 2 M2 -E 2 N2 ä , (2.14)
where F p,q denotes the estimated expectation obtained with the quantization-based cubature and quantization levels of p and q for X 1 and X 2 respectively. In other words, F p,q is defined by

F p,q := E F " X 1 p , " X 2 q
where " X 1 p , " X 2 q are quantizers of levels p and q for X 1 and X 2 respectively. In Equation (2.14), E Mi and E Ni denote the quadratic quantization error of level M i and N i for X i .

3S t o c h a s t i c c a l c u l u s w i t h r e s p e c t t o m B m

From now on and until the end of the work, we fix our mBm to b e the well-balanced multifractional Brownian motion defined in Section 1. In addition, we will always assume that h is a C 1 function with derivative bounded on R .

Some backgrounds on white noise theory

Define the probability space as Ω := S ′ (R) and let F := B(S ′ (R)) be the σ-algebra of Borel sets. There exists a probability measure µ such that, for every f in L 2 (R),t h em a p •,f :Ω→ R defined by •,f (ω)= ω, f (where ω, f is by definition ω(f ), i.e the action of the distribution ω on the function f ) is a centred Gaussian random variable with variance equal to f 2 L 2 (R) under µ.F o re v e r yn in N,d e n o t e e n (x) := (-1)

n π -1/4 (2 n n!) -1/2 e x 2 /2 d n dx n (e -x 2 ) the nth Hermite function. Let (|•| p ) p∈Z be the family of norms defined by |f | 2 p := +∞ k=0 (2k +2) 2p f, e k 2 L 2 (R) ,f o ra l l(p, f ) in Z × L 2 (R).T h eo p e r a t o rA defined on S (R) by A := -d 2 dx 2 + x 2 +1
admits the sequence (e n ) n∈N as eigenfunctions and the sequence (2n +2) n∈N as eigenvalues. As is customary, we denote (L 2 ) the space L 2 (Ω, G,µ) where G is the σ-field generated by ( •,f ) f ∈L 2 (R) .F o r every random variable Φ of (L 2 ) there exists, according to the Wiener-Itô theorem, a unique sequence (f n ) n∈N of functions f n in L 2 (R n ) such that Φ can be decomposed as Φ= I n (A ⊗n f n ),w h e r eA ⊗n denotes the nth tensor power of the operator A (see [START_REF] Janson | Gaussian Hilbert spaces[END_REF]Appendix E] for more details about tensor products of operators). The operator Γ(A) is densely defined on (L 2 ).I t i s i n v e r t i b l e a n d i t s i n v e r s e Γ(A)

-1 is bounded. Let us denote ϕ 2 0 := ϕ 2 (L 2 ) for ϕ in (L 2 ) and let Dom(Γ(A) n ) be the domain of the nth iteration of Γ(A).D e fi n et h ef a m i l yo fn o r m s( • p ) p∈Z by:

Φ p := Γ(A) p Φ 0 = Γ(A) p Φ (L 2 ) , ∀p ∈ Z, ∀Φ ∈ (L 2 ) ∩ Dom(Γ(A) p ).
For any p in N,l e t(S p ) := {Φ ∈ (L 2 ): Γ ( A) p Φ exists and belongs to (L 2 )} and define (S -p ) as the completion of the space (L 2 ) with respect to the norm • -p .A si n[ 5 2 ] ,w el e t(S) denote the projective limit of the sequence ((S p )) p∈N and (S) * the inductive limit of the sequence ((S -p )) p∈N .T h es p a c e(S) is called the space of stochastic test functions and (S)

* the space of Hida distributions. One can show that, for any p in N,t h ed u a ls p a c e(S p ) * of S p is (S -p ).T h u sw ew i l lw r i t e(S -p ),i nt h es e q u e l ,t od e n o t et h e space (S p ) * .N o t ea l s ot h a t(S) * is the dual space of (S).W ew i l ln o t e • , • the duality bracket between (S)

* and (S).I fΦ belongs to (L 2 ) then we have the equality Φ,ϕ = Φ,ϕ (L 2 ) = E[Φ ϕ].S i n c ew eh a v e defined a topology given by a family of norms on the space (S) * it is possible to define a derivative and an integral in (S) * .Af u n c t i o nΦ:R → (S) * is called a stochastic distribution process, or an (S) * -process, or aH i d ap r o c e s s . The Hida process Φ is said to be differentiable at t 0 if lim For f in L 2 (R) and Φ := •,f ,i ti se a s yt ov e r i f yt h a texp Φ exists and coincides with : e •,f : defined at the beginning of this section.

process. If Φ is (S) * -integrable over R then for all η in S (R), S( R Φ(u) du)(η)= R S(Φ(u))(η) du.I fΦ is (S) * -differentiable over R then for all η in S (R), S[ dΦ dt (t)](η)= d dt î [SΦ(t)](η) ó .

Fracti o nal and mul ti fracti o nal W hi te no i s e

Operators M H and ∂MH ∂H . Let H belong to (0, 1).F o l l o w i n g [ 3 4 ] ,t h eo p e r a t o rM H is defined in the Fourier domain by

◊ M H (u)(y) := √ 2π cH |y| 1/2-H u(y), ∀y ∈ R * .
This operator is well defined on the homogeneous Sobolev space of order 1/2 -H noted L 2 H (R) and defined by

L 2 H (R) := {u ∈S ′ (R): u = T f ; f ∈ L 1 loc (R) and u H < +∞}.T h en o r m • H derives from the inner product •, • H defined on L 2 H (R) by: u, v H := 1 c 2 H R |ξ| 1-2H " u (ξ)" v (ξ)
dξ where c H is defined right after Definition 1.1. The definition of the operator ∂MH ∂H is quite similar [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF]. Precisely, define, for H in (0, 1),thespaceΓ

H (R) := {u ∈S ′ (R): u = T f ; f ∈ L 1 loc (R) and u δH (R) < +∞},w h e r et h en o r m • δH (R) derives from the inner product •, • δH defined on Γ H (R) by u, v δH := 1 c 2 H R (β H + ln |ξ|) 2 |ξ| 1-2H " u (ξ) dξ. The operator ∂MH ∂H ,f r o m Ä Γ H (R), •, • δH (R) ä to Ä L 2 (R), •, • L 2 (R) ä
, is defined in the Fourier domain by:

ÿ ∂MH ∂H (u)(y) := -(β H + ln |y|) √ 2π cH |y| 1/2-H u(y), ∀y ∈ R * .
Fracti o nal and mul ti fracti o nal W hi te no i s e .

For any measurable function h : R → (0, 1),i ti se a s i l ys e e nt h a tt h ep r oc e s sB h :=

Ä B h t ä t∈R defined by ∀(ω, t) ∈ Ω × R,B h t := +∞ k=0 t 0 M h(t) (e k )(s)ds •,e k is an mBm. Assuming that h is differentiable, we define the (S) * -valued function W h := (W h t ) t∈R by W h t := +∞ k=0 ï d dt Å t 0 M h(t) (e k )(s) ds ãò •,e k . ( 3.2)
The following theorem states that, for all real t,t h er i g h t -h a n ds i d eo f( 3 . 2 )d o e si n d e e db e l o n gt o(S) * and is exactly the (S) * -derivative of B h at point t.

Theorem-Definition 3.1 ([53, Theorem-definition 5.1]). Let h : R → (0, 1) be a C 1 deterministic function such that its derivative function h ′ is bounded. The process W h defined by (3.2) is an (S) * -process which verifies the following equality (in (S) * ):

W h t = +∞ k=0 M h(t) (e k )(t) •,e k + h ′ (t) +∞ k=0 Å t 0 ∂MH ∂H (e k )(s) H=h(t) ds ã •,e k . (3.3)
Moreover the process B h is (S) * -differentiable on R and verifies dB h dt (t)=W h t in (S) * .

When the function h is constant, identically equal to H,wewilldenoteW H := W H t t∈R and call the process W h fractional white noise. This process was defined and studied in [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF][START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF].

Stochastic integral with respect to mBm

We recall the definition of the Wick-Itô sto chastic integral with respect to mBm from [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF]

:= R 1 I (s) Y s d B h s .
When the function h is constant over R,e q u a lt oH,t h em u l t i f rac t i on alW i c k -I t ôi n t e gralcoi n c i d e sw i t ht h e fractional Wick-Itô integral defined in [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF], [START_REF] Biagini | An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion[END_REF], [?]a n d[ ?]. In particular, when Y is adapted and when the function h is identically equal to 1/2,( 3. 4 )i sn ot h i n gb u tt h ecl assi c alIt ôi n t e gralw i t hrespe c tt oBro wn i an motion. The multifractional Wick-Itô integral verifies the following properties: Proposition 3.2. Let B h be an mBm and I be an interval of R.

-F o ra l l(a, b) in R 2 such that a<b, b a 1 d B h u = B h b -B h a almost surely. -L e tX : I → (S) * be a d B h -integrable process over I. If I X s d B h s belongs to (L 2 ),t h e nE[ I X s d B h s ]= 0.

Multifractional Wick-Itô integral of deterministic elements

In order to solve differential equations driven by an mBm that will be encountered below, it is necessary to know the exact nature of multifractional Wick-Itô integrals of deterministic elements. For H in (0, 1) and f in S (R),d e fi n et h ef u n c t i o ng f : R × (0, 1) → R by g f (t, H) :

= t 0 M H (f )(x)dx
where M H is the operator defined in at the beginning of Section 3.1. It has been shown that g f belongs to C ∞ (R × (0, 1), R) (cf. [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF]Lemma 5.5]). The main result on the multifractional Wick-Itô integral of deterministic elements is the following: Theorem 3.3. ([53, Theorem 5.25]) Let h : R → (0, 1) be a C 1 deterministic function and let f : R → R be a measurable deterministic function which belongs to L 1 loc (R).L e tZ := (Z t ) t∈R be the process defined by

Z t := t 0 f (s) d B h s .
T h e nZ is an (S) * -process which verifies the following equality in (S)

* t 0 f (s) d B h s = +∞ k=0 t 0 f (s) d ds [g e k (s, h(s))] ds •,e k . (3.5)
Moreover Z is a (centred) Gaussian process if and only if

+∞ k=0 Ä t 0 f (s) d ds [g e k (s, h(s))] ds ä 2 < +∞,f o ra l lt in R.
In this case we have, for every t in R,

Z t = t 0 f (s) d B h s L ∼N 0, +∞ k=0 t 0 f (s) d ds [g e k (s, h(s))] ds 2 . (3.6)
In particular, Z is a Gaussian process when f belongs to C 1 (R; R).

Deriving the quantity E Z 2 t in the previous theorem might be complicated using Equation (3.6). However, when f is a C 1 function, thanks to the Itô formula with respect to mBm [53, Theorem 6.9], we obtain the following integration-by-parts formula

t 0 f (s) d B h s (L 2 ) = f (t) B h t - t 0 f ′ (s) B h s ds, (3.7)
which leads to

E î Z 2 t ó = f (t) 2 t 2h(t) + t 0 t 0 f ′ (s) f ′ (u) R h (s, u) ds du -2f (t) t 0 f ′ (s) R h (t, s) ds. ( 3
I f t : g → f (t)g(t) - t 0 f ′ (s)g(s) ds .
(3.9)

Stochastic differential equations

Mixed multifractional Brownian S.D.E.

Let us consider the following mixed multifractional stochastic differential equation, where γ 1 and γ 2 are positive constants and B t is a Brownian motion:

dX t = X t Ä γ 1 d B t + γ 2 d B h t ä , X 0 = x 0 ∈ R. (3.10)
Of course (3.10) is a shorthand notation for the equation

X t = x 0 + γ 1 t 0 X s d B s + γ 1 t 0 X s d B h s ,X 0 = x 0 ∈ R,
where the previous equality holds in (S) * .A s o l u t i o n o f t h i s e q u a t i o n w i l l b e c a l l e d geometric mixed multifractional Brownian motion.R e w r i t i n gt h ep r e v i o u se q u a t i o ni nt e r m so fd e r i v a t i v e si n(S) * ,w eg e t :

dX t dt = X t Ä γ 1 W 1/2 t + γ 2 W h t ä ,x 0 ∈ R. (3.11)
Theorem 3.5 (Geometric mixed multifractional Brownian motion). The (S) * -process (X t ) t∈[0,T ] defined by

X t := x 0 exp Ä γ 1 B t + γ 2 B h t ä , (3.12)
is the unique solution of (3.11) in (S) * .

Proof: Applying the S-transform to both sides of Equation (3.11) and denoting by y η the map t → S(X t )(η), for every η in S (R),w eg e t :

y ′ η (t)=y η (t) Å γ 1 M 1/2 (η)(t)+γ 2 d dt [g η (t, h(t))] ã ,y η (0) = x 0 .
This equation admits a unique solution which verifies y η (t)=x 0 exp {γ

1 t 0 M 1/2 (η)(u)du + γ 2 t 0 d du [g η (u, h (u)
)]du}. Using (i) and (ii) of [53, Theorem 5.12]w eh e n c eg e t ,f o re v e r yη in S (R),

y η (t)=x 0 exp {γ 1 S(B t )(η)+γ 2 S(B h t )(η)} = S Ä x 0 exp {γ 1 B t + γ 2 B h t } ä (η).
The injectivity of the S-transform allows us to conclude that

X t = x 0 exp ¶ γ 1 B t + γ 2 B h t © for every t in [0,T].
Remark 3.6. (i) Using [47, Equality (3.16)], one sees that X is an (L 2 )-valued process that may be represented as:

X t = x 0 exp ¶ γ 1 B t + γ 2 B h t -1 2 Ä γ 2 1 t + γ 2 2 t 2h(t)
ä© .

(ii) The theorem is also a consequence of [START_REF] Holden | Stochastic Partial Differential Equations, A Modeling, White Noise Functional Approach[END_REF]Theorem 3.1.2].

Mixed multifractional Ornstein-Uhlenbeck S.D.E.

Let us now consider the following mixeds t o c h a s t i cd i ff e r e n t i a le q u a t i o n :

® dU t = θ(µ -U t )dt +(α 1 d B t + α 2 d B h t ) U 0 = u 0 ∈ R, (3.13)
where (B t ) t∈R and Ä B h t ä t∈R are independent, θ 0 and µ, α 1 ,α 2 belong to R.As o l u t i o no ft h i se q u a t i o n will be called a mixed multifractional Ornstein-Uhlenbeck process. Theorem 3.7 (Mixed multifractional Ornstein-Uhlenbeck process). The L 2 (Ω)-valued process (U t ) t∈R defined by

U t := u 0 e -θt + µ Ä 1 -e -θt ä + α 1 t 0 e θ(s-t) d B s + α 2 t 0 e θ(s-t) d B h s , (3.14)
is the unique solution of the stochastic differential equation (3.13).

Proof: The proof that the process U defined by (3.14) is the unique solution of (3.13) is very similar to the one of Theorem 3.5. Indeed, setting y η (t) := S(U t )(η) for every (t, η) in R × R and applying the S-transform to both sides of (3.13) we get, for every η in S (R),t h eo r d i n a r yd i ff e r e n t i a le q u a t i o n

y ′ η (t)=θ(µ -y η (t)) + α 1 M 1/2 (η)(t)+α 2 d dt [g η (t, h(t))],y η (0) = u 0 . (3.15)
Its unique solution is

y η (t)=u 0 e -θt + e -θt t 0 e θs Ä θµ + α 1 M 1/2 (η)(s)+α 2 d ds [g η (s, h(s))] ä ds, y η (0) = u 0 .
Again, one concludes using the injectivity of the S-transform.

Quantization of solutions of S.D.E. driven by mBm

Quantizing a Gaussian process X often yields as well a quantization of the solutions of stochastic differential equations driven by X:i n d e e d ,i nm a n yc a s e s ,t h e s es o l u t i o n sm a yb ee x p r e s s e da sf u n c t i o n a l so fX.A quantizer of the solution can then be obtained by simply plugging the quantizer of X into the functional.

In the one-dimensional setting, under rather general conditions on the diffusion coefficients and if X is a continuous semimartingale, this functional is easily determined using the Lamperti transform (see [START_REF] Luschgy | Functional quantization of a class of Brownian diffusions: A constructive approach[END_REF]). In this case, the corresponding quantizer of the stochastic differential equation is obtained by plugging the Gaussian quantizer in the S.D.E. written in the Stratonovich sense, leading to a finite set of ordinary differential equations. This leads to a simple and general constructive method to build a functional quantization of the solution of an S.D.E.

Unfortunately, no such result is available in the case of an S.D.E. driven by multifractional Brownian motion (or even by fractional Brownian motion). However, in some situations, and in particular when an explicit solution is known, one may sometimes still use the procedure just described: if the functional giving the solution is regular enough, quantization-based cubatures can then be used. This is for instance the case of geometric mixed multifractional Brownian motion defined in Section 3.3, which is a simple functional of a Brownian motion and a multifractional Brownian motion (see Remark 3.6 and section 2.3). We describe two other favourable situations in the next subsections.

The case of a Wiener integral

An easy case is the one of a Wiener integral t 0 f (s)d B h s where f is a C 1 deterministic function. The integration-by-parts formula for mBm (3.7) reads

t 0 f (s)d B h s a.s. = f (t)B h t - t 0 f ′ (s)B h s ds.T h u s ,f o rp 1, the stochastic process t → t 0 f (s)d B h
s ,s e e na sar a n d o mv a r i a b l ev a l u e di nL p (0,T),i st h ei m ageofB h by the map

J f : L p ([0,T]) → L p ([0,T]) g → f (•)g(•) - • 0 f ′ (s)g(s)ds.
In other words we have (dt-almost everywhere) J f (g)(t)=I f t (g) where I f t was defined in Remark 3.4. Proposition 3.8 (L p -regularity of the Wiener map). For every p 1,t h em a pJ f is Lipschitz continuous on L p ([0,T]).

Proof: It is straightforward that for (g 1 ,g 2 ) ∈ L p ([0,T]) 2 J f (g 1 ) -J f (g 2 ) p f (g 1 -g 2 ) p + • 0 f ′ (s)(g 1 (s) -g 2 (s))ds p ( f ∞ + f ′ ∞ T ) g 1 -g 2 p .
In Appendix VIII, we prove that if h is C 1 ,t h eK a r h u n e n -L o è v ee i g e n f u n c t i o n so faw e l l -b a l a n c e dm B m B h have bounded variations, and thus stationary quantizers of B h have bounded variations as well (because they lie on a subspace of L 2 ([0,T]) spanned by a finite number of Karhunen-Loève eigenfunctions, as already mentioned). In this setting, another integration by parts gives I ) is the functional that maps g ∈ L p ([0,T]) to the unique solution in L p ([0,T]) of Equation (3.17). In [START_REF] Luschgy | Functional quantization of a class of Brownian diffusions: A constructive approach[END_REF], the map Ψ β p is showed to be Lipschitz continuous in L p ([0,T]). More precisely, one has Mixed multifractional Ornstein-Uhlenbeck process, defined in Section 3.3, is of the form (3.17), with β(s, u)= θ(µ -u) and X = α 1 B + α 2 B h .

4M u l t i f r a c t i o n a l H u l l & W h i t e s t o c h a s t i c v o l a t i l i t y m o d e l

We assume that, under the risk-neutral measure, the forward price of a risky asset is the solution of the S.D.E. ß dF t = F t σ t dW t , d ln(σ t )=θ (µln(σ t )) dt + γ h d B h t + γ σ dW σ t ,σ 0 > 0,

where θ 0 and where W and W σ are two standard Brownian motions and B h is a well-balanced multifractional Brownian motion independent of W and W σ with functional parameter h assumed to be continuously differentiable. We assume that W is decomposed into ρdW σ t + 1ρ 2 dW F t ,w h e r eW F is a Brownian motion independent of W σ .H e n c e , ( 5. Actually, in [START_REF] Luschgy | Functional quantization of a class of Brownian diffusions: A constructive approach[END_REF], the Lamperti transform is used to reduce a general Brownian diffusion to this case.

In other words, ln(σ t ) is a mixed multifractional Ornstein-Uhlenbeck process. Note that, although the volatility process is not a semimartingale, the process (F t ) t∈[0,T ] remains a (positive) F F,σ,h -local martingale, and thus a super-martingale. The same proof as in [START_REF] Jourdain | Loss of martingality in asset price models with lognormal stochastic volatility[END_REF] shows that, if ρ =0,t h i sl o c a lm a r t i n g a l ei si n d e e d am a r t i n g a l e . N u m e r i c a le x p e r i m e n t ss e e mt oindicate that this property still holds for ρ<0,af a c tt h a t remains to be proved. We now consider the problem of pricing a forward start call option (the put case is handled similarly). The payoff of this option writes Ä The following decomposition holds: Conditioning by F σ,h Appendix B shows that χ h i has bounded variations. This entails that σ i is a semimartingale. Define σ i ,W σ τ,T := σ i ,W σ Tσ i ,W σ τ ,w h e r e •, • denotes the semimartingale bracket and let us denote by T τ σ i s • dW σ s the Stratonovich integral of σ i .T h e n ,I σ i τ,T reads

F t =
T yields E Ä FT Fτ -K ä + = E E Ä FT Fτ -K ä + F σ,h T = E E Ä F τ,T exp Ä 1 -ρ 2 t 0 σ t dW F t -1-ρ 2
I σ i τ,T = T τ σ i s • dW σ s - 1 2 σ i ,W σ τ,T .
Itô's formula yields and (p h i p σ j ) 1 i N1,1 j N2 are the paths and weights of a stationary quantizer of the mixed multifractional Ornstein-Uhlenbeck process ln(σ).T h er e s u l t ss t a t e di nS e c t i o n3 . 4a l l o wu st o control its quadratic quantization error with the quantization error of W σ and " B h .W et h e na p p l y( 2 . 1 0 )t o get an upper bound for the L 2-quantization error of the process σ on [0,T],f o ra n y >0 .

5M u l t i f r a c t i o n a l S A B R m o d e l

We now assume that, under the risk-neutral measure, the forward price of a risky asset is the solution of the S.D.E. where W and W σ are two standard Brownian motions and B h is a well-balanced multifractional Brownian motion independent of W and W σ with functional C 1 parameter h.W ea s s u m et h a tW is decomposed into ρdW σ t + 1ρ 2 dW F t ,w h e r eW F is a Brownian motion independent of W σ .W eu s et h es a m en o t a t i o n sa s in the previous section for F σ , F F , F h , F σ,h and F F,σ,h .H e n c e ,( 5 . 1 )w r i t e s

®
dF t = F t σ t Ä ρdW σ t + 1 -ρ 2 dW F t ä dσ t = σ t Ä γ h d B h t + γ σ dW σ t ä ,σ 0 > 0. (5.
2)

The solution of the stochastic differential equation verified by σ,e s t a b l i s h e di nT h e o r e m3 . 5 ,i s 6N u m e r i c a l e x p e r i m e n t s

σ t = σ 0 exp Ä γ σ W σ t + γ h B h t ä = σ 0 exp Å γ σ W σ t + γ h B h t - 1 2 Ä γ 2 σ t + γ 2 h t 2h(t) ä ã . ( 5

Variance reduction method for the quantization-based cubature

Numerical experiments carried out in [START_REF] Corlay | Some aspects of optimal quantization and applications to finance[END_REF] showed that, in the case of vanilla options, computing the implied volatility using the estimated forward instead of the theoretical forward in the Black & Scholes formula improves the accuracy. The counterpart of this method in the frame of forward start options is to replace the "1"a p p e a r i n gi nF o r m u l a( V I I . 3 )b yt h eq u a n t i t y I N1,N2 :=

1 i N1, 1 j N2 p h i p σ j F i,j τ,T (6.1)
This also holds when using Richardson-Romberg extrapolation: in this case, one uses the extrapolated value of I N1,N2 instead of 1 in Formula (VII.3). These methods were used to generate the numerical results presented below.

Numerical results

We present results on the multifractional Hull & White mo del. We have computed the price as a function of strike for different maturities: 1, 2.5, 5 and 10 years. Driving noises were chosen in the class of fBms and mBms. More precisely, we display results of our experiments with:

1. An fBm with H =0.2.

2. An fBm with H =0.5.

3. An fBm with H =0.75.

4. An fBm with H =0.9.

5. An mBm with h = h 1 =0. [START_REF] Falconer | Multifractional, multistable, and other processes with prescribed local form[END_REF] The results displayed below provide an experimental justification to the claims made in the introduction. Indeed, one sees that, for the short maturity T =1year, in the fractional Hull & White model (i.e. with h constant), the smiles are more pronounced for small H and decrease as H increase, while the reverse is true for all maturities larger than one year (Figure 4.4). Thus, stronger correlations in the driving noise do translate in this model into a slower decrease of the smile as maturities increase, asn o t e di n[ 2 3 ] . H o w e v e r , with such an fBm-based model, an H larger than 1/2 is needed to ensure long-range dependence and thus am o r er e a l i s t i ce v o l u t i o no ft h es m i l ea sc o m p a r e dt ot h eB r o w n i a nc a s e . A sm e n t i o n e da b o v e ,t h i si sn o t compatible with empirical graphs of the volatility which show a very irregular behaviour, and would require as m a l lH.I na d d i t i o n ,t h el o c a lr e g u l a r i t yo ft h ev o l a t i l i t ye v o l v e si nt i m e ,c a l l i n gf o rav a r y i n gH, i.e. an mBm. Another aspect is that a fixed H,a si nam o d e l l i n gw i t hf B m ,d o e sn o ta l l o wt oc o n t r o li n d e p e n d e n t l y the shape of the smiles at different maturities. This is possible with mBm, where the smile at maturity T depends on a weighted average of the values of h up to time T ,a sc a nb ei n f e r r e df r o me q u a l i t i e s( 3 . 7 )a n d (4.3). This is apparent on Figure 4.5. We have compared fBms and mBms at various maturities T ,whereH and h are chosen such that h(t)=H,or,f ort h ebot t omri gh tp l ot ,h 1 (t)=h 4 (t).O n es e e st h a tt h es h a p eo f the smile depends on a weighted average of past values of h.F o ri n s t a n c e ,i nt h eb o t t o ml e f tp l o t ,t h ev a l u e s of h before T =2 .5 are in average smaller than 0.9,r e s u l t i n gi nafl a t t e rs m i l e . T h ef a c tt h a taweighted average must be considered is apparent on the bottom right plot: indeed, the smile is more pronounced for h 1 ,a l t h o u g ht h ea v e r a g ef r o m0 to 5 of this function is smaller than the one of h 4 .I nc o n t r a s t ,t h ev a l u e s in the immediate past of t =5are larger for h 1 than for h 4 ,a sm a yb ec h e c k e do nF i g u r e4 . 3 . A na d e q u a t e choice of h may thus allow one to better approximate a whole implied volatility surface. This topic will be addressed in a future work. Finally, we display on Figure 4. 6 Top left: fBm with H =0 .9 and mBm with function h 1 at T =5(h 1 (5) = 0.9). Top right: fBm with H =0.2 and mBm with function h 2 at T =5(h 2 (5) = 0.2). Bottom left: fBm with H =0.9 and mBm with function h 3 at T =2.5 (h 3 (2.5) = 0.9). Bottom right: mBm with function h 1 and mBm with function h 4 at T =5(h 1 (5) = h 4 (5) = 0.9).

VII Appendix

The implied forward start volatility

The forward start option price.

Let W be a standard Brownian motion on [0,T] and τ ∈ (0,T).L e tu sc o n s i d e rt h es t o c h a s t i cd i ff e r e n t i a l equation dS t = S t σ t dW t (with (σ t ) t∈ The implied forward start volatility.

In the Black & Scholes model, where the asset price follows a geometric Brownian motion with a constant volatility, the forward start Call (or Put) option price is an increasing function of the volatility (if the strike is not zero). Conversely, for a given forward start Call (or Put) option price, the Black & Scholes implied volatility is the unique value of the volatility for which the Black & Scholes formula recovers the price; in other words, the implied forward start volatility associated with a given forward F 0 ,af o r w a r ds t a r td a t eτ , am a t u r i t yT>τ ,as t r i k eK,a n da no p t i o np r i c eP is defined by P = FSPrimeBS (ImpliedFSVolBS (τ, T, K, P) ,τ,T,K) . We show that F i has bounded variations for every i in {1, 2, 3}.T h ec a s e so fF 1 , F 2 and F 3 are similar, and we only treat here F 1 .L e t(t i ) 0 i N be a sequence of elements of [0,T] such that 0=t 0 <t 1 < We hence have shown that

∀i ∈{1; •••; N },G i =:K7 (1 + T )( 1+2 H 2 )( 1+K 4 )( 1+K 6 ) |t i -t i-1 | + ti ti-1 Ä x 2H2-1 -x 2H1-1 ä dx .
(VIII.4) Using (VIII.3) and (VIII.4) we finally obtain

N i=1 |F 1 (t i ) -F 1 (t i-1 )| 2K 7 Å 1+ 1 2H 1 ã Ä T + T 2H1 + T 2H2 ä < +∞,
which ends the proof.

E[| W(A)| 2 ]=E[W 1 (A) 2 ]+E[W 2 (A) 2 ]=λ(A) and E[ W(A) 2 ]=E[W 1 (A) 2 ] -E[W 2 (A) 2 ]=0.
Consider the set of complex-valued functions h := h 1 + ih 2 defined on R such that h 1 and h 2 are real-valued functions which satisfy, for every real ξ , h 1 (ξ)=h 1 (-ξ), h 2 (ξ)=-h 2 (-ξ) and R h i (ξ) = R F (f )(ξ) W(dξ) for every f in L 2 R (du). Remark 1.1. Intuitively it seems natural to set I(h) := R (h 1 (ξ)+ih 2 (ξ))(W 1 (dξ)+iW 2 (dξ)) := R (h 1 (ξ)+ ih 2 (ξ))W 1 (dξ)+ R (ih 1 (ξ) -h 2 (ξ))W 2 (dξ).W eh e n c ew o u l dh a v e

I(h)= R h 1 (ξ)W 1 (dξ) - R h 2 (ξ)W 2 (dξ)+i Ä R h 2 (ξ)W 1 (dξ) =:R2,1 - R h 1 (ξ)W 2 (dξ) =:R1,2
ä .

However, since R-h 2 (ξ)W 1 (dξ)= R+ h 2 (-u)W 1 (-du)=-R+ h 2 (u)W 1 (du) and R-h 1 (ξ)W 2 (dξ)=

R+ h 1 (-u)W 2 (-du)=-R+ h 1 (u)W 2 (du) almost surely, we get that R 2,1 = R 1,2 =0almost surely and thus, finally, I(h)= R h 1 (ξ)W 1 (dξ) -R h 2 (ξ)W 2 (dξ),f o re v e r yω as stated the definition.

2S e c o n d m e t h o d : [ 7 5 ] a n d r e f e r e n c e s t h e r e i n

Let Ψ := (ψ λ ) λ∈Λ be an orthonormal basis of the Hilbert space L 2 R (du) with inner product <, > defined by <f ,g> := R f (u) g(u) du for all (f, g) in L 2 R (du) × L 2 R (du).F o re x a m p l ew ec a nt a k eΨ to be a wavelet basis, namely ψ λ (u) := 2 -j/2 ψ(2 -j u -k),w h e r eλ =(j, k), j, k belongs to Z and where the "mother wavelet "ψ is such that Ψ is an orthonormal basis. The image of the space L 2 R (du) with respect to the Fourier transform is nothing but the space " L 2 C (dξ) defined at the begining of section 1. Moreover, for any h and l in " L 2 C (dξ),d e n o t e<h ,l> := R h(ξ)l(ξ) dξ the corresponding inner product. Now, for any f in L 2 R (du) and h in " L 2 C (dξ),d e fi n et h es t o c h a s t i ci n t e g r a l s

I (f )= R f (u) W (du) := λ∈Λ <f,ψ λ >Z λ , (2.1)
J (h)= R h(ξ) W (dξ) := λ∈Λ <h, ψ λ >Z λ , (2.2)
where Z := (Z λ ) λ∈Λ is some fixed collection of independent and identically distributed standard Gaussian random variables on (Ω, F , P).T h e s es e r i e sc o n v e r g ei nt h eL 2 -sense, for all f in L 2 R (du) and h in " L 2 C (dξ). Moreover one has, for all f and g in L 2 R (du), E[I (f )I (g)] = <f,g> = < f, g> = E[J ( f )J ( g)],

(

2.3)

There is a "stochastic Parseval identity "relating the stochastic integrals R f (u) W (du) and R h(ξ) W (dξ) in ( 2

3L i n k s b e t w e e n t h e t w o m e t h o d s

From second to first metho d

Starting from the second method framework, it is easy to build gaussian measures W 1 , W 2 and W on R, satisfying the assumptions of the first method and hence recover the integral I.M o r ep r e c i s e l y ,w ec o n s i d e r I 1 and I 2 as in (2.1) as well as J 1 and J 2 as in (2.2) such that, for every i in {1; 2}

I i (f )= R f (u) W i (du) := λ∈Λ <f,ψ λ >Z (i) λ , J i (h)= R h(ξ) " W i (dξ) := λ∈Λ <h, ψ λ >Z (i) λ ,
where Z (1) := (Z

λ ) λ∈Λ and Z (2) := (Z

λ ) λ∈Λ are two fixed independent collections of independent and identically distributed standard Gaussian random variables on (Ω, F , P). We then define the three maps W 1 , W 2 and W from B h∈ "

L 2 C (dξ)
have the same distribution. In particular, R h(ξ) W (dξ)

(d) = R h(ξ) W(dξ).
Proof. It is clear that the process W 1 := {W 1 (A); A ∈B (λ 1/4 ) 0

} is gaussian. Moreover, for every A in B (λ 1/4 ) 0 ,w eh a v eW 1 (A)

N (0; 1/4||1 A + 1 -A +( √ 2 -2)1 A∩(-A) || 2 L 2 (R) ) and hence W 1 (A) N (0; λ(A) 2 ) since ||1 A + 1 -A +( √ 2 -2)1 A∩(-A) || 2 L 2 (R) =2λ(A). Moreover, for every A 1 ,A 2 , ••• ,A k in (B (λ 1/4 ) 0
) k such that the A i are disjoints and all in R + ,t h ei d ependence of W 1 (A 1 ), W 1 (A 2 ), ••• , W 1 (A k ) results from the fact that E[W 1 (A i )W 1 (A j )] = 1 4 < 1 Ai + 1 -Ai , 1 Aj + 1 -Aj >=0 .F i n a l l y ,t h ee q u a l i t y ,i nL 2 (Ω, F , P),b e t w e e nW( ∪

  Ecole doctorale de l'Ecole Centrale Paris Grande Voie des Vignes 92 295 Châtenay-Malabry Cedex École doctorale Paris centre Case 188 4p l a c eJ u s s i e u 75 252 Paris cedex 05 iii Je dédie cette thèse à ma mère qui m'a appris à ne pas céder sur mon désir et à la mémoire de mon père qui m'a initié à l'étrange beauté des mathématiques. r o d u c t i o n 1 1L e m o u v e m e n t b r o w n i e n f r a c t i o n n a i r e ( f B m ) . . . . . . . . . . . . . . . . . . . . . . . . . . .1 2L e m o u v e m e n t b r o w n i e n m u l t i f r a c t i o n n a i r e ( m B m ) . . . . . . . . . . . . . . . . . . . . . . . .6 3P r é s e n t a t i o n d e s t r a v a u x d e t h è s e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2W h i t e N o i s e -b a s e d S t o c h a s t i c C a l c u l u s w . r . t . m B m 1 7 1B a c k g r o u n d a n d M o t i v a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 2W h i t e n o i s e t h e o r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 2.1 White noise measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Properties of Hermite functions and space S ′ (R) ..................... 2 1 2.3 Space of Hida distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 S-transform and Wick product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3T h e o p e r a t o r s M H and their derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Study of M H ......................................... 2 5 3.2 Study of ∂MH ∂H ......................................... 2 8 4W i e n e r i n t e g r a l w i t h r e s p e c t t o m B m o n R ............................ 2 9 4.1 Wiener integral with respect to fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Wiener integral with respect to mBm . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5S t o c h a s t i c i n t e g r a l w i t h r e s p e c t t o m B m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 5.1 Fractional White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Multifractional White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.3 Generalized functionals of mBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.4 S-Transform of mBm and multifractional white noise . . . . . . . . . . . . . . . . . . . 37 5.5 The multifractional Wick-Itô integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.6 Multifractional Wick-Itô integral versus Wiener integral with respect to mBm . .. . . 40 6I t ô F o r m u l a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 6.1 Itô Formula for generalized functionals of mBm on an interval [a, b] with 0 <a<b .. 4 3 6.2 Itô Formula in (L 2 ) ..................................... 4 4 7T a n a k a f o r m u l a a n d e x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 8 7.1 Tanaka formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.2 Itô formula for functions h such that d dt [R h (t, t)] = 0 ................... 4 9 7.3 Itô formula for functions h such that d dt [R h (t, t)] = 1 ................... 4 9 8C o n c l u s i o n a n d f u t u r e w o r k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0 9A c k n o w l e d g m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 XA p p e n d i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 X.A Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 X.B Proof of proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 xi Chapter 1 Introduction Contents 1L e m o u v e m e n t b r o w n i e n f r a c t i o n n a i r e ( f B m ) . . . . . . . . . . . . . . . . . . . .1 2L e m o u v e m e n t b r o w n i e n m u l t i f r a c t i o n n a i r e ( m B m ) . . . . . . . . . . . . . . . .6 3P r é s e n t a t i o n d e s t r a v a u x d e t h è s e . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1L e m o u v e m e n t b r o w n i e n f r a c t i o n n a i r e ( f B m )

Figure 1 . 1 -Figure 1 . 2 -

 1112 Figure 1.1 -Trajectoire d'un fBm de paramètre de Hurst H =0.3

Figure 1 . 3 -

 13 Figure 1.3 -Trajectoire d'un fBm de paramètre de Hurst H =0.8

Dépendance de long terme Définition 1. 2 .

 2 On dit qu'une suite de variables aléatoires stationnaires (Y n ) n∈N présente une dépendance de long terme si la fonction d'autocovariance ρ : N → R,d éfini ep a rρ(n) := Cov(Y k ,Y k+n ) satisfait la condition lim n→+∞ ρ(n) cn -α =1, (1.3) pour une certaine constante c et un certain réel α de ]0; 1[. Dans ce cas la dépendance entre Y k et Y k+n décroît lentement lorsque n tend vers l'infini. On a de plus +∞ n=1 ρ(n)=+∞.

  Hurst H, le processus défini, à une constante dépendant de H près, par ‹ Y (t) := R (e itξ -1) |ξ| H+1/2 W (dξ), qui n'est autre que √ 2π Γ(H+1/2) ‹ Y (1,1) (t) pour δ(H) := 1.E n fi n ,onap pel l erep ré s e n t at i onàm o y e n n em ob i l ed u fBm le processus Y (1,0) défini par (1.6) pour (a + ,a -)=(1, 0) et δ(H) := 1.O nad o n c ,

  H-1/2 dv si 1/2 <H<1, et où, pour tout H de (0, 1),α H := Ä 2H (1-2H)β(1-2H,H+1/2) ä 1/2 .Calcul stochastique par rapport au fBm L'indice de variation du fBm d'indice de Hurst H est défini par,I(B H , [0,T]) := inf{p>0,ϑ p (B H , [0,T]) < +∞}, où l'on a défini ϑ p (B H , [0,T]) := sup π n i=1 |B H t k -B H t k-1 | p et où π := {0=t 0 <t 1 <t 2 < ••• <t n = T }désigne une partition finie de [0,T].D a n s[ 7 2

Figure 1 . 4 -Figure 1 . 5 -

 1415 Figure 1.4 -Trajectoire d'un mBm de paramètre fonctionnel h 1 : t → 0, 1+0, 8t

Définition 2 .Figure 1 . 6 -

 216 Figure 1.6 -Trajectoire d'un mBm de paramètre fonctionnel h 3 : t → 0, 3+0, 3(1 + e -100(t-0,7) ) -1

Proposition 2 . 4 .

 24 ([75, corollaire 6.1]) Soient (a + ,a -) et (b + ,b -) deux couples de réels. On suppose que |b + | = |b -|.C o n s i d é r o n sY (a + ,a -) et Y (b + ,b -) deux mBm de même paramètre fonctionnel h : R → (0, 1).O n al e sr é s u l t a t ss u i v a n t s (a) Si a + b -= a -b + , alors pour tout réel t et s dans R * ,o na

  pour un certain ε strictement positif. En particulier, les mBm Y hχ (a + ,a -) et Y hχ (b + ,b -) de paramètre fonctionnel h χ ne peuvent pas se déduire l'un de l'autre par simple multiplication d'une fonction déterministe de t.

. 8 )

 8 Ap a r t i rd ' e x e m p l e sd ' i n t é g r a l e sd ep r oc e s s u ss i m p l e sp a rr a p p o r tàu n es o m m ed ef B me te nf a i s a n tt e n d r e n vers +∞ dans (3.8) on s'aperçoit qu'une définition raisonnable de l'intégrale stochastique d'un processus Y par rapport au mBm, au sens de la méthode (M) est nécessairement de la forme :

  et appartient à F , où B α désigne un fBm de coefficient de Hurst α. Le critère suivant permet de savoir, pour une méthode d'intégration par rapport au fBm donnée et notée (M),s il as u i t ed ef o n c t i o n n e l l e s(L n ) n∈N ,d é fi n i ee n3 . 9 ,c o n v e r g es i m p l e m e n ts u rF tout entier. Théorème 3.4. Soit (a n ) n∈N une suite croissante d'entiers strictement positifs telle que 2 n 0 k n-1 a k 2 2 n pour tout n in N et telle que lim n→+∞ (n(a n -1)) (

  (t, t)] = 1 ................. 8C o n c l u s i o n a n d f u t u r e w o r k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9A c k n o w l e d g m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XA p p e n d i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X.A Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 X.B Proof of proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1B a c k g r o u n d a n d M o t i v a t i o n s

Definition 2 . 1 .

 21 Let (|| p ) p∈Z be the family of norm defined by

2 0 := ||g|| 2 L 2

 222 (R) for any g in L 2 (R).F o r p in Z,l e tA p denote the p th iteration of the operator A,i fp belongs to N,a n do fA -1 otherwise. Then Dom(A p )=S p (R) and Dom(A -p )=L 2 (R) where Dom(U ) denotes the domain of the operator U and p belongs to N.M o r e o v e r ,f o re v e r yq in Z and every

+∞ n=0 In! |A ⊗n f n | 2 0

 n=02 n (f n ) satisfying the condition +∞ n=0 < +∞,d e fi n et h ee l e m e n tΓ(A)(Φ) of (L 2 ) by Γ(A)(Φ) := +∞ n=0

2 0 := ||ϕ|| 2 (L 2 )Definition 2 . 2 .

 22222 for any random variable ϕ in (L 2 ) and, for n in N,l e tDom(Γ(A) n ) be the domain of the n th iteration of Γ(A).T h es p a c eo fH i d ad i s t r i b u t i o n si sd e fi n e di naw a ya n a l o g o u st o the one that allowed to define the space S ′ (R): Define the family of norms (|| || p ) p∈Z by:

p∈N(

  S p ) (resp. (S) * = ∪ p∈N (S -p ))

Remark 2 . 3 . 2 n 2 n 2 -

 23222 Ac o n s e q u e n c eo ft h ep r e v i o u ss u b s e c t i o ni st h a tf o re v e r ye l e m e n tf := +∞ n=0 a n e n in S ′ (R) where (a n ) n∈N belongs to R N ,t h e r ee x i s t sp 0 in N such that f belongs to S -p0 (R).M o r e o v e r i f we define Φ := +∞ n=0 a n < ., e n >,t h e nΦ belongs to (S -p0 ) ⊂ (S) * and we have ||Φ|| 2 -p0 = |f | 2 -p0 = +∞ n=0 a (2n+2) 2p 0 < +∞.C o n v e r s e l y ,e v e r ye l e m e n tΦ, written as Φ := +∞ n=0 b n < ., e n > where (b n ) n∈N belongs to R N ,b e l o n g st o(S) * if and only if there exists an integer p 0 such that +∞ n=0 b (2n+2) 2p 0 < +∞. In this case the element f := +∞ n=0 b n e n belongs to S -p0 (R) and hence is a tempered distribution which verifies |f |

  and w g H (0) := 0.D e fi n et h et e m p e r e d distribution v g H by v g H := 1 2π ˇ w g H ,w h e r ef o ra l lt e m p e r e dd i s t r i b u t i o nT ,b yd e fi n i t i o n ,< Ť,f >:=<T, f> for all functions f in S (R) and where f

  a s s u m i n gt h i si spo s s i b l e . B yd o i n g so, we define a new operator, denoted ∂MH ∂H ,f r o mac e r t a i ns u b s e to fL 2 H (R) to L 2 (R).O fc o u r s e ,i no r d e rt o compute the derivative at H 0 of H → ◊ M H (u)(y),w en e e dt oc o n s i d e ran e i g h b o u r h o o dV H0 of H 0 in (0, 1) and thus consider only elements u which belong to

  h(s) and H 2 := max s∈[a,b] h(s).T h i sy i e l d st h es e c o n dc o n d i t i o no ft h e o r e m2 . 8a n ds h o w s that

  t 0 ||τ n (s)τ m (s)|| -p0 ds t 0 +∞ k=m+1 f (s) d ds [g e k (s, h(s))]< ., e k > -p0 ds = t 0 ||τ (s)-τm(s)|| -p 0 ds M Ü +∞ k=m+1 sup s∈[0,t]

  yt h e o r e m5 . 2 5 ,Φ f = +∞ k=0 < f, M H (e k ) >< ., e k > where the equality holds in (S)

Theorem 6 . 1 .

 61 Let p ∈ N, a and b two reals with 0 <a<b,a n dl e tF be an element of C 1 ([a, b], S -p (R)) such that both maps ∂F ∂x and ∂ 2 F ∂x 2 ,f r o m[a, b] into S -p (R),a r ec o n t i n u o u s .T h e nt h ef o ll o w i n ge q u a l i t yh o l d s in (S) * :

Lemma 6 . 5 .

 65 Let f : R → R be a continuous function such that there exists a couple (C, λ) in R × R + with |f (y)| Ce λy 2 ,f o ra l lr e a ly.L e tg : R → R * + be a measurable function such that lim t→0 g(t)=0and define L f on R * + × R by L f (u, x) := R f (y)γ(u, x -y) dy.T h e n lim (t,x)→(0 + ,x0)

2 |η| 2 0

 22 f o ra l lt in [0,b].S i n c e || : e <.,η> : || 0 = e 1 ,C a u c h y -S c h w a r zi n e q u a l i t yy i e l d s ,f o ra l lt in [0,b] and η in S (R)

3 )

 3 ,η> : || 0 (1 + M ) It then just remains to show that lim t→0+ S[F (B (h) (t))](η)=F (0).T h a n k st ot h e o r e m5 . 1 2a n dl e m m a6 . 5 , we get

  S) * ,l e tu su s et h e o r e m2 . 8 . F r o mt h e o r e mdefinition 2.2, we know that F ′ (B (h) (t)) W (h) (t)) belongs to (S) * for every t in (0,b].M o r e o v e r u s i n g lemma 2.5 we get, for η ∈ S (R) and t ∈ (0,b],

  6 )a n dt h ed o m i n a t e dc o n v e r g e n c et h e o r e m ,i ti se a s yt os h o wt h a t lim n→+∞ S(Ψ n )(η)= 0.H e n c et h e o r e m8.6 of

  Note that the "local time" part disappears in this equality. Solving the differential equation d dt [R h (t, t)] = 0 yields two families of functions, denoted (h 1,λ ) λ∈R * + and (h 2,λ ) λ∈R * -: ∀λ>0,h 1,λ :( -∞, -e λ ) ∪ (e λ , +∞) → (0, 1) t → λ ln |t| , and ∀λ<0,h 2,λ :( -e λ , 0) ∪ (0,e λ ) → (0, 1) t → λ ln |t| .

  Figures 1 and 2 display examples of mBm with functions h 1 (t) := 1 ln t defined on [e +10 -3 , 100] and h 2 (t) := -1 ln t defined on [10 -3 , 1/e -10 -2 ].

Figure 2 .

 2 Figure 2.1: t → B (h1) (t) with h 1 (t) := 1 ln t on [e +10 -3 , 100].Figure 2.2: t → B (h2) (t) with h 2 (t) := -1 ln t on [1.10 -3 , 1/e -10 -3 ].

Figure 2 . 2 :

 22 Figure 2.1: t → B (h1) (t) with h 1 (t) := 1 ln t on [e +10 -3 , 100].Figure 2.2: t → B (h2) (t) with h 2 (t) := -1 ln t on [1.10 -3 , 1/e -10 -3 ].

Figures 3 2 +10 - 3 ,

 323 Figures 3 and 4 display examples of multifractional Brownian motion with regularity functions h 3 (t) := 1 2 ln(t-1) ln t defined on [2 + 10 -3 , 5] and h 4 (t) := 1 2

Figure 2 . 3 :

 23 Figure 2.3: t → B (h3) (t) with h 3 (t) := 1 2 ln(t-1) ln t on [2 + 10 -3 , 5].Figure 2.4: t → B (h4) (t) with h 4 (t) := 1

Figure 2 . 4 :+10 - 3 ,

 243 t → B (h4) (t) with h 4 (t) := -10 -3 ]. Note that the case c =0yields the constant function h c ≡ 1/2, i.e. standard Brownian motion. Moreover, since lim t→+∞ h c (t)=1 /2 for every c,w es e et h a tt h ef a m i l yo ff u n c t i o n sh c behaves, asymptotically, like the constant function equal to 1/2.H o w e v e rt h i sd o e sn o tm e a nt h a tt h e r ei sc o n v e r g e n c ei nl a wo fB (hc) to Brownian motion; in fact one needs to scale B (hc) .F o re v e r yt in R * ,d e fi n et h ep r o c e s sX t on R + by X t (u) := B (hc ) (tu) √ t .T h e n {X t (u); u ∈ R + } L ----→ t→+∞ {B(u); u ∈ R + } where B still denotes a Brownian motion and L denotes convergence in law.

Theorem X. 2 .

 2 Let p ∈ N and (Φ n ) n∈N be a sequence of processes from I to (S) * such that Φ n (u) ∈ (S -p ) for almost every u ∈ I and for every n.A s s u m em o r e o v e rt h a tΦ n is Bochner-integrable on I for every n and that lim (n,m)→(+∞,+∞) I ||Φ m (s) -Φ n (s)|| -p ds =0. Then there exists an (S) * -process (almost surely (S -p )-valued), denoted Φ,d e fi n e da n dB o c h n e r -i n t e g r a b l e on I such that lim n→+∞ I ||Φ(s) -Φ n (s)|| -p ds =0. (X.1)

  p in the sequel. The sequence (x (n) t ) n∈N is increasing and converges to t as n tends to +∞.B e s i d e s ,d e fi n ef o re v e r yn in N,t h ef u n c t i o nh n :[a, b] → (0, 1) by setting h n (b)=h(b) and, for any t in [a, b), h n (t) := h(x (n) t ).T h es e q u e n c eo ff u n c t i o n s(h n ) n∈N converges pointwise to h on [a, b]. Define, for t in [a, b] and n in N,t h ep r oc e s s

( i )

 i We pro ceed as usual in two steps (see for examples [16, 71]), a):fi n i t e -d i m e n s i o n a lc o n v e r g e n c ea n db): tightness of the sequence of probability measures (P • B hn ) n∈N . a) Finite dimensional convergence Since the process B h and, for every n in N,t h ep r o c e s sB hn defined by (2.1) are centered and Gaussian it is sufficient to prove that lim n→∞ E î (s, t) in [a, b]

  be the unique element of the partition C (i) p which contains H.T h ed i a m e t e r of C (i) p (H) is by definition ∆ i (C (i) p (H)) := sup (H,H ′ )∈Cp(H) 2

. 10 )

 10 Using (2.7) and (2.10), inequality (2.5) then becomes J τ,m n <εfor every n N ,w h i c he n d st h ep r o o f .

t and hence 1 0

 1 Y t d (M) B hn t will belong to L 2 (Ω) if (M) denotes the Skohorod integral, whereas 1 0 Y t d (M) B H t and hence 1 0 Y t d (M) B hn t belong to the space (S) * of stochastic distributions when (M) denotes the integral in the sense of white noise theory.

. 13 )

 13 Inequalities (3.12) and(3.13) show that (J (1) n ) n∈N tends to 0 in L 2 (Ω) which ends the proof. Remark 3.3. (i) One can also show that the sequence ( 1 0 1 dB hn t dt) n∈N converges almost surely. (ii) Proposition (3.

  k+1 ] for every n m.D e fi n en o wS n := 1 0 Y t d (M1) B hn t for every ineteger n in N.U s i n g( 3 . 5 )w eg e t ,f o re v e r yn m,

  soon as Y is integrable w.r.t. fBm of Hurst index H in sense of M 2 .T h u si fw ed e fi n eT n := 1 0 Y t d (M2) B hn t for every n in N,w eh a v eT n := 1 0 Y t d (M2) B hn t = S n .K e e p i n gt h es a m en o t a i o n sa si nt h ep r e v i o u se x a m p l e , in particular for the simple process Y := (Y t ) t∈[0,1] defined by Y t := qm-1 i=0

  and belongs to F , where B α denotes the fractional Brownian motion of Hurst index α.D e n o t ea l s oH E = α∈h([0,1]) H α E . It is of practical interest to get a criterion which allow us to know when a process Y of H α E is such that its integral w.r.t mixed fBm (definition 3.1) converges in F .T h i st h ep u r p o s eo ft h en e x ts u b s e c t i o n . Ag e n e r a lc r i t e r i o nf o rt h ec o n v e r g e n c eo ft h es e q u e n c e( [0,1] Y t dB hn t ) n∈N We will always assume that there exists a subset Λ E of H E (maybe equal to H E )w h i c hm a yb ee n d o w e d with a norm ΛE such that (Λ E , ΛE ) is complete and which satisfies the following property: there exists M>0 and a real χ such that for all partitions of [0, 1] in intervals A 1 ,...,A n of equal size 1 n , Y. A1 ΛE + •••+ Y. An ΛE Mn χ Y ΛE . (3.22)

n+1 ) t å .

 å Using assumptions (3.22) and (3.24), one obtains

Definition 3 . 2 (

 32 Integral with respect to mBm in sense of (M)). Assume (M) fulfills condition (3.24) and let

2 L 2 ( 2 0

 222 +∞ n=0 I n (f n ), where L 2 (R n ) denotes the set of all symmetric functions f in L 2 (R n ) and I n (f ) denotes the nth multiple Wiener-Itô integral of f with the convention that I 0 (f 0 )=f 0 for constants f 0 .M o r e o v e r w e h a v e t h e equality E[Φ 2 ]= +∞ n=0 n! ||f n || R n ) where E denotes the expectation with respect to µ.F o r a n yΦ := +∞ n=0 I n (f n ) satisfying the condition +∞ n=0 n! |A ⊗n f n | < +∞,d e fi n et h ee l e m e n tΓ(A)(Φ) of (L 2 ) by Γ(A)(Φ) := +∞ n=0

  nt h eF o u r i e rd o m a i n ,b y : ÿ ∂MH ∂H (u)(y) := -(β H + ln |y|) √ 2π cH |y| 1/2-H u(y),f o re v e r yy in R * .T h er e a d e ri n t e r s t e di nt h ep r o p e r t i e so f M H and ∂MH ∂H can refer to [53, section 3.2].

  4 )d oe si n d e e db e l o n gt o(S) * and is exactly the (S) * -derivative of B h at point t of [0, 1].

Remark 5 . 3 .

 53 1 ([0, 1],dt).M o r e o v e r ,e 2 D(r 0 -p 0 ) < 1,f o re v e r yr 0 p 0 +2.T h e o r e m5 . 1t h e na l l o w st oc o n c l u d et h a tt → Y t W H t is Bochner integrable of index r 0 . Of course if we make more assumptions on the (S) * -process Y they may lead to [0,1] Y s d B H s belongs to (L 2 ).S e e[ 5 3 ,s e c t i o n s6 and 7]f o rs o m ee x a m p l e s .

Lemma 5 . 6 .

 56 (M 1 ) fulfills conditions (3.22) and (3.24). Lemma 5.7. for every process Y

  dt. Since L(Y ) and M (Y ) both belong to (S -s0 ),i ti ss u ffi c i e n tt os h o wt h a tt h e yh a v et h es a m eS-transform. Let η be fixed in S (R),u s i n g[ 5 2 ,t h e o r e m8 . 6 ]w eg e t S(L(Y ))(η) = lim n→∞ S

  cH .L e tτ be fixed in (0,c) and M := e ln 2

  ed e n o t eb yE N,p (X, |•|) the minimal L p quantization error for the random variable X and the norm |•|: E N,p (X, |•|) = min ¶ |X -Y | p ,Ymeasurable with respect to X and |Y (Ω)| N ©

  o t h e rp o s s i b i l i t yi st o use a product quantization, that is to use the Cartesian product of the optimal quadratic quantizers of the standard one-dimensional Gaussian distributions N Ä 0,λ X j ä 1 j d X (N )

. 3 )

 3 As o l u t i o no f( 2 . 3 )i sc a l l e da no p t i m a lK -Lp r o d u c tq u a n t i z e r . T h i sp r o b l e mc a nb es o l v e db yt h e" b l i n d optimization procedure", which consists in computing the criterion for everypossibledecompositionN1 ו • •× N d with N 1 ••• N d .T h er e s u l to ft h i sp r o c e d u r ec a nb es t o r e df o rf u t u r eu s e .O p t i m a ld e c o m p o s i t i o n s for a wide range of values of N for both Brownian motion and Brownian bridge are available on the web site www.quantize.maths-fi.com[START_REF] Pagès | W ebsitedevotedto optimal quantization[END_REF]. Another fact on quadratic functional product quantization is that it is shown to be rate-optimal under certain assumptions on the K-L eigenvalues (see Theorem 2.1). Quadratic product quantizers of fBms and well-balanced mBms for different H and h are displayed on Figures 4.1 and 4.2. A fixed product decomposition is used for simplicity.

8 Figure 4 . 1 :

 841 Figure 4.1: Quadratic 5 × 2 × 2-product quantizer of fBm on [0, 1] with H =0.25 (left) and H =0.75 (right).

8 Figure 4 . 2 :

 842 Figure 4.2: Quadratic 5 × 2 × 2-product quantizer of mBm on [0, 1] with h(t) := 0.1+0 .8t (left) and h(t) := 0.9 -0.8t (right).

t

  close to 1,t h ec o r r e s po n d i n gm B mh a sl a r g e rl oc a lH a u s d o r ffd i m e n s i o n . S i m i l a rr e m a r k sh o l df o rt h ec a s e where h is an increasing function (right pane of Figure 4.2).

  i n a l l y ,x n n→∞ y n means that x n = O(y n ) and y n = O(x n ) as n →∞.Sharp rates based on asymptotics of Karhunen-Loève eigenvaluesRecall the following well-known definition: Definition 2.1 (Regularly varying function at infinity). Am e a s u r a b l ef u n c t i o nφ :(s, ∞) → (0, ∞),( s>0) is regularly varying at infinity with index b ∈ R if for every t>0, lim x→∞ φ(tx) φ(x) = t b .

  ) = t b .

.

  Define the map φ : C([0,T], R) → C([0,T], R) by φ(f ) := T 0 e 2p p|f (s)| ds.I ti se a s i l ys h o w nt h a tφ is convex and continuous on (C([0,T], R), • ∞ ).H e n c e ,I n e q u a l i t y( 2 . 9 )y i e l d s

2 2

 2 and E D 2 F (Y N )(X -Y N ) ⊗2 has been established yet. Still, Equation (2.11) suggests to use a Richardson-Romberg extrapolation with respect to the quantization error E 2 N := X -Y N 2 .T h et w o -s t e p se x t r a p o l a t i o nb e t w e e nN = k and N = l leads to approximate E[F (X)] by the quantity E

+∞ n=0 In! f n 2 L 2 ( 2 0

 n=0222 n (f n ),w h e r e L 2 (R n ) denotes the set of all symmetric functions f in L 2 (R n ) and I n (f ) denotes the nth multiple Wiener-Itô integral of f with the convention that I 0 (f 0 )=f 0 for constants f 0 .M o r e o v e rw eh a v et h ee q u a l i t yE[Φ 2 ]= +∞ n=0 R n ) where E denotes the expectation with respect to µ.F o r a n y Φ := +∞ n=0 I n (f n ) satisfying the condition +∞ n=0 n! |A ⊗n f n | < +∞,d e fi n et h ee l e m e n tΓ(A)(Φ) of (L 2 ) by Γ(A)(Φ) := +∞ n=0

  r→0r -1 (Φ(t 0 + r) -Φ(t 0 )) exists in (S)* .M o r e o v e r we may also define an integral of an Hida process:Theorem 3.1 (Integral in (S) * ).Assume that Φ:R → (S) * is weakly in L 1 (R,dt),i . e . a s s u m et h a tf o ra l l ϕ in (S),t h em a p p i n gu → Φ(u),ϕ from R to R belongs to L 1 (R,dt).T h e n ,t h e r ee x i s t sau n i q u ee l e m e n t in (S) * ,d e n o t e db y R Φ(u)du,s u c ht h a t ),ϕ du for all ϕ in (S). (3.1) One says that Φ is (S) * -integrable on R or integrable on R in the Pettis sense. For every f in L 2 (R),d e fi n et h eWick exponential of •,f ,n ot ed: e •,f :,ast h e(L 2 ) random variable equal to e •,f -1 2 |f | 2 0 .T h eS-transform of an element Φ of (S * ),n o t e dS(Φ),i sd e fi n e dt ob et h ef u n c t i o nf r o m S (R) to R given by S(Φ)(η) := ¨¨Φ, : e •,η ∂∂ for every η in S (R).F i n a l l yf o re v e r y(Φ, Ψ) ∈ (S) * × (S) * , there exists a unique element of (S) * ,c a l l e dt h eW i c kp r o d u c to fΦ and Ψ and noted Φ Ψ,s u c ht h a t S(Φ Ψ)(η)=S(Φ)(η) S(Ψ)(η);f o re v e r yη in S (R). The map S :Φ → S(Φ),f r o m(S) * to (S) * ,i si n j e c t i v e . F u r t h e r m o r e ,l e tΦ:R → (S) * be a fixed (S) *

For

  any Φ in (S) * and k in N * ,letΦ k denote the element k times Φ ••• Φ of (S) * .O n ec a ng e n e r a l i z et h ed e fi n i t i o n of exp to the case where Φ belongs to (S) * .I n d e e d ,f o ra n yΦ in (S) * such that the sum +∞ k=0 Φ k k! converges in (S) * ,d e fi n et h ee l e m e n texp Φ of (S) * by setting exp Φ := +∞ k=0 Φ k k! .I ti sc a l l e dW i c ke x p o n e n t i a lo fΦ.

  c([β] Lip ,T) g 1 -g 2 p p Ψ β p (g 1 ) -Ψ β p (g 2 ) p p C([β] Lip ,T) g 1 -g 2 p p , with c([β] Lip ,T)= 1 2 p-1 (1-[β] p Lip T p ) and C([β] Lip ,T)=e 2 p-1 [β] Lip T p-1 .

FT Fτ -K ä +FT

 ä for some fixed maturity τ ∈ [0,T].W en e e dt oc o m p u t et h e risk-neutral expectation E Ä

=T τ σ s dW σ s -ρ 2 2 T τ σ 2 σ 2 s+ γ σ t 0 e

 2220 E PrimeBS F τ,T , (1ρ 2 ) 1 τ, K , (4.4) where F τ,T := exp Ä ρ s ds ä and PrimeBS is the closed-form expression for the price of a Call option in the Black & Scholes model, detailed in Appendix VII. The aim is to estimate the expectation (4.4) by a quantization-based cubature associated with the functional quantization of B h and W σ .W et h u s need to write the terms F τ,T and T τ σ 2 s ds as explicit functionals of the paths of W σ and B h in L 2 ([0,T]). Recall that σ is the exponential of a mixed multifractional Ornstein-Uhlenbeck process: σ t =exp Ä ln(σ 0 )e -θt + µ Ä 1 -e -θt ä + γ σ e -θt I e θ• t (W σ )+γ h e -θt I e θ• t (B h ) ds as a function of the paths of W σ and B h .D e n o t e (p h j ) 1 j N1 and (χ h j ) 1 j N1 the weights and the paths of the quantizer " B h of B h ,a n d(p σ j ) 1 j N2 and (χ σ j ) 1 j N2 the weights and the paths of the quantizer W σ of W σ .C o n d i t i o n a l l yo nB h = χ h i ,o n eh a s I σ τ,T = I σ i τ,T ,w h e r e I σ i τ,T := =exp ln(σ 0 )e -θt + µ Ä 1 -e -θt ä θ(s-t) dW σ s + γ h e -θt I g t

. 3 )

 3 Reasoning as in the case of the Hull & White model presented in Section 4, it can be shown that F is an F F,σ,h -martingale for ρ =0.I na d d i t i o n ,t h es a m en u m e r i c a lp r o c e d u r e sa sa b o v em a yb eu s e d .

3 Figure 4 . 3 :

 343 Figure 4.3: left: functions h 1 and h 4 ;m i d d l e : f u n c t i o n h 2 ;r i g h t : f u n c t i o n h 3 .

8 . 5 Ä t + 5 4 ää +0. 7 .

 8547 An mBm with h = h 4 = -0.2 sin Ä 6π The four functions are plotted on Figure 4.3. The values of the other parameters are γ h =0 .3, γ σ = ρ =0 (except for the experiments displayed on Figure 4.6), θ =0.3, µ = ln(0.2), σ 0 =0.2 and F 0 =100.

Figure 4 . 5 :

 45 Figure 4.5: Comparisons of vanilla option volatility smiles for various fBm and mBm at several maturities.Top left: fBm with H =0 .9 and mBm with function h 1 at T =5(h 1 (5) = 0.9). Top right: fBm with H =0.2 and mBm with function h 2 at T =5(h 2 (5) = 0.2). Bottom left: fBm with H =0.9 and mBm with function h 3 at T =2.5 (h 3 (2.5) = 0.9). Bottom right: mBm with function h 1 and mBm with function h 4 at T =5(h 1 (5) = h 4 (5) = 0.9).

Figure 4 . 6 :

 46 Figure 4.6: Vanilla option volatility smiles in the multifractional Hull & White model, with γ h =0 .3, γ σ =0 .3 ρ = -0.5, θ =0 .3, µ = ln(0.2), σ 0 =0 .2 and F 0 =1 0 0 ,a n dh = h 2 for maturities T =1 , T =2 .5 and T =5.

√ T -τ 2 +√ T -τ 2 and σ 2 := 1 T -τ T τ σ 2 s

 22212 [0,T ] ad e t e r m i n i s t i cp r o c e s s )w h o s es o l u t i o ni sag e o m e t r i cB r o w n i a n motion S t = S 0 exp Ä t 0 σ s dW s -ef o r w a r ds t a r tC a l lo p t i o np r i c eF S P r i m e B S (σ, τ, T , K ) is given by FSPrimeBS(σ, τ, T , K )=E Å S T S τ -K ã + = N (d 1 ) -KN (d 2 ), where d 1 := σ ln(K) σ √ T -τ , d 2 := d 1 -σ ds .I no t h e rw o r d s ,w eh a v e FSPrimeBS(σ, τ, T , K )=PrimeBS(1,σ,Tτ, K), (VII.1) where (S 0 , Vol, Mat, Strike) → PrimeBS(S 0 , Vol, Mat, Strike) is the closed-form expression for the vanilla Call option price in the Black & Scholes model.

(VII. 2 )

 2 Using Equation (VII.1), this yields ImpliedFSVolBS (τ, T, K, P)=ImpliedVolBS (1,Tτ, K, P) ,(VII.3)where ImpliedVolBS (Fwd, maturity, Strike, Price) is the Black & Scholes implied volatility a certain forward, maturity, strike and option price.VIII Variations of the Karhunen-Loève eigenfunctions of mBmLet R h denote the covariance function of a normalized mBm B h with functional C 1 parameter h and e h k be the kth Karhunen-Loève eigenfunction of B h .F o rk in N,d e fi n et h em a pI k : [0,T] → R by I k (t) := T 0 R h (t, s) e h k (s) ds = λ h k e h k ,w h e r eλ h k is the eigenvalue associated with e h k . Theorem VIII.1. For every integer k,t h em a pe h k has bounded variations on [0,T]. Proof: For every fixed (k, t) in N × [0,T], t) c h(s) |t -s| 2ht,s e h k (s) ds =: F 1 (t)+F 2 (t) -F 3 (t). (VIII.1)

,

  we easily get that∀s ∈ [0,T], |C i (s)| 2H 2 ti ti-1 Ä x 2H2-1 -x 2H1-1 ä dx.Define the family of maps(g α ) α∈R * + from R + to R + ,b yg α (x) := α x if x>0 and g α (x) := 1 if x =0 .L e t K 5 := sup α∈[0,T ]| ln(α)| (e 2H1 ln(α) + e 2H2 ln(α) ).T h em e a n -v a l u et h e o r e ma p p l i e dt og α yields∀s ∈ [0,T], |D i (s)| 2 -1 K 5 |2h ti,s -2h ti-1,s | K 5 sup u∈[0,T ] |h ′ (u)|| t i -t i-1 | =: K 6 |t i -t i-1 |.

2 2 L 2 C 2 C

 2222 dξ < +∞ for i ∈{1; 2}.N o t et h a th belongs to " L 2 C (dξ).W ea r en o wa b l et og i v et h ef o l l o w i n g Definition 1.1. For any h in " L 2 C (dξ),w ed e fi n et h es t o c h a s t i cW i e n e ri n t e g r a lI(h)= R h(ξ) W(dξ) of h byI(h) := R h 1 (ξ)W 1 (dξ) -R h 2 (ξ)W 2 (dξ).The process{I(h); h ∈ " L 2 C (dξ)} is a centred Gaussian process whose covariance function Σ is given, for all (h, g) in " L 2 C (dξ) × " L 2 C (dξ),b yΣ(h, g)= <h ,g> L 2 C (R) .I np a r t i c u l a rI(h) N (0; ||h|| (R) ) for every h in " L (dξ). Let us note f the Fourier transform of any element f in L 1 R (du) ∪ L 2 R (du) defined, for every ξ ∈ R,b y f (ξ) := 1 √ 2π R e iξu f (u)du,i ff belongs to L 1 (R) and extended by continuity for abitrary f in L 2 R (du). It is then clear that, for every Gaussian random measure W 0 on R with control measure λ 1/2 ,w eh a v et h e equality in law: R f (u)W(du)(d)

(λ 1/ 4

 4 ) 0 to L 2 (Ω, F , P) by W 1 (A) := 1 2 J 1 (1A + 1 -A + ( √ 2 -2)1 A∩(-A) ), W 2 (A) := -1 2 J 2 (i1 A -i1 -A + √ 21 A∩(-A) ) and W := W 1 + W 2 .W eh e n c ed e fi n e ,f o r every h in " L 2 C (dξ), I(h) := R h 1 (ξ)W 1 (dξ) -R h 2 (ξ)W 2 (dξ).W eh a vet h ef o l l o w i n gr e s u l t s : Properties 3.1. (i) W 1 and W 2 are independent Gaussian random measure on R,i n d e p e n d e n t l ys c a t t e r e d on R + ,w i t hc o n t r o lm e a s u r eλ 1/4 which satisfy W 1 (A)=W 1 (-A) and W 2 (A)=-W 2 (-A) for any A in B (λ) 0 . (ii) For any h := h 1 + ih 2 in " L 2 C (dξ) and i in {1; 2},t h et w oG a u s s i a np r o c e s s e s(J i (h)) h∈L 2 C (dξ) and (I(h))

2 =

 2 i∈N A i ) and i∈N W(A i ) +∞ i=N +1 λ(A i ) -→ N →+∞ 0.T hes e r i e so fi n d e p e n d e n tr a n d o mv a r iables i∈N W(A i ) converges in L 2 (Ω, F , P) sense as well as almost surely.

  ) où δ : R → R est une fonction de classe C 1 .N o u sn o u spe r m e t t o n sd ' e m p l o y e ri c il e sm ê m e sn o t a t i o n sq u ' e n (1.6) puisque, lorsque la fonction h est constante, égale à H,l e sd e u xd é fi n i t i o n sc o i n c i d e n t .D ep l u s ,s il ' o n note ‹Y (a + ,a -) le processus défini, pour tout réel t,p a r

	‹ Y (a + ,a -) (t) := δ(h(t))	Γ(H +1/2) √ 2π	R	(e itξ -1) |ξ| h(t)+1/2 U (a + ,a -) (ξ; h(t)) W (dξ),	(2.4)
	où W est construite à partir de W ,commeindiquédanslasection2del'annexeAdecemémoire,alorsona
		‹ Y (a

+ ,a -) (t) p.s.

  4 et 5 on considère le cas particulier où (M) désigne le calcul de Malliavin et la théorie du bruit blanc. On y montre ainsi notamment le Théorème 3.5. Soit Y =(Y t ) t∈[0,1] un processus Bochner intégrable d'indice p 0 ∈ N (cf. chapitre 3 définition 5.1 pour une définition). Alors Y est intégrable par rapport au mBm harmonisable à la fois au sens de la méthode du bruit blanc et au sens défini dans le chapitre 2.D ep l u sc e sd e u xi n t é g r a l e s [0,1] Y t d (M1) B h t et

	[0,1] Y t d B h t coincident dans l'espace (S

* ) des distributions stochastiques.

  une certaine fonctionnelle ψ de W σ et B h .N o u su t i l i s o n sa l o r su n em é t h o d ed ec u b a t u r e ,f o n d é es u r une quantification fonctionnelle de W σ et B h afin d'évaluer l'espérance E[ψ(Wσ ,B h )]. Nous évaluons ensuite le prix de forward start calls et donnons enfin, en utilisant une méthode d'extrapolation et de cubature les smiles obtenus pour différentes fonctions h et pour quatre maturités différentes. Contents 1B a c k g r o u n d a n d M o t i v a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2W h i t e n o i s e t h e o r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 White noise measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Properties of Hermite functions and space S ′ (R) ................... 2.3 Space of Hida distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 S-transform and Wick product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3T h e o p e r a t o r s MH and their derivatives . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Study of MH ....................................... 4W i e n e r i n t e g r a l w i t h r e s p e c t t o m B m o n R ..................... 4.1 Wiener integral with respect to fBm . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Wiener integral with respect to mBm . . . . . . . . . . . . . . . . . . . . . . . . . 30 5S t o c h a s t i c i n t e g r a l w i t h r e s p e c t t o m B m . . . . . . . . . . . . . . . . . . . . . . 5.1 Fractional White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Multifractional Wick-Itô integral versus Wiener integral with respect to mBm . . 40 6I t ô F o r m u l a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	3.2	Study of ∂M H

∂H ...................................... Multifractional White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.3 Generalized functionals of mBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.4 S-Transform of mBm and multifractional white noise . . . . . . . . . . . . . . . . . 37 5.5 The multifractional Wick-Itô integral . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.6 6.1 Itô Formula for generalized functionals of mBm on an interval [a, b] with 0 <a<b 43 6.2 Itô Formula in (L 2 ) ................................... 7T a n a k a f o r m u l a a n d e x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Tanaka formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.2 Itô formula for functions h such that d dt [R h (t, t)] = 0 ................. 7.3 Itô formula for functions h such that d dt [R h

  For a function f which is not in L 2 (R),w em a ys t i lld e fi n e|f | p by allowing |f | p to be infinite.

	Remark 2.2.
	.6)
	For p in N,d e fi n et h es p a c e sS p (R) := {f ∈ L 2 (R), |f | p < +∞} and S -p (R) as being the completion of the space L 2 (R) with respect to the norm || -p .

  Definition 2.5. For any Φ in (S) * such that the sum

				.11)
	Lemma 2.4.	1. Let p be a positive integer and Φ be an element of (S -p ).T h e n |S(Φ)(η)| ||Φ|| -p e 1 2 |η| 2 p , for any η in S (R).	(2.12)
	2. Let Φ :=	+∞ k=0 a k < ., e k > belong to (S)	* .T h ef o l l o w i n ge q u a l i t yh o l d sf o re v e r yη in S (R): +∞
		S(Φ)(η)=	a k <η,e k > L 2 (R) .	(2.13)
			k=0
	Proof. Item 1 is proved in [52] p.79.I t e m2i sa ne a s yc a l c u l a t i o nl e f tt ot h er e a d e r .
	Another useful tool in white noise analysis is the Wick product:
	Theorem-Definition 2.2 ([52] p.92). For every (Φ, Ψ) ∈ (S) (S) * ,c a l l e dt h eW i c kp r o d u c to fΦ and Ψ and noted Φ Ψ,s u c ht h a t ,f o re v e r yη in S (R), ,t h e r ee x i s t sau n i q u ee l e m e n to f
		S(Φ Ψ)(η)=S(Φ)(η) S(Ψ)(η).	(2.14)
	Lemma 2.5. For any (p, q) ∈ N 2 , X ∈ (S -p ) and Y ∈ (S -q ),
		|S(X Y )(η)| ||X|| -p ||Y || -q e |η| 2 max{p;q} .	(2.15)
	Proof. The proof is easy since, for every η in S (R),u s i n gl e m m a2 . 4 , |S(X Y )(η)| = |S(X)(η)||S(Y )(η)| ||X|| -p e 1 2 |η| 2 p ||Y || -q e 1 2 |η| 2 q ||X|| -p ||Y || -q e |η| 2 max{p;q} .
	For any Φ in (S)	k times Φ ••• Φ of (S) * .W ec a ng e n e r a l i z et h ed e fi n i t i o n
	of exp to the case where Φ belongs to (S) +∞
				k=0

* × (S) * * and k in N * let Φ k denote the element * :

  p.39 and [43] p.280-281 for proofs.

	Lemma 2.7. The S-transform verifies the following properties: (i) The map S :Φ → S(Φ),f r o m(S) * into F unction (S (R); R),i si n j e c t i v e . * be an (S) * process. If Φ is (S) * -integrable over R then (ii) Let Φ:R → (S) S( Φ(u) du)(η)= S(Φ(u))(η) du ,f o ra l lη in S (R).
	R (iii) Let Φ:R → (S) [SΦ(u)](η) is differentiable at t and verifies S[ dΦ R * be an (S) * -process differentiable at t.T h e n ,f o re v e r yη in S (R) the map u → dt (t)](η)= d dt î S[Φ(t)](η) ó .

  Proof. We shall apply theorem 2.8. The measurability of t → S[F (t, B (h) (t))](η) results from (iii) of theorem 5.12, the continuity of the two maps t → F (t) and t → γ Ä t 2h(t) ,.-

	t 0 M h(t) (η)( u) du	ä	and
	theorem 5.17. Since h is bounded on [a, b],l e m m a2 . 4a n d( 5 . 2 7 )y i e l d
	|S[F (t, B (h

R) and t 0 ∈ I. If both maps t → F (t) and t → ϕ(t, .) are continuous (respectively differentiable) at t 0 ,t h e nt h ef u n c t i o nt →<F(t),ϕ(t, .) > is continuous (respectively differentiable) at t 0 . Theorem 5.18. Let p ∈ N, 0 <a<band let F :[a, b] → S -p (R) be a continuous map. Then the stochastic distribution process F (t, B (h) (t)) is both (S) * -integrable and dB (h) -integrable over [a, b].

  h(t))], thanks to (ii) of theorem 5.12. Since the function t → d dt [g η (t, h(t))] is continuous (by lemma 5.5), the measurability of t → S[F (t, B (h) (t)) W (h) (t)](η) for every function η in S (R) follows. Moreover, for every integer p 0 max{p, 2}, F (t, B (h) (t)) and W (h) (t) belong to (S -p0 ) for all t in [a, b]. Using lemma 2.5 and (5.27), we may write, for all t in [a, b],

  )-process, noted Z,d e fi n e do nR + such that Z is differentiable on R + and verifies equation(3.11) in (S * ).A si n[ 1 1 ] ,i ti se a s yt o g u e s st h es o l u t i o no f(3.11) if we replace Wick products by ordinary products. Once we have a solution of (3.11), we replace ordinary products by Wick products. This heuristic reasoning leads to defining the process Z := (Z(t) t∈R+ ) by

*

  tseem sn at u ralt o give the following definition.

	Definition 5.2. (Wiener integral with respect to mBm)
	For an mBm B (h) ,d e fi n et h eG a u s s i a ns p a c eΘ h

  t h a ti st h en th derivative in S ′ (R),o ft h et e m p e r e dd i s t r i b u t i o nF (t).H e n c ew em a yc o n s i d e r the map t → ∂ n F ∂x n (t) from [a, b] to S ′ (R).M o r e o v e rf o ra n yt 0 in [a, b],w ew i l ln o t e ∂F ∂t (t 0 ) the quantity lim

		F (t0+r)-F (t0)
	r→0	r

  et h a tw h e nr =1we have card({h(t 1 ); •••; h(t n )})=1(treated in the first case) and when r = n we have card({h(t 1 ); •••; h(t n )})=n and then h(t 1 ) >h (t 2 ) (treated in the second case). We hence assume from now that 2 k 1 n -1 and 2 r n -1. Define the sets I 1 ,I 2 , ••• ,I r by I 1

  Theorem 2.1.1 p. 33]. D e fi n ea l s oD := Λ (η (b -a) β ) δ where Λ and δ are the constants appearing in (H) and η in (2.2). Let N be the smallest integer n such that max

							that verify
	lim sup n→+∞	P	Ç®	max 0 i m	sup t∈[ti,ti+1)	|B hn t -B hn ti | >τ ´å <ε.	(2.3)
	Let us then fix (ε, τ ) in (R * + ) 2 .D e fi n e[H 1 ,H 2 ] := [ inf u∈[a,b] h(u), sup u∈[a,b]	h(u)] and set, until the end of this proof,
	q n := 2 2 n ,n ∈ N.D e fi n eF := [a, b] × [H 1 ,H 2 ].T h ep r o c e s s(B(t, H)) (t,H)∈F is Gaussian and the space of continuous real-valued functions defined on F endowed with the sup-norm is a separable Banach space.
	Fernique theorem (see [30, theorem 2.6 p.37]) applies to the effect that there exists a positive real α such that A α := E ñ exp ® α sup (t,H)∈F B(t, H) 2 ´ô < +∞.S e tG := L +∞ p=0 2 p/2 qp δ/2 ,w h e r eL is the universal constant
	in [76,

  et h a tt h i se n t a i l st h a t ,f o ra l ln larger than N , h n (t i )=h(t i ).B e s i d e s ,a l s oa ss o o na sn N , h n (t) belongs to the set h([t i ,t i+1 ]) when t ∈ [t i ,t i+1 ] (draw ap i c t u r e ) . L e tJ τ,m

	n

  Note that lemma 2.2 implies that (τ/8) 2 < (τ/4 -µ i ) 2 .L e t ' sg ob a c kt ot h ep r o o fo ft i g h t n e s s .[ 2 ,(2.6) p.43]y i e l d s ,f o ra l ln N ,

	1/2 δβ/2 N q	<
	τ/8.

  Theorem 5.10. Let Y =( Y t ) t∈[0,1] be a Bochner integrable process of index p 0 ∈ N.T h e nY is integrable with respect to mBm in both sense (5.12) and (5.13).M o r e o v e r [0,1] Y t d (M1) B h t and [0,1] Y t d B h t are equal in (S * ).

	equal, assuming they both exist.	[0,1] Y t d (M1) B h t and [0,1] Y t d B h t are

  Proof of proposition 3.1 in the case of B 1 and B 2Once again it is sufficient to establish the proof forB 1 .L e t[a, b] × [c, d] be fixed in R × (0, 1) and (t, s, H, H ′ ) be fixed in [a, b] 2 × [c, d] 2 .W ek n o w ,( s e e[ 3 4 ,s e c t i o n2]f orexam p l e)t h at ,a.s B 1 (t, H) :=< ., M H (1 [0,t] ) >, for every (t, H) in R × (0, 1).T h a n k st o[5 3 ,L e m m a5.5,P r o p o s i t i o n3.8 and remark 3.9]w ek n o wt h a t t the mapH → M H (1 [0,t] ) is C 1 ,f o re v er yr e a lt,f r o m(0, 1) to L 2 (R).I t i s t h e n c l e a r t h a t t h e m a p is H → B 1 (t, H) is C 1 for every real t and that its derivative, noted∂B1 ∂H is such that: almost surely, ∂MH ∂H (1 [0,t] ) > for every (t, H) in (0, 1).N o t em o r e o v e rt h a tt h ep r e v i o u se q u a l i t ya l s oh o l d s in L 2 (Ω) and that the process ( ∂B1 ∂H (t, H)) (t,H)∈R×(0,1) is Gaussian and centred. Now and still using notations of[START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF], we getE[ ∂B1 ∂H (t, H) ∂B1 ∂H (s, H ′ )] = < ∂MH ∂H (1 [0,t] ), ∂MH ∂H (1 [0,s] ) > L 2 (R),f o re v e r y(t, H) ∈ R × (0, 1).W e hence have

	J :=E	Ä	∂B1 ∂H (t, H) -∂B1 ∂H (s, H)

∂B1

∂H (t, H)=< .,

  by matching their local regularity. This is important e.g. for purposes of detection orr e a l -t i m ec o n t r o l . T h e price to pay is of course that one has to deal with the added complexity brought by having a functional parameter instead of a single number. In general, the increments of multifractional Brownian motion are neither independent nor stationary. Since an mBm B h is an fBm of Hurst index H when h is constant and equal to H, there is no risk of confusion by denoting B H the fractional Brownian motion with Hurst index H.The quantization of a random variable X valued in a reflexive separable Banach space (E, |•|) consists in its approximation by a random variable Y that is measurable with respect to X and that takes finitely many values in E.T h er e s u l t i n ge r r o ro ft h ed i s c r e t i z a t i o ni su s u a l l ym e a s u r e db yt h eL p norm of the difference |X -Y |.I fw es e t t l eo nafi x e dm a x i m u mc a r d i n a lN for Y (Ω),t h em i n i m i z a t i o no ft h ee r r o rr e d u c e st ot h e following optimization problem:As Y is supposed to be measurable with respect to X,t h e r ee x i s t saB o r e lm a pProj : E → E valued in a finite subset Γ of E such that Y =P r o j ( X).T h e fi n i t es u b s e t Γ is called the codebook. Hence if Γ={γ 1 , ••• ,γ N },t h e r ee x i s t saB o r e lp a r t i t i o

	2F u n c t i o n a l q u a n t i z a t i o n o f m u l t i f r a c t i o n a l B r o w n i a n m o t i o n
	2.1 Computation of the quantization
	min	¶	|X -Y | p ,Y:Ω→ E measurable with respect to X, card(Y (Ω)) N	©	.	(2.1)

  the following, for two positive sequences (x n ) n∈N and (y n ) n∈N ,w ew r i t ex n ∼

	symbol x n	n→∞	y n means that lim	n→∞	y n if lim n→∞	xn yn =1 .T h e

  √ Mx H1∧ βδ 2 .W eh a v ed e n o t e d ,f o rn in N, κ n the number such that E Y 2n = κ n E Y 2 n for the centred Gaussian random variable Y .I ti sc l e a rt h a tφ ρ is regularly varying with index H 1 ∧ βδ 2 ,w h i c hi sp o s i t i v e .T h er e s u l tt h e nf o l l o w sf r o mT h e o r e m2 . 3 . Remark 2.6. Corollary 2.5 extends to every process V

:

  Definition 3.1 (Multifractional Wick-Itô integral). Let B h be a normalized multifractional Brownian motion and Y : R → (S) * be a process such that the process t → Y t W h -integrable on R or integrable on R with respect to mBm B h . Moreover, the integral on R of Y with respect to B h is defined by For an interval I of R, I Y s d B h s

	R	Y s d B h s :=	R	Y s W h s ds.	(3.4)

t is (S) * -integrable on R.T h ep r o c e s sY is said to be d B h

  stands for the signed measure associated with the function of bounded variation s → " B h s (ω).Another easy case is the one of an S.D.E. of the formY t = y 0 + Y s )ds + X t ,(3.16) where β(s, y) is assumed to be Lipschitz continuous in y uniformly in s.T h i ss e t t i n gi sa d d r e s s e di n[ 5 8 ,p . 20-21] 5 ,w h e r et h ea u t h o r sc o n s i d e rt h ea s s o c i a t e di n t e g r a le q u a t i o n y(t)=y 0 + L p ([0,T]) is fixed. The existence and uniqueness in L p ([0,T]) of a solution for the integral equation (3.17) follows from the same approach used for ordinary differential equations. Then the solution of the associated S.D.E. (3.16) simply reads U t =Ψ β p (X) t ,w h ereΨ β p : L p ([0,T]) → L p ([0,T]

	f t s (ω) The case of certain simple diffusions Ä " B h ä = t 0 f (s)d " B h s where d " B h
	t
	0 β(s, t
	β(s, y(s))ds + g(t),	(3.17)
	0
	where g ∈

  4 . 1 )w r i t e s® dF t = F t σ t Ä ρdW σ t + 1ρ 2 dW F t ä d ln(σ t )=θ (µln(σ t )) dt + γ h d B h t + γ σ dW σ t ,σ 0 > 0.We denote resp ectively by F σ , F F and F h the natural filtrations of W σ , W F and B h .W e d e fi n e t h e filtration F σ,h by F σ,h

											(4.2)
	t	= σ	Ä	F σ t , F h t	ä	and F F,σ,h by F F,σ,h t	= σ	Ä	F F t , F σ t , F h t	ä	.
	The unique solution of (4.1) reads
	F t = F 0 exp σ s =exp Ä ln(σ 0 )e -θs + µ Ä t 0 σ s dW s -1 2 Ä 1 -e -θs t 0 σ 2 ä s ds

  F 0 exp ρ

	0	t	σ s dW σ s -	ρ 2 2	0	t	σ 2 s ds	exp	» 1 -ρ 2	0	t	σ s dW F s -	1 -ρ 2 2	0	t	σ 2 s ds .
	Measurable with respect to F σ,h t

  The cubature formula is then fully explicit and one finally obtains the following approximation: + γ σ e -θt I e θ• t (χ σ j )+γ h e -θt I e θ•

	Moreover,			τ	T	" σ i t d W σ t =	" σ i T -σ τ γ σ	-	1 γ σ	τ	T	" σ i t θ	Ä	µ -ln	Ä " σ i t	ää	dt -	γ h γ σ	τ	T	" σ i t dχ h i (t).
	This shows that approximate I σ i τ,T by I σ i T τ σ i t • dW σ t may be approximated by τ,T := T τ " σ i s d W σ s -γσ 2 T τ " σ i s ds.	T τ	" σ i t d W σ t and	T τ σ i t dt by	T τ	" σ i t dt.T h u sw e
	E	Å F T F τ	-K	ã +	≈	N1 i=1	N2 j=1	p h i p σ j PrimeBS F i,j τ,T ,	Ä 1 -ρ 2	ä 1 T -τ	τ	T	Ä	σ i,j (s) ä 2	ds	1 2	,T -τ, K ,
	where				F i,j τ,T =exp ρ		τ	T	σ i,j (s)dχ σ j (s) -ργ σ 1 2	τ	T	σ i,j (s)ds -	ρ 2 2	τ	T	Ä	σ i,j (s) ä 2	ds ,
	and Ä ln(σ i,j ) ä	σ i,j (t) := exp 1 i N1,1 j N2	Ä	ln(σ 0 )e -θt + µ	Ä 1 -e -θt	t (χ h j ) ä	.
			τ	T	σ i t dW σ t =	σ i T -σ τ γ σ	-	1 γ σ	τ	T	σ i t θ	Ä	µ -ln	Ä σ i t	ää	dt -	γ h γ σ	τ	T	σ i t dχ h i (t)	-	γ σ 2	τ	T	σ i t dt	.
																						=		T τ	σ i t • dW σ t	= 1 2 σ i ,W σ τ,T

ä

  dF t = F t σ t dW t , dσ t = σ t Ä γ h d B h t + γ σ dW σ

	t	ä	,σ 0 > 0,	(5.1)

  Figure 4.4: Comparisons of vanilla option volatility smiles for fBm with H =0 .2,H =0 .5,H =0 .7 and H =0 .9 at different maturities. Top left: T =1 .T o pr i g h t :T =2 .5.B o t t o ml e f t :T =5 .B o t t o mr i g h t : T =10.

	6. NUMERICAL EXPERIMENTS											107
	0.3							0.3
		H = 0.2
	0.28	H = 0.5					0.28
		H = 0.75
	0.26							0.26	H = 0.9
	0.24							0.24	H = 0.75
	0.22							0.22	H = 0.5
		H = 0.9
	0.2							0.2
									H = 0.2
	0.18							0.18
	0.16							0.16
	0	50	100	150	200	250	300	0	50	100	150	200	250	300
	0.3							0.3
	0.28							0.28
	0.26							0.26
	0.24							0.24
		H = 0.9							H = 0.9
	0.22	H = 0.75						0.22	H = 0.75
	0.2							0.2
		H = 0.5							H = 0.5
	0.18	H = 0.2						0.18	H = 0.2
	0.16							0.16
	0	50	100	150	200	250	300	0	50	100	150	200	250	300

  an example with ρ =0for illustration purposes.

		CHAPTER 4. MULTIFRACTIONAL STOCHASTIC VOLATILITY MODELS
	0.3							0.3
	0.28							0.28
	0.26							0.26
	0.24							0.24
	0.22	H = 0.9						0.22	mBm with h2
	0.2							0.2
		mBm with h1
	0.18							0.18	H = 0.2
	0.16							0.16
	0	50	100	150	200	250	300	0	50	100	150	200	250	300
	0.3							0.3
	0.28							0.28
	0.26							0.26
		H = 0.9
	0.24							0.24
	0.22							0.22	mBm with h1
	0.2	mBm with h3						0.2	mBm with h4
	0.18							0.18
	0.16							0.16
	0	50	100	150	200	250	300	0	50	100	150	200	250	300

  ••• <t N = T . For any i in {1, ••• ,N} we get, |F 1 (t i ) -F 1 (t i-1 )| h(t) is C 1 as soon as h is C 1 ,t h em e a n -v a l u et h e o r e my i e l d s || t i -t i-1 | =: K 2 |t i -t i-1 |,where f ′ s (t) denotes, for every s in [0,T],thederiv ativ e,atpoin tt,oft h em apt →L i K 2 |t i -t i-1 | ds T (1 + K 2 ) |t i -t i-1 | Ä e 2H1T + e 2H2T ä =: K 3 |t i -t i-1 |. (VIII.3)

										=:K1
						sup s∈[0,T ]	e k (s) c h(s)	0	T	c 2 ht i ,s c h(ti)	t	2ht i ,s i	-	c 2 ht i-1 ,s c h(ti-1)	t	2ht i-1 ,s i-1	ds
						K 1	Å T 0	c 2 ht i ,s c h(ti)		t	2ht i ,s i	-t	2ht i-1 ,s i-1	ds	+	0	T	c 2 ht i ,s c h(ti)	-	c 2 ht i-1 ,s c h(ti-1)	t	2ht i-1 ,s i-1	ds	ã .
																	=:Gi	=:Li
																			(VIII.2)
	Since the map (s, t) →	c 2 h t,s
	ï	ò	c 2 ht i ,s c h(ti)	-	c 2 ht i-1 ,s c h(ti-1)			sup s∈[0,T ]	|f ′ s (t)c 2 h t,s c h(t) .S e t t i n g[H 1 ,H 2 ] :=
	inf u∈[0,T ] h(u), sup u∈[0,T ] h(u)			,o n eg e t s :
		0	T	t	2ht i-1 ,s i-1
	Besides, G i	sup (t,s)∈[0,T ] 2		c 2 h t,s c h(t)		T 0	t i	2ht i ,s -t	2ht i-1 ,s i-1	ds =: K 4	T 0	t	2ht i ,s i	-t	2ht i-1 ,s i-1	ds.
	Now, writing
						t i	2ht i ,s -t	2ht i-1 ,s i-1	= t i	2ht i ,s -t	2ht i ,s i-1	-t	2ht i ,s i-1 -t	2ht i-1 ,s i-1
																			:=Ci(s)	:=Di(s)

c

  .1) and (2.2), namely:Proposition 2.1. For al l f in L 2 R (du) and h in " L 2 C (dξ),w eh a v et h a t :Proof. The equality beetwen the right-hand sides of (2.1) and (2.2) results simply from the Parseval identity.Remark 2.2. We now have R f (u) W (du)a.s = R f (ξ) W (dξ) for every f in L 2 R (du).M o r e o v e r{J (h); h ∈ " L2C (dξ)} is a centred Gaussian process, and once again, J (h) N (0; ||h|| 2

	a.s =	h(ξ) W (dξ), if and only if f (ξ)=h(ξ), a.e.	(2.4)
	R
		L 2 C (R) ) for every h in " L 2 C (dξ).

R f (u) W (du)

it is a straightforward computation to check that the conditions of this proposition are verified by L

Of course one can define an integral w.r.t. mBm in sense of definition 3.2, with (M): =( M 1 ),f o ro t h e rf r a c t i o n a lfi e l d s B and not only for B = B 1 .

Such a fractional Black and Scholes model raises some financial and economical issues, see[START_REF] Bender | Arbitrage with fractional Brownian motion? Theory Stoch[END_REF][START_REF] Björk | A note on Wick products and the fractional Black-Scholes model[END_REF].

An alternative definition would be to start from a family of fBms (B H ) H∈(0,1) (i.e. B H := (B H t ) t∈R is an fBm for every H in (0, 1))a n dd e fi n ef r o mi tt h efi e l d(B(t, H)) (t,H)∈[0,T ]×(0,1) by B(t, H): =B H t .H o w e v e ri ti sn o tt r u e ,i ng e n e r a l ,t h a t the field (B(t, H)) (t,H)∈[0,T ]×(0,1) obtained in this way is Gaussian.

Remerciements

Acknowledgments

The authors are very thankful to Bruno Dupire, Gilles Pagès and Marc Yor for helpful remarks and comments.

Chapter 3

Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions Joint work with E.Herbin and J. Lévy-Véhel Abstract Stochastic integration w.r.t. fractional Brownian motion( f B m )h a sr a i s e dh u g ei n t e r e s ti nr e c e n ty e a r s , motivated in particular by applications in finance and Internet traffic modeling. Since fBm is not a semimartingale, stochastic integration required specific developments. Multifractional Brownian motion (mBm) generalizes fBm by letting the Hölder exponent vary in time. The aim of this work is to explore some avenues in order to give firstly a new definition of mBm that generalizes the already existing one and secondly to define a stochastic calculus w.r.t. mBm from stochastic integrals w.r.t. fBm. keywords: Brownian motion, multifractional Brownian motion, Gaussian processes, Skorohod integral white noise theory, S-transform, Wick-Itô integral, stochastic differential equations.

1I n t r o d u c t i o n a n d B a c k g r o u n d

Fractional Brownian motion (fBm) is a centered Gaussian pro cess with features that make it a useful mo del in various applications such as financial and Internet traffic modeling, image analysis and synthesis, physics, geophysics and more. These features include self-similarity, long range dependence and the ability to match any prescribed constant local regularity. Fractional Brownian motion is a centred Gaussian process indexed by a parameter, usually denoted H,t h a tb e l o n g st o(0, 1),a n di sc a l l e dt h eH u r s te x p o n e n t . I t sc o v a r i a n c e function R H reads:

where γ H is a positive constant. An fBm with γ H =1is termed normalized. Obviously, when H = 1 2 , fBm reduces to standard Brownian motion. Various integral representations of fBm are known, including the harmonizable and moving average ones [73], as well as representations by integrals over a finite domain [3,[START_REF] Boufoussi | Local time and Tanaka formula for a Volterra-type multifractional Gaussian process[END_REF]. The fact that most of the properties of fBm are governed by the single real H restricts its application in some situations. In particular, its Hölder exponent remains the same all along its trajectory. This does not seem to be adapted to describe adequately natural terrains, for instance. In addition, long range dependence requires H>1/2,a n dt h u si m p o s e sp a t h ss m o o t h e rt h a nt h eo n e so fB r o w n i a nm o t i o n . M u l t i f r a c t i o n a l Brownian motion was introduced to overcome these limitations. The basic idea is to replace the real H by af u n c t i o nt → h(t) ranging in (0, 1). Several definitions of multifractional Brownian motion exist. The first ones were proposed in [START_REF] Peltier | Multifractional brownian motion[END_REF] and in [7]. A more general one was introduced in [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF]. In this work, we shall use a new definition that includes all previously known ones and which, in our opinion, is both more flexible and retains the essence of this class of processes. We first need to define a fractional Brownian field: Definition 1.1 (Fractional Brownian field). Let (Ω, F ,P) be a fixed probability space. A fractional Brownian field on R×(0, 1) is a Gaussian field, noted (B(t, H)) (t,H)∈R×(0,1) ,s u c ht h a t ,f o re v e r yH in (0, 1),thepr o c ess

Multifractional stochastic volatility models

Joint work with S.Corlay and J. Lévy-Véhel

Abstract

The aim of this work is to advocate the use of multifractional Brownian motion (mBm) as a relevant model in financial mathematics. Multifractional Brownian motion is an extension of fractional Brownian motion where the Hurst parameter is allowed to vary in time. This enables the possibility to accommodate for varying local regularity, and to decouple it from long-range dependence properties. While we believe that mBm is potentially useful in a variety of applications in finance, we focus here on a multifractional stochastic volatility Hull & White model that is an extension of the model studied in [START_REF] Comte | Long memory continuous-time stochastic volatility models[END_REF]. Using the stochastic calculus with respect to mBm developed in [START_REF] Lebovits | White noise-based stochastic calculus with respect to multifractional brownian motion[END_REF], we solve the corresponding stochastic differential equations. Since the solutions are of course not explicit, we take advantage of recently developed numerical techniques, namely functional quantization-based cubature methods, to get accurate approximations. This allows us to test the behaviour of our model (as well as the one in [START_REF] Comte | Long memory continuous-time stochastic volatility models[END_REF]) with respect to its parameters, and in particular its ability to explain the smile effect of implied volatility. An advantage of our model is that it is able both to fit smiles at different maturities, and to take into account volatility persistence in a more precise way than in [START_REF] Comte | Long memory continuous-time stochastic volatility models[END_REF].

keywords: Hull & White model, functional quantization, vector quantization, Karhunen-Loève, Gaussian process, fractional Brownian motion, multifractional Brownian motion, white noise theory, S-transform, Wick-Itô integral, stochastic differential equations.

would thus require to choose a "small" H, i.e. H<1/2,whichisnotcompatiblewithlong-rangedependence properties. In contrast, mBm has at each time t local regularity h(t),a n d ,n om a t t e rt h ev a l u eo fh in (0, 1), always display long-range dependence as long h is not constant. In addition, as we will show from numerical experiments, the model (written in a risk-neutral setting):

where B h t is an mBm, yields shapes of the smile at maturity T that are governed by a weighted average of the values of the function h up to time T :t h u s ,b ya d e q u a t e l yc h o o s i n gh,o n em a ym i m i cag i v e ni m p l i e d volatility surface more faithfully than with a Hull & White model driven by fBm (the calibration of h for this purpose will be addressed in a forthcoming work). In order to give a rigorous meaning to the model above, as t o c h a s t i ci n t e g r a lw i t hr e s p e c tt om B mm u s tb e defined. Multifractional and fractional Brownian motion are not semimartingales, thus classical Itô theory does not apply to them. At the time [START_REF] Comte | Long memory continuous-time stochastic volatility models[END_REF] was written, no theory for integration with respect to fBm was available yet. Various approaches have been developed since. Among these, the one based on white noise theory is well fitted for an extension to mBm. In particular, it allows to deal with any H ∈ (0, 1) and to obtain Itô formulas. This integral was developed in [34, ?,1 3 ] ,a n da p p l i e dt oo p t i o np r i c i n gi naf r a c t i o n a l Black and Scholes model in [START_REF] Elliott | A general fractional white noise theory and applications to finance[END_REF] 1 .T h ew h i t en o i s eb a s e ds t o c h a s t i ci n t e g r a lw a se x t e n d e dt om B mi n [ 5 3 ] . This is the theory we will use in order to study precisely our stochastic volatility models. While we focus here on the multifractional stochastic volatility model (0.2) (we also briefly consider a multifractional SABR model with β =1in Section 5), we would like to mention that mBm is useful in a variety of applications in finance (see [1] for a partial list of articles dealing with mBm in this field). In order to assess the relevance of our model, we computen u m e r i c a l l yt h es m i l e sa td i ff e r e n tm a t u r i t i e s . Since the solution cannot be written in an explicit form, we need to resort to approximations. In our case, this is made possible by recent advances in the theory off u n c t i o n a lq u a n t i z a t i o nof Gaussian processes. Functional quantization of Gaussian processes has become an active field of research in recent years since the seminal article [START_REF] Luschgy | Functional quantization of Gaussian processes[END_REF]. As far as applications are concerned, cubature methods [START_REF] Pagès | Functional quantization for numerics with an application to option pricing[END_REF][START_REF] Corlay | Some aspects of optimal quantization and applications to finance[END_REF] and variance reduction methods [START_REF] Corlay | Functional quantization-based stratified sampling methods[END_REF][START_REF] Lejay | A variance reduction technique using a quantized Brownian motion as a control variate[END_REF] based on functional quantization have been proposed. However, as the numerical use of functional quantizers requires the evaluation of the Karhunen-Loève eigenfunctions, this method was restricted to processes for which a closed-form expression for this expansion is known, such as Brownian motion. In [START_REF] Corlay | The Nyström method for functional quantization with an application to the fractional Brownian motion[END_REF], a numerical method was proposed to perform numerical quadratic functional quantization of more general Gaussian processes, which will be applied here to multifractional Brownian motion. We show that we can handle a fast and accurate forward start option pricing in this mo del thanks to a functional quantization-based cubature method similar to the one proposed in [START_REF] Pagès | Functional quantization for numerics with an application to option pricing[END_REF] and in [START_REF] Corlay | Some aspects of optimal quantization and applications to finance[END_REF]. This allows us to study the dependency of the smile dynamics on the functional parameter of the considered mBm. The remaining of this paper is organized as follows. We recall in Section 1 basic facts about mBm. In Section 2, we explain how to perform functional quantization of mBm and investigate the rate of decay of the corresponding quantization error. Quantization-based cubature is also addressed in this section. Section 3i sd e v o t e dt or e c a l l so nt h ew h i t en o ise based stochastic integral with respect to mBm. It also shows how to solve some stochastic differential equations (S.D.E.) in this frame and presents general remarks on the quantization of solutions of S.D.E. A detailed treatment of the Hull & White and SABR models are proposed in Sections 4 and 5. Numerical experiments and conclusions are gathered in Section 6.

1R e c a l l s o n m u l t i f r a c t i o n a l B r o w n i a n m o t i o n

Fractional Brownian motion (fBm) [START_REF] Kolmogorov | Wienersche spiralen und einige andere interessante kurven im hilbertschen raume[END_REF][START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF] is a centred Gaussian pro cess with features that make it a useful model in various applications such as financial and teletraffic modelling, image analysis and synthesis, geophysics and more. These features include self-similarity, long-range dependence and the ability to match any prescribed constant local regularity. Fractional Brownian motion depends on a parameter, usually denoted by H and called the Hurst exponent, that belongs to (0, 1).I t sc o v a r i a n c ef u n c t i o nR H reads:

where γ H is a positive constant. A normalized fBm is one for which γ H =1.O b v i o u s l y ,w h e nH = 1 2 ,f B m reduces to standard Brownian motion. While fBm is a useful model, the fact that most of its properties are governed by the single number H restricts its application in some situations. In particular, its Hölder exponent remains the same all along its trajectory. Thus, for instance, long-range dependent fBm, which require H> 1 2 ,m u s th a v es m o o t h e rp a t h st h a nB r o w n i a nm o t i o n . M u l t i f r a c t i o n a lB r o w n i a nm o t i o n[ 6 8 ,7 ] was introduced to overcome these limitations. The basic idea is to replace the real H by a function t → h(t) ranging in (0, 1). The construction of mBm is best understood through the use of a fractional Brownian field. Fix a probability space (Ω, F ,P) and a positive real T .Af r a c t i o n a lB r o w n i a nfi e l do n[0,T]×(0, 1) is a Gaussian field, denoted (B(t, H)) (t,H)∈[0,T ]×(0,1) , such that for every H in (0, 1) the process (B H t ) t∈[0,T ] ,w h e r eB H t := B(t, H), is a fractional Brownian motion with Hurst parameter H 2 .F o r a d e t e r m i n i s t i c c o n t i n u o u s f u n c t i o n h : [0,T] → (0, 1),w ec a l lm u l t i f r a c t i o n a lB r o w n i a nm o t i o nw i t hf u n c t i o n a lp a r a m e t e rh the Gaussian process

.W e s a y t h a t h is the regularity function of the mBm. The fractional field (B(t, H)) (t,H)∈[0,T ]×(0,1) is termed normalized when, for all H in (0, 1), (B H t ) t∈[0,T ] is a normalized fBm. In this case we will also say that B h is normalized. In order for mBm to posses interesting properties, we need some regularity of B(t, H) with respect to H. More precisely, we will always assume that B(t, H) satisfies the following condition:

(H) Under this assumption, and if the functional parameter h is continuous, then the associated mBm has a continuous modification. The class of mBm is rather large, since there is some freedom in choosing the correlations between the fBms composing the fractional field B(t, H).F o rd e fi n i t e n e s s ,w ew i l lo f t e nc o n s i d e ri nt h i sw o r kt h es ocalled "well-balanced" version of multifractional Brownian motion. Essentially the same analysis could be conducted with other versions. More precisely, a well-balanced mBm is obtained from the field B(t, H) :=

|u| H+1/2 W (du) where W denotes a complex-valued Gaussian measure (cf. [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF] for more details). We show in Prop osition 2.8 that assumption H is satisfied by the well-balanced fractional Brownian field (in fact, it is verified by all mBms considered so far in the literature). The proof of the following proposition can be found in [5]: Proposition 1.1 (Covariance function of well-balanced mBm). The covariance function R h of well-balanced mBm is given by

where h t,s := h(t)+h(s)

The other main properties of mBm are as follows: the pointwise Hölder exponent at any point t of B (h) is almost surely equal to h(t) ∧ β h (t),w h e r eβ h (t) is the pointwise Hölder exponent of h at t [START_REF] Herbin | From n-parameter fractional Brownian motions to n-parameter multifractional Brownian motions[END_REF][START_REF] Herbin | Stochastic 2 micro-local analysis[END_REF]. For as m o o t hh,o n et h u sm a yc o n t r o lt h el o c a lr e g u l a r i t yo ft h ep a t h sb yt h ev a l u eo fh.I n a d d i t i o n , t h e increments of mBm display long range dependence for all non-constant h(t) [5]. Finally, when h is C 1 ,m Bm is tangent to fBm with exponent h(u) in the neighbourhood of any u in the following sense [START_REF] Falconer | Multifractional, multistable, and other processes with prescribed local form[END_REF]:

Appendix A Let (Ω, F , P) be a probability space and L 0 (Ω) be the set of all real random variables on Ω.L e t(E, E,m) be a measured space, and E There are two equivalent ways to understand integral with respect to Gaussian measure. We recall in two following sections these two ways and explain in the third section the links between them.

Real and complex Gaussian measures

1F i r s t m e t h o d : [ 7 3 ] a n d r e f e r e n c e s t h e r e i n

Define firstly W := W 1 +iW 2 where W 1 and W 2 are independent Gaussian measures, independently scattered on R + ,withcon trolmeasureλ 1/4 satisfying W 1 (A)=W 1 (-A) and W 2 (A)=-W 2 (-A) for any A in B

From first to second metho d

Conversly, let us start from the first method framework. We define for every function h in " L 2 C (dξ), J (h) := I(h)= R h 1 (ξ)W 1 (dξ) -R h 2 (ξ)W 2 (dξ),a n df o re v e r yf in L 2 R (du), R f (u)W (du) := J ( f ).I ti sc l e a r that (2.4) is verified. Remark 3.1. Since we have shown that the two methods of [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications[END_REF] and [73] are equivalent, we use, in the sequel, notations of both methods.