S Vakeroudis
email: stavros.vakeroudis@etu.upmc.fr

On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity

Keywords: Planar Brownian motion, Ornstein-Uhlenbeck process, winding process, Bougerol's identity, exit time from a cone

Some identities in law in terms of planar complex valued Ornstein-Uhlenbeck processes (Z t = X t + iY t , t ≥ 0) including planar Brownian motion are established and shown to be equivalent to the well known Bougerol identity for linear Brownian motion (β t , t ≥ 0): for any fixed u > 0:

= β( u 0 ds exp(2βs)) , with ( βt , t ≥ 0) a Brownian motion, independent of β. These identities in law for 2-dimensional processes allow to study the distributions of hitting times T θ c ≡ inf{t :

and more specifically of T θ -c,c ≡ inf{t : θ t / ∈ (-c, c)}, (c > 0) of the continuous winding processes θ t = Im( t 0 dZs Zs ), t ≥ 0 of complex valued Ornstein-Uhlenbeck processes.

Introduction

The conformal invariance of planar Brownian motion has deep consequences as to the structure of its trajectories (see, e.g., Le Gall [START_REF] Gall | Some properties of planar Brownian motion[END_REF]). In particular, a number of articles have been devoted to the study of its continuous winding process (θ t , t ≥ 0): Spitzer [START_REF] Spitzer | Some theorems concerning two-dimensional Brownian Motion[END_REF], Williams [START_REF] Williams | A simple geometric proof of Spitzer's winding number formula for 2-dimensional Brownian motion[END_REF], Durrett [START_REF] Durrett | A new proof of Spitzer's result on the winding of 2-dimensional Brownian motion[END_REF], Messulam-Yor [START_REF] Messulam | On D. Williams' "pinching method" and some applications[END_REF], Pitman-Yor [START_REF] Pitman | Asymptotic Laws of planar Brownian Motion[END_REF], Le Gall-Yor [START_REF] Gall | Etude asymptotique des enlacements du mouvement brownien autour des droites de l'espace[END_REF], Bertoin-Werner [START_REF] Bertoin | Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process[END_REF], Yor [START_REF] Yor | Generalized meanders as limits of weighted Bessel processes, and an elementary proof of Spitzer's asymptotic result on Brownian windings[END_REF], Pap-Yor [START_REF] Pap | The accuracy of Cauchy approximation for the windings of planar Brownian motion[END_REF], Bentkus-Pap-Yor [START_REF] Bentkus | Optimal bounds for Cauchy approximations for the winding distribution of planar Brownian motion[END_REF]. In this paper, we take up again the study of the first hitting times:

T θ -d,c ≡ inf{t : θ t / ∈ (-d, c)}, (c, d > 0),
this time in relation with Bougerol's well-known identity (see Bougerol [START_REF] Ph | Exemples de théorèmes locaux sur les groupes résolubles[END_REF], Alili-Dufresne-Yor [START_REF] Alili | Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift. In Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers[END_REF] and Yor [START_REF] Yor | Exponential Functionals of Brownian Motion and Related Processes[END_REF]): for fixed u > 0:

sinh(β u ) (law)
= β( u 0 ds exp(2βs)) ,

where ( βt , t ≥ 0) is a Brownian motion ‡ , independent of β.

In particular, it turns out that: for fixed c > 0:

θ T β c (law) = C a(c) , (⋆)
where β is a BM ‡ independent of (θ u , u ≥ 0), T β c = inf{t : βt = c}, (C t , t ≥ 0) is a standard Cauchy process and a(c) = arg sinh(c) ≡ log c + √ 1 + c 2 , c ∈ R. The identity (⋆) yields yet another proof of the celebrated Spitzer theorem:

2 log t θ t (law) -→ t→∞ C 1 ,
with the help of Williams' "pinching method" (see Williams [START_REF] Williams | A simple geometric proof of Spitzer's winding number formula for 2-dimensional Brownian motion[END_REF] and Messulam-Yor [START_REF] Messulam | On D. Williams' "pinching method" and some applications[END_REF]). Moreover, we study the distributions of T θ -∞,c and T θ -c,c . In particular, we give explicit formulae for the density function of T θ -c,c and for the first moment of ln T θ -c,c .

The last section of the paper is devoted to developing similar results when planar Brownian motion is replaced by a complex valued Ornstein-Uhlenbeck process. We note that Bertoin-Werner [START_REF] Bertoin | Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process[END_REF] already made discussions of windings for planar Brownian motion using arguments related to Ornstein-Uhlenbeck processes.

Firstly, we obtain some analogue of (⋆) when T β c is replaced by T for λ large and for λ small.

The Brownian motion case 2.1 A reminder on planar Brownian motion

Let (Z t = X t +iY t , t ≥ 0) denote a standard planar Brownian motion, starting from x 0 + i0, x 0 > 0, where (X t , t ≥ 0) and (Y t , t ≥ 0) are two independent linear Brownian motions, starting respectively from x 0 and 0.

As is well known (see e.g. Itô-McKean [START_REF] Itô | Diffusion Processes and their Sample Paths[END_REF]), since x 0 = 0, (Z t , t ≥ 0) does not visit a.s. the point 0 but keeps winding around 0 infinitely often. In particular, the continuous winding process θ t = Im( t 0 dZs Zs ), t ≥ 0 is well defined.

Furthermore, there is the skew product representation:

log |Z t | + iθ t ≡ t 0 dZ s Z s = (β u + iγ u ) u=Ht= t 0 ds |Zs| 2 , (1)
where (β u + iγ u , u ≥ 0) is another planar Brownian motion starting from log x 0 + i0. For a study of the Bessel clock H, see Yor [START_REF] Yor | Loi de l'indice du lacet Brownien et Distribution de Hartman-Watson[END_REF].

Rewriting (1) as:

log |Z t | = β Ht ; θ t = γ Ht , (2)
we easily obtain that the total σ-fields σ{|Z t | , t ≥ 0} and σ{β u , u ≥ 0} are identical, whereas (γ u , u ≥ 0) is independent from (|Z t | , t ≥ 0). A number of studies of the properties of the first hitting time (see Figure 1

(b)) T θ -d,c ≡ inf{t : θ t / ∈ (-d, c)}, (c, d > 0),
have been developed, going back to Spitzer [START_REF] Spitzer | Some theorems concerning two-dimensional Brownian Motion[END_REF].

In particular, it is well known (Spitzer [START_REF] Spitzer | Some theorems concerning two-dimensional Brownian Motion[END_REF], Burkholder [START_REF] Burkholder | Exit times of Brownian Motion, Harmonic Majorization and Hardy Spaces[END_REF], Revuz-Yor [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] Ex. 2.21/page 196) that:

E (T θ -d,c ) p < ∞ if and only if p < π 2(c + d) . (3)
Moreover, Spitzer's asymptotic theorem (see e.g. Spitzer [START_REF] Spitzer | Some theorems concerning two-dimensional Brownian Motion[END_REF]) states that:

2θ t log t (law) -→ t→∞ C 1 (law) = γ T β 1 , (4)
where C 1 is a standard Cauchy variable.

On the Laplace transform of the distribution of the hitting time

T θ c ≡ T θ -∞,c
Now, we use the representation (2) to access the distribution of T θ c (see Figure 1(a)). We define T γ c ≡ inf{t : γ t / ∈ (-∞, c)} the hitting time associated to the Brownian motion (γ t , t ≥ 0). Note that, from ( 2):

H T θ c = T γ c , hence: T θ c = H -1 u u=T γ c
, where

H -1 u ≡ inf{t : H t > u} = u 0 ds exp(2β s ) := A u . (5)
Thus, we have obtained:

T θ c = A T γ c , (6)
where (A u , u ≥ 0) and T γ c are independent, since β and γ are independent. We can write:

β s = (log x 0 ) + β (0) s , with (β (0)
s , s ≥ 0) a standard onedimensional Brownian motion starting from 0. Then, we deduce from ( 6) that:

T θ c = x 2 0 T γ c 0 ds exp(2β (0) s ) . (7)
From now on, for simplicity, we shall take x 0 = 1, but this is really no restriction, as the dependency in x 0 , which is exhibited in [START_REF] Billingsley | Probability and Measure[END_REF], is very simple. We shall also make use of Bougerol's identity [START_REF] Ph | Exemples de théorèmes locaux sur les groupes résolubles[END_REF][START_REF] Alili | Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift. In Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers[END_REF] and [START_REF] Yor | Exponential Functionals of Brownian Motion and Related Processes[END_REF] (p. 200), which is very useful to study the distribution of A u (e.g. [START_REF] Matsumoto | On Bougerol and Dufresne's identities for exponential Brownian functionals[END_REF][START_REF] Matsumoto | Exponential functionals of Brownian motion, I: Probability laws at fixed time[END_REF]). For any fixed u > 0:

sinh(β u ) (law) = βAu = β( u 0 ds exp(2βs)) , (8)
where on the right hand side, ( βt , t ≥ 0) is a Brownian motion, independent of A u ≡ u 0 ds exp(2β s ). Thus, from ( 8) and ( 6), and as is well known [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], the law of β T γ c is the Cauchy law with parameter c, i.e., with density:

h c (y) = c π(c 2 + y 2 ) ,
we deduce that:

Proposition 2.1 For fixed c > 0, there is the following identity in law:

sinh(C c ) (law) = β(T θ c ) , (9)
where, on the left hand side, (C c , c ≥ 0) denotes a standard Cauchy process and on the right hand side, ( βu , u ≥ 0) is a one-dimensional BM, independent from T θ c .

We may now identify the densities of the variables found on both sides of (9),i.e.: on the left hand side:

1 √ 1+x 2 h c (arg sinh x) = 1 √ 1+x 2 h c (a(x)) ; on the right hand side: E 1 √ 2πT θ c exp -x 2 2T θ c
, where a(x) = arg sinh(x). Thus, we have obtained the following:

Proposition 2.2
The distribution of T θ c may be characterized by:

E 1 2πT θ c exp - x 2T θ c = 1 √ 1 + x c π(c 2 + log 2 ( √ x + √ 1 + x)) , x ≥ 0. ( 10
)
The proof of Proposition 2.2 follows from: a(y) = arg sinh(y) ≡ log(y + 1 + y 2 ) and by making the change of variable y 2 = x. Let us now define the probability:

Q c = πc 2 2T θ c • P .
The fact that Q c is a probability follows from [START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes[END_REF] by taking x = 0. Thus we obtain that c E[ π/2T θ c ] = 1, and we may write:

E Qc exp - x 2T θ c = 1 √ 1 + x 1 1 + 1 c 2 log 2 ( √ x + √ 1 + x)
, ∀x ≥ 0, [START_REF] Comtet | [END_REF] which yields the Laplace transform of 1/T θ c under Q c . Let us now take a look at what happens if we make c → ∞. If we denote by T β 1 ≡ inf{t : β t = 1} the first hitting time of level 1 for a standard BM β and by N a standard Gaussian variable N (0, 1), from equation [START_REF] Comtet | [END_REF], we obtain:

lim c→∞ E Qc e -x/2T θ c = E e -xN 2 /2 = E e -x/2T β 1 , (12)
which means that :

T θ c (law) -→ c→∞ T β 1 .
(At this point, one may wonder whether there is some kind of convergence in law involving (θ u , u ≥ 0), under Q c , as c → ∞, but, we shall not touch this point).

From Proposition 2.2 we deduce the following: Corollary 2.3 Let ϕ(x) denote the Laplace transform [START_REF] Comtet | [END_REF], that is the Laplace transform of 1/2T θ c under Q c . Then, the Laplace transform of 1/2T θ c under P is:

E exp - x 2T θ c = ∞ x dw √ w -x ϕ(w). ( 13
)
Proof of Corollary 2.3 From Fubini's theorem, we deduce from [START_REF] Comtet | [END_REF] that:

E exp - x 2T θ c = ∞ 0 dy √ y E 1 2πT θ c exp - x + y 2T θ c = ∞ 0 dy √ y ϕ(x + y) y=xt = √ x ∞ 0 dt √ t ϕ(x(1 + t)) v=1+t = √ x ∞ 1 dv √ v -1 ϕ(xv) w=xv = ∞ x dw √ w -x ϕ(w),
which is formula (13).

Some related identities in law

This subsection is strongly related to [START_REF] Dufresne | A two-dimensional extension of Bougerol's identity in law for the exponential functional of Brownian motion: the story so far[END_REF]. A slightly different look at the combination of Bougerol's identity [START_REF] Ph | Exemples de théorèmes locaux sur les groupes résolubles[END_REF] and the skew-product representation (1) lead to the following striking identities in law:

Proposition 2.4 Let (δ u , u ≥ 0) be a 1-dimensional Brownian motion independent of the planar Brownian motion (Z u , u ≥ 0), starting from 1 + i0.

Then, for any b ≥ 0, the following identities in law hold:

(i) H T δ b (law) = T β a(b) (ii) θ T δ b (law) = C a(b) (iii) θT δ b (law) = |C a(b) |,
where C A is a Cauchy variable with parameter A and θu = sup s≤u θ s .

Proof of Proposition 2.4 From the symmetry principle (see [START_REF] André | Solution directe du problème résolu par M. Bertrand[END_REF] for the original Note and [START_REF] Gallardo | Mouvement Brownien et calcul d[END_REF] for a detailed discussion), Bougerol's identity may be equivalently stated as: sinh( βu )

(law) = δAu(β) . (14)
Consequently, the laws of the first hitting times of a fixed level b by the processes on each side of ( 14) are identical, that is:

T β a(b) (law) = H T δ b , which is (i).
(ii) follows from (i) since:

θ u (law)
= γ Hu , with (γ s , s ≥ 0) a Brownian motion independent of (H u , u ≥ 0) and (C u , u ≥ 0) may be represented as (γ T β u , u ≥ 0). (iii) follows from (ii), again with the help of the symmetry principle.

2

Remark 2.5 Proposition 2.2 may also be derived from (iii) in Proposition 2.4. Indeed, for c > 0, starting from the LHS of (iii), and letting N ∼ N (0, 1) independent from T θ c :

P θT δ b < c = P T δ b < T θ c = P b < δT θ c = P b < T θ c |N| = P b T θ c < |N| = 2 π E ∞ b/ √ T θ c dy e -y 2 /2 , (15)
while, on the RHS of (iii):

P |C a(b) | < c = 2 c 0 a(b) dy π(a 2 (b) + y 2 ) y=a(b)h = 2 π c/a(b) 0 dh 1 + h 2 . ( 16
)
Taking derivatives in ( 15) and ( 16) with respect to b and changing the variables b = √ x, we obtain Proposition 2.2.

Recovering Spitzer's theorem

The identity (ii) in Proposition 2.4 is reminiscent of Williams' remark (see [START_REF] Williams | A simple geometric proof of Spitzer's winding number formula for 2-dimensional Brownian motion[END_REF][START_REF] Messulam | On D. Williams' "pinching method" and some applications[END_REF]), that:

H T R r (law) = T δ log r , (17)
where here R starts from 1 and δ starts from 0 (in fact, this is a consequence of ( 2) ). For a number of variants of [START_REF] Gallardo | Mouvement Brownien et calcul d[END_REF], see [START_REF] Yor | Une décomposition asymptotique du nombre de tours du mouvement brownien complexe[END_REF][START_REF] Mansuy | Aspects of Brownian Motion[END_REF]. This was D. Williams' starting point for a non-computational proof of Spitzer's result (4). We note that in (ii), T δ b is independent of the process (θ u , u ≥ 0) while in ( 17) T R r depends on (θ u , u ≥ 0). Actually, we can mimic Williams' "pinching method" to derive Spitzer's theorem ( 4) from (ii) in Proposition 2.4.

Proposition 2.6 (A new proof of Spitzer's theorem) As t → ∞, θ T δ √ t
θ t converges in law, which implies that:

1 log t θ T δ √ t -θ t (P ) -→ t→∞ 0, (18)
and, in turn, implies Spitzer's theorem (see formula ( 4) ):

2 log t θ t (law) -→ t→∞ C 1 .
Proof of Proposition 2.6 From equation (ii) of Proposition 2.4 we note:

1 log b θ T δ b (law) = C a(b) log b (law) -→ b→∞ C 1 .
So, for b = √ t we have:

2 log t θ T δ √ t (law) -→ b→∞ C 1 .
On the other hand, following Williams' "pinching method", we note that:

1 log t θ T δ √ t -θ t (law) -→ t→∞ 0, since Z u = x 0 + Z (0)
u and also, as we change variables u = tv and we use the scaling property, we obtain:

θ T δ √ t -θ t ≡ Im T δ √ t t dZ u Z u (law) -→ t→∞ Im T δ 1 1 dZ (0) v Z (0) v .
Here, the limit variable is -in our opinion-of no other interest than its existence which implies [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF], hence (4). Proposition 2.7 The asymptotic equivalence:

(log t) P (T θ c > t) t→∞ -→ (4c)/π , (19)
holds.

As a consequence, for

η > 0, E[(log T θ c ) η + ] < ∞ if and only if η < 1 (where (•) + denotes the positive part).
Proof of Proposition 2.7 a) We rely upon the asymptotic distribution of

H t ≡ t 0 ds |Zs| 2 which is given by [32]: 4H t (log t) 2 (law) -→ t→∞ T β 1 ≡ inf{t : β t = 1}, (20)
or equivalently:

log t 2 √ H t (law) -→ t→∞ |N| , ( 21
)
where N is a standard Gaussian variable N (0, 1). We note that, from the representation (2) of θ t , the result ( 20) is equivalent to Spitzer's theorem [START_REF] Spitzer | Some theorems concerning two-dimensional Brownian Motion[END_REF]:

2θ t log t (law) -→ t→∞ C 1 (law) = γ T β 1 , (22)
where C 1 is a standard Cauchy variable. b) We shall now use this, in order to deduce Proposition 2.7. We denote S θ t ≡ sup s≤t θ s ≡ S γ Ht and we note that (from scaling):

P (T θ c ≥ t) = P (S γ Ht ≤ c) = P ( H t S γ 1 ≤ c), (23)
since γ and H are independent. Thus, we have (since S γ 1 (law)

= |N| and by making the change of variable x = cy √ Ht ):

P (T θ c ≥ t) = 2 π E c/ √ Ht 0 dx e -x 2 2 = 2 π c E 1 0 dy √ H t exp - c 2 y 2 2H t . (24)
Thus, we now deduce from ( 21) that:

log t 2 P (T θ c ≥ t) t→∞ -→ 2 π c E [|N|] = 2 π c. ( 25
)
which is precisely [START_REF] Ishiyama | Methods for Evaluating Density Functions of Exponential Functionals Represented as Integrals of Geometric Brownian Motion[END_REF].

It is now elementary to deduce from (25) that: for η > 0:

E[(log T θ c ) η + ] < ∞ ⇔ 0 < η < 1, since (25)
is equivalent to:

u P (log T θ c > u) u→∞ -→ 4c π . (26)
Consequently, Fubini's theorem yields:

E (log T θ c ) η + = ∞ 0 du η u η-1 P (log T θ c > u),
and from [START_REF] Matsumoto | On Bougerol and Dufresne's identities for exponential Brownian functionals[END_REF] this is finite if and only if:

∞ • du u η-2 < ∞ ⇔ η < 1. So, E[(log T θ c ) η + ] < ∞ ⇔ 0 < η < 1. 2
Now we give several examples of random times T : C(R + , R) → R + which may be studied quite similarly to T θ c . For such times T , it will always be true that:

H T (θ) = T (γ) is equivalent to T (θ) = A T (γ)
, defined with respect to Z, issued from x 0 = 0. Using Bougerol's identity, we obtain:

sinh(β T (γ) ) (law) = βA T (γ) = β(T(θ)) . ( 27
)
where ( βu , u ≥ 0) is a 1-dimensional Brownian motion independent of (β, γ) (or equivalently, of Z). Consequently, denoting by h T the density of β T (γ) , we deduce from [START_REF] Matsumoto | Exponential functionals of Brownian motion, I: Probability laws at fixed time[END_REF] that:

E 1 2πT (θ) exp(- x 2 2T (θ) ) = 1 √ 1 + x 2 h T (log(x + √ 1 + x 2 )), (28)
or equivalently, changing x in √ x, we obtain:

E 1 2πT (θ) exp(- x 2T (θ) ) = 1 √ 1 + x h T (log( √ x + √ 1 + x)). (29)
In a number of cases, h T is known explicitly, for example: (i)

T (γ) = T γ -d,c ⇔ T (θ) = T γ -d,c 0 ds exp (2β s ) = T θ -d,c .
Thus:

E   1 2πT θ -d,c exp(- x 2T θ -d,c )   = 1 √ 1 + x h -d,c (log( √ x + √ 1 + x)), (30)
where h -d,c is the density of the variable β T γ -d,c . The law of β T γ -d,c may be obtained from its characteristic function which is given by [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], page 73:

E exp(iλβ T γ -d,c ) = E exp(- λ 2 2 T γ -d,c ) = cosh( λ 2 (c -d)) cosh( λ 2 (c + d))
.

In particular, for c = d, we recover the very classical formula:

E exp(iλβ T γ -c,c ) = 1 cosh(λc)
.

It is well known that [START_REF] Dugué | 158 Random Functions: General Theory with Special Reference to Laplacian Random Functions by Paul Lévy[END_REF][START_REF] Biane | Valeurs principales associées aux temps locaux browniens[END_REF]:

E exp(iλβ T γ -c,c )) = 1 cosh(λc) = 1 cosh(πλ c π ) = ∞ -∞ e i( λc π )x 1 2π 1 cosh( x 2 ) dx y= cx π = ∞ -∞ e iλy 1 2π π c cosh( yπ 2c ) dy = ∞ -∞ e iλy 1 2c 1 cosh( yπ 2c ) dy. (31)
Hence, the density of β T γ -c,c is:

h -c,c (x) = 1 2c 1 cosh( xπ 2c ) = 1 c 1 e xπ 2c + e -xπ 2c , and
h -c,c log( √ x + √ 1 + x) = 1 c 1 ( √ x + √ 1 + x) ζ + ( √ x + √ 1 + x) -ζ ,
where ζ = π 2c . However using:

( √ x + √ 1 + x) -ζ = ( √ 1 + x - √ x) ζ , (32)
we obtain:

h -c,c log( √ x + √ 1 + x) = 1 c 1 ( √ x + √ 1 + x) ζ + ( √ 1 + x - √ x) ζ . (33)
So we deduce that (for c = d):

E   1 2πT θ -c,c exp(- x 2T θ -c,c )   = 1 c 1 √ 1 + x 1 ( √ x + √ 1 + x) ζ + ( √ 1 + x - √ x) ζ . ( 34
)

2

(ii) As a second example of a random time T , let us consider the time introduced in [START_REF] Vallois | Amplitude du mouvement Brownien et juxtaposition des excursions positives et ngatives[END_REF], [START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes[END_REF], exercise 6.2, p. 178 (we use a slightly different notation). Let (β t , t ≥ 0) be a real valued Brownian motion and define, for c > 0:

T (θ) ≡ T θ c = inf t : sup s≤t θ s -inf s≤t θ s = c , T (γ) ≡ T γ c = inf t : sup s≤t γ s -inf s≤t γ s = c .
Thus, from the skew-product representation (1), θ u ≡ γ Hu , by replacing u = T θ c , we obtain:

H T θ c = T γ c ⇒ T θ c = T γ c 0 ds exp (2β s ) ≡ A T γ c .
Thus:

E   1 2πT θ c exp(- x 2T θ c )   = 1 √ 1 + x h c (log( √ x + √ 1 + x)), (35)
where h c is the density of the variable β T γ c . The law of β T γ c may be obtained from its characteristic function which is given by [START_REF] Biane | Valeurs principales associées aux temps locaux browniens[END_REF][START_REF] Chaumont | Exercises in Probability: A Guided Tour from Measure Theory to Random Processes[END_REF]:

E exp(iλβ T γ c ) = E exp(- λ 2 2 T γ c ) = 1 (cosh(λ c 2 )) 2 = 1 cosh(πλ c 2π ) 2 = ∞ -∞ e i( λc 2π )x 1 2π x sinh( x 2 ) dx y= cx 2π = ∞ -∞ e iλy 1 2π 2πy c sinh( πy c ) 2π c dy = ∞ -∞ e iλy 2π c 2 y sinh( πy c ) dy. (36)
So, the density of β T γ c is:

h c (y) = 2πy c 2 1 sinh( πy c ) = 4π c 2 y e πy c -e -πy c , and
h c log( √ x + √ 1 + x) = 4π c 2 log( √ x + √ 1 + x) ( √ x + √ 1 + x) ζ -( √ x + √ 1 + x) - ζ ,
where ζ = π c . Thus:

E   1 2πT θ c exp(- x 2T θ c )   = 4π c 2 1 √ 1 + x log( √ x + √ 1 + x) ( √ x + √ 1 + x) ζ -( √ 1 + x - √ x) ζ . (37)
We note that this study may be related to [START_REF] Pitman | Infinitely divisible laws associated with hyperbolic functions[END_REF]; and more precisely β T γ c and T γ c correspond to the variables C 2 and Ĉ2 respectively (see e.g. Table 6 in p. 312).

2

Let us now return to the case of T θ -c,c (example (i)). More specifically, we shall obtain its density function f (t).

Proposition 2.8 The density function f of T θ -c,c is given by:

f (t) = 1 √ 2c ∞ k=0 (-1) k Γ(ν k ) Γ(2ν k ) 1 √ t e -1 4t M1 2 ,ν k ( 1 2t ), ( 38
)
where M a,b (•) is the Whittaker function with parameters a, b. Equivalently:

f (t) = √ 2 c ∞ k=0 (-1) k 1 √ t e -1 2t 1 2t ν k + 1 2 ν k ∞ n=0 Γ(ν k + n) Γ(2ν k + n + 1) 1 n! 1 2t n , ( 39
)
where ν k = π 4c (2k + 1). Proof of Proposition 2.8 The following calculation relies upon a private note by A. Comtet [START_REF] Comtet | [END_REF].We denote:

ϕ ζ (x) = ( √ x + √ 1 + x) ζ + ( √ 1 + x - √ x) ζ .
Noting:

√ 1 + x = cosh y 2 ⇐⇒ y = 2 arg cosh( √ 1 + x), (40)
we get:

ϕ ζ (x) = (sinh y 2 + cosh y 2 ) ζ + (cosh y 2 -sinh y 2 ) ζ = 2 cosh yζ 2 .
Thus, from (34), we have:

II := E   1 2πT θ -c,c exp(- x 2T θ -c,c )   = 1 ψ 1 cosh y 2 1 cosh πy 2ψ , (41)
where ψ = 2c. However, expanding cosh πy 2ψ , we get:

1 cosh πy 2ψ = 2 e -πy 2ψ 1 + e -πy ψ = 2 ∞ k=0 -e -πy ψ k e -πy 2ψ ,
and from (41), we deduce that:

II = ∞ k=0 2 ψ (-1) k cosh y 2 e -π 2ψ (2k+1)y = ∞ k=0 4(-1) k ψ √ 2 2 sinh y 2 cosh y 2 sinh y 2 cosh y 2 e -ν k y = ∞ k=0 4(-1) k ψ √ 2 2 sinh y 2 cosh y 2 tanh y 2 e -ν k y ,
where ν k = π 2ψ (2k + 1). From (40), we have 1 + x = cosh 2 y 2 ⇐⇒ x = sinh 2 y 2 , thus:

(tanh y 2 ) 1/2 = sinh y 2 cosh y 2 = √ x √ 1 + x 1/2 = x 1 + x 1/4
. Moreover, we know that (see [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables[END_REF], equation 8.6.10, or [START_REF] Lebedev | Special Functions and their Applications[END_REF]):

i π 2 sinh y e -ν k y = Q 1/2 ν k -1/2 (cosh y),
where {Q a b (•)} is the family of Legendre functions and cosh y = 2x + 1. So, we deduce:

II = ∞ k=0 4(-i) ψ √ π (-1) k x 1 + x 1/4 Q 1/2 ν k -1/2 (2x + 1). ( 42
)
By using formula 7.621.9, page 864 in [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF]:

∞ 0 e -sw M l,ν k (w) dw w = 2Γ(1 + 2ν k ) e -iπl Γ( 1 2 + ν k + l) s -1 2 s + 1 2 l/2 Q l ν k -1/2 (2s), ( 43
) with: l = 1 2 , ν k = π 2ψ (2k + 1), s = x + 1 2 and M •,• (•) denoting the Whittaker function, which is defined as: M a,b (w) = w b+ 1 2 e -1 2 w Γ(2b + 1) Γ( 1 2 + b -a) ∞ n=0 Γ( 1 2 + b -a + n) Γ(2b + 1 + n) w n n! .
we have:

-2i Γ(1 + 2ν k ) Γ(1 + ν k ) x 1 + x 1/4 Q 1/2 ν k -1/2 (2x + 1) = ∞ 0 e -sw M 1/2,ν k (w) dw w . (44)
From (42) and by changing the variable w = 1 2t , we deduce:

II = ∞ k=0 2 ψ √ π (-1) k Γ(ν k + 1) Γ(2ν k + 1) ∞ 0 dw w exp -w(x + 1 2 ) M 1/2,ν k (w) = ∞ k=0 ∞ 0 dt t 2 ψ √ π (-1) k Γ(ν k + 1) Γ(2ν k + 1) exp - 1 4t - x 2t M 1/2,ν k ( 1 2t
).

(45)

By using the equations ( 41) and (45), we conclude:

E   1 2πT θ -c,c exp - x 2T θ -c,c   = ∞ k=0 ∞ 0 dt t 2 ψ √ π (-1) k Γ(ν k + 1) Γ(2ν k + 1) exp - 1 4t - x 2t M1 2 ,ν k ( 1 2t ) = ∞ k=0 ∞ 0 dt t 2 ψ √ π (-1) k Γ( π 4c (2k + 1) + 1) Γ(2 π 4c (2k + 1) + 1) exp - 1 4t - x 2t M1 2 , π 4c (2k+1) ( 1 2t
).

(46)

Thus, the density function f of T θ -c,c is given by:

f (t) = 2 √ 2 ψ ∞ k=0 (-1) k Γ(ν k + 1) Γ(2ν k + 1) 1 √ t e -1 4t M1 2 ,ν k ( 1 2t ) (47) = √ 2 c ∞ k=0 (-1) k Γ( π 4a (2k + 1) + 1) Γ( π 2a (2k + 1) + 1) 1 √ t e -1 4t M1 2 , π 4a (2k+1) ( 1 2t ) (48) = √ 2 c ∞ k=0 (-1) k ν k Γ(ν k ) 2ν k Γ(2ν k ) 1 √ t e -1 4t M1 2 ,ν k ( 1 2t ), ( 49
)
where the Whittaker function M1 2 ,ν k ( 1 2t ) is:

M1 2 , π 4c (2k+1) ( 1 2t ) = 1 2t π 4c (2k+1)+ 1 2 e -1 4t Γ( π 2c (2k + 1) + 1) Γ( π 4c (2k + 1)) ∞ n=0 Γ( π 4c (2k + 1) + n) Γ( π 2c (2k + 1) + 1 + n) 1 n! 1 2t n = 1 2t ν k + 1 2 e -1 4t Γ(2ν k + 1) Γ(ν k ) ∞ n=0 Γ(ν k + n) Γ(2ν k + 1 + n) 1 n! 1 2t n = 1 2t ν k + 1 2 e -1 4t (2ν k ) Γ(2ν k ) Γ(ν k ) ∞ n=0 Γ(ν k + n) (2ν k + n)Γ(2ν k + n) 1 n! 1 2t n . ( 50
)
Thus, from (49) and (50), we deduce (39).

2

Next, we present the graphs of different approximations f K,N (t) of f (t), in [START_REF] Yor | Exponential Functionals of Brownian Motion and Related Processes[END_REF], where f K,N denotes the sum in the series in [START_REF] Yor | Exponential Functionals of Brownian Motion and Related Processes[END_REF] of the terms for k ≤ K, and n ≤ N. Remark 2.9

• Figure 2 represents the approximation of the density function f with respect to the time t (for K and N ≤ 9), with c = 2π, whereas Figure 3 represents the approximation of f with respect to the time t for several values of k and n, with c = 2π.

• From Figure 3, we may remark that the approximation K and N ≤ 9

is sufficiently good (comparing to the one for K and N ≤ 100).

• For the case K and N ≤ 9 it seems that locally, in a small area around 0, f (t) < 0 which is not right. This is due to the first negative (k = 1) term of the sum and due to the fact that we have omitted many terms. However, this is not a problem because it appears only locally. Similar irregularities have already been observed in previous articles [START_REF] Ishiyama | Methods for Evaluating Density Functions of Exponential Functionals Represented as Integrals of Geometric Brownian Motion[END_REF] p.275.

On the first moment of ln T θ -c,c

This subsection is related to a result in [START_REF] Comtet | Exponential functionals of Brownian motion and disordered systems[END_REF].

Proposition 2.10 The first moment of ln T θ -c,c has the following integral representation:

E ln T θ -c,c = 2 ∞ 0 dz cosh πz 2 ln (sinh (cz)) + ln (2) + c E , (51)
where c E = -Γ ′ (1) is the Euler-Mascheroni constant (also called Euler's constant).

Proof of Proposition 2.10 Let us return to equations ( 2) and ( 6). So, for t = T θ -c,c , we have:

θ T θ -c,c = γ H T θ -c,c ⇐⇒ H T θ -c,c = T γ -c,c ⇐⇒ T θ -c,c = A T γ -c,c . (52)
Thus, for ε > 0:

E T θ -c,c ε = E A T γ -c,c ε .
Consider (δ u , u ≥ 0) a Brownian motion, independent of A t . Then, Bougerol's identity and the scaling property yield (G a denotes a gamma variable with parameter a, and N 2 (law) = 2G 1/2 ):

E (sinh (B t )) 2ε = E (δ At ) 2ε = E A ε t (δ 1 ) 2ε = E [A ε t ] E 2G 1/2 ε = E [A ε t ] (2 ε ) Γ 1 2 + ε Γ 1 2 , because E G 1/2 ε = ∞ 0 x ε+ 1 2 -1 e -x Γ 1 2 dx = Γ 1 2 + ε Γ 1 2 .
Thus, for t = T γ -c,c , we have:

E sinh B T γ -c,c 2ε = E A ε T γ -c,c (2 ε ) Γ 1 2 + ε Γ 1 2 . (53)
Recall that [START_REF] Dugué | 158 Random Functions: General Theory with Special Reference to Laplacian Random Functions by Paul Lévy[END_REF][START_REF] Biane | Valeurs principales associées aux temps locaux browniens[END_REF]:

E exp(iλB T γ -c,c )) = E exp(- λ 2 2 T γ -c,c )) = 1 cosh(λc) ,
and the density of β T γ -c,c is:

h -c,c (y) = 1 2c 1 cosh( yπ 2c ) = 1 c 1 e yπ 2c + e -yπ 2c .
Thus, on the left hand side of (53), we have:

E sinh B T γ -c,c 2ε = ∞ -∞ dy 2c 1 cosh( πy 2c ) (sinh (y)) 2ε = ∞ 0 dy c 1 cosh( πy 2c ) (sinh y) 2ε = ∞ 0 dz 1 cosh( πz 2 ) (sinh(cz)) 2ε ,
where we have made the change of variable z = y c . Hence, from (53), by writing:

E A ε T γ -c,c = E T θ -c,c ε = E e ε ln(T θ -c,c ) ,
we deduce:

Γ 1 2 + ε Γ 1 2 E e ε ln(T θ -c,c ) = 1 2 ε ∞ 0 dz cosh( πz 2 ) (sinh(cz)) 2ε ,
and by removing 1 from both sides, we obtain:

Γ 1 2 + ε Γ 1 2 E e ε ln(T θ -c,c ) -1 = ∞ 0 dz cosh( πz 2 ) (sinh(cz)) 2ε 2 ε -1 . ( 54
)
On the left hand side, we apply the trivial identity ab -

1 = a(b -1) + a -1 with a = Γ( 1 2 +ε) Γ( 1 2 ) and b = E e ε ln(T θ -c,c
) , we divide by ε and we take the limit for ε → 0. Thus:

a(b -1) ε = Γ 1 2 + ε Γ 1 2 E e ε ln(T θ -c,c ) -1 ε ε→0 -→ E ln T θ -c,c
, and:

a -1 ε = 1 ε Γ 1 2 + ε Γ 1 2 -1 = 1 Γ 1 2 Γ 1 2 + ε -Γ 1 2 ε ε→0 -→ 1 √ π Γ ′ 1 2 = 1 √ π - √ π (c E + 2 ln 2) = -(c E + 2 ln 2) .
On the right hand side of (54), we have:

1 ε (sinh(cz)) 2 2 ε -1 = 1 ε exp ε ln (sinh(cz)) 2 2 -1 , hence: 1 ε ∞ 0 dz cosh( πz 2 ) (sinh(cz)) 2ε 2 ε -1 ε→0 -→ ∞ 0 dz cosh( πz 2 ) ln (sinh(cz)) 2 2 = -ln (2) + 2 ∞ 0 dz cosh( πz 2 ) (ln (sinh(cz))) ,
which finishes the proof.

2

Remark 2.11 a) We denote now:

F (c) ≡ ∞ 0 dz cosh πz 2 ln (sinh (cz)) . (55)
Thus:

F (c) -ln(c) ≡ ∞ 0 dz cosh πz 2 ln sinh (cz) c c→0 -→ ∞ 0 dz ln(z) cosh πz 2 ≈ -0.7832. (56)
b) More generally, we denote:

F (c, δ) ≡ ∞ 0 dz cosh (δz) ln (sinh (cz)) , (57)
and, changing the variables: z = π 2δ u, we obtain:

F (c, δ) = π 2δ ∞ 0 du cosh π 2 u ln sinh c π 2δ u = π 2δ F c π 2δ . (58)
3 The Ornstein-Uhlenbeck case 3.1 An identity in law for Ornstein-Uhlenbeck processes, which is connected to Bougerol's identity

Consider the complex valued Ornstein-Uhlenbeck (OU) process:

Z t = z 0 + Zt -λ t 0 Z s ds, (59)
where Zt is a complex valued Brownian motion (BM), z 0 ∈ C and λ ≥ 0 and

T (λ) c ≡ T θ Z -c,c ≡ inf t ≥ 0 : θ Z t = c (θ Z t
is the continuous winding process associated to Z) denoting the first hitting time of the symmetric conic boundary of angle c for Z. It is well known that [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]:

Z t = e -λt z 0 + t 0 e λs d Zs = e -λt (B αt ) , (60)
where, in the second equation, with the help of Dambis-Dubins-Schwarz Theorem, (B t , t ≥ 0) is a complex valued Brownian motion starting from z 0 and α t = t 0 e 2λs ds = e 2λt -1 2λ .

We are interested in the study of the continuous winding process θ Z t = Im( t 0 dZs Zs ), t ≥ 0. By applying Itô's formula to (60), we have: dZ s = e -λs (-λ)B αs ds + e -λs d (B αs ) .

We divide by Z s and we obtain:

dZ s Z s = -λ ds + dB αs B αs , hence: Im dZ s Z s = Im dB αs B αs ,
which means that:

θ Z t = θ B αt .
Thus, the following holds: Proposition 3.1 Using the previously introduced notation, we have:

θ Z t = θ B αt , (61)
and:

T (λ) c = 1 2λ ln 1 + 2λT θ B -c,c , (62)
where T θ B -c,c is the exit time from a cone of angle c for the complex valued BM B.

Proof of Proposition 3.1 We define

T (λ) c ≡ T θ Z -c,c ≡ inf t ≥ 0 : θ Z t = c = inf t ≥ 0 : θ B αt = c . (63)
Thus, we deduce that α T (λ)

c = T θ B -c,c ≡ T θ -c,c
. However, T θ -c,c (the exit time from a cone for the BM) has already been studied in the previous chapter and we know the explicit formula of its density function (Proposition 2.8). Thus:

T (λ) c = α -1 T θ B -c,c = α -1 T θ -c,c , (64)
where α -1 (t) = 1 2λ ln (1 + 2λt). Consequently:

T (λ) c = 1 2λ ln 1 + 2λT θ -c,c ,
and:

E T (λ) c = 1 2λ E ln 1 + 2λT θ -c,c , (65)
which finishes the proof.

2

From now on, for simplicity, we shall take z 0 = 1 (but this is really no restriction, as the dependency in z 0 , which is exhibited in [START_REF] Billingsley | Probability and Measure[END_REF], is very simple).

The following Proposition may be considered as an extension of the identity in law (ii) in Proposition 2.4, which results from Bougerol's identity.

Proposition 3.2 Consider (Z λ t , t ≥ 0) and (U λ t , t ≥ 0) two independent Ornstein-Uhlenbeck processes, the first one complex valued and the second one real valued, both starting from a point different from 0, and call

T (λ) b (U λ ) = inf t ≥ 0 : e λt U λ t = b .
Then, an Ornstein-Uhlenbeck extension of identity in law (ii) in Proposition 2.4 is the following:

θ Z λ T (λ) b (U λ ) (law) = C a(b) , (66)
where a(x) = arg sinh(x).

Proof of Proposition 3.2 Let us consider a second Ornstein-Uhlenbeck process (U λ t , t ≥ 0) independent of the first one. Then, taking equation (60) for U λ t , we have:

e λt U λ t = δ ( e 2λt -1 2λ
) ,

where (δ t , t ≥ 0) is a complex valued Brownian motion starting from z 0 = 1. Thus:

T (λ) b (U λ ) = 1 2λ ln 1 + 2λT δ b . (68)
Equation (61) for t = 1 2λ ln 1 + 2λT δ b , equivalently: α(t) = T δ b becomes (we suppose that z 0 = 1):

θ Z λ T (λ) b (U λ ) = θ Z λ 1 2λ ln(1+2λT δ b ) = θ B u=T δ b (law) = C a(b) .
T (λ) c ≡ T θ Z -c,c ≡ inf t ≥ 0 : θ Z t = c ,
and its first moment.

Proposition 3.3 Asymptotically for λ large, for z 0 = 1, we have:

2λ E T (λ) c -ln (2λ) λ→∞ -→ E ln T θ -c,c , (69)
and:

E ln T θ -c,c = 2 ∞ 0 dz cosh πz 2 ln (sinh (cz)) + ln (2) + c E , (70)
where c E is Euler's constant. For c < π 8 , we have the asymptotic equivalence:

1 λ E T (λ) c -E sinh B T γ -c,c 2 λ→0
-→ -

1 3 E sinh B T γ -c,c 4
.

Equivalently:

d dλ λ=0 E T (λ) c = lim λ→0 1 λ E T (λ) c -E T (0) c = - 1 3 E sinh B T γ -c,c 4 . (72)
Moreover:

E sinh B T γ -c,c 4 = ∞ 0 dz cosh πz 2 (sinh (cz)) 4 . (73)
More precisely, for c < π 8 :

E sinh B T γ -c,c 4 = 1 8 1 cos(4c) -4 1 cos(2c) + 3 , (74)
and asymptotically:

E sinh B T γ -c,c 4 ≃ c→0 5c 4 . (75)
Proof of Proposition 3.3 λ large Let us return to equation (65). For λ → ∞, we have:

E T (λ) c = 1 2λ E ln 1 + 2λT θ -c,c = 1 2λ E ln 2λ T θ -c,c + 1 2λ = ln (2λ) 2λ + 1 2λ E ln T θ -c,c + 1 2λ .
Thus:

2λ E T (λ) c -ln (2λ) λ→∞ -→ E ln T θ -c,c
, which is precisely (69). Moreover, by the integral representation (51) for E ln T θ -c,c , we deduce (70). λ small We shall now study the case λ → 0. We have that:

T (λ) c = 1 2λ ln 1 + 2λT θ -c,c .
For c < π 8 , from Spitzer ( 3), (at least) the first two positive moments of

T θ -c,c are finite: E T θ -c,c
p < ∞, (p = 1, 2). We make the elementary computation:

1 λ ln (1 + 2λx) 2λ -x = 1 λ 1 2λ 1+2λx 1 dy y -x y=1+a = 1 2λ 2 2λx 0 1 1 + a -1 da a=2λb = -2 x 0 b db 1 + 2λb λ→0 -→ -x 2 .
Consequently, by replacing x = T θ -c,c , we have:

1 λ E T (λ) c -E T θ -c,c = E -2 T θ -c,c 0 b db 1 + 2λb
.

We may now use the dominated convergence theorem [START_REF] Billingsley | Probability and Measure[END_REF], since the (db) integral is majorized by (T θ -c,c ) 2 , which is integrable. Thus:

1 λ E T (λ) c -E T θ -c,c λ→0 -→ -E (T θ -c,c ) 2 .
Following the proof of Proposition 2.10, Bougerol's identity and the scaling property yield:

E (sinh (B t )) 2 = E (δ At ) 2 = E A t (δ 1 ) 2 = E [A t ] E (δ 1 ) 2 = E [A t ] .
Thus, for t = T γ -c,c , we have:

E A T γ -c,c = E sinh B T γ -c,c 2
.

Similarly:

E (sinh (B t )) 4 = E (δ At ) 4 = E (A t ) 2 (δ 1 ) 4 = E (A t ) 2 E (δ 1 ) 4 = 3E (A t ) 2 .
Thus, for t = T γ -c,c , we have:

E A T γ -c,c 2 = 1 3 E sinh B T γ -c,c 4 . So, because A T γ -c,c = T θ -c,c
, we deduce (71). In order to prove (72), it suffices to remark that:

E T (0) c = E T θ -c,c = E A T γ -c,c = E sinh B T γ -c,c 2
.

On the one hand, by using the density of B T γ -c,c : which is precisely (74) and this is finite if and only if c < π 8 . Moreover, asymptotically for c → 0, by using the scaling property, we have: 4 , since E T γ -1,1 2 = 5/3 (see [START_REF] Pitman | Infinitely divisible laws associated with hyperbolic functions[END_REF]; by using the notation of this paper, Table 3:

E sinh B T γ -c,c 4 = ∞ -∞ dy
E sinh B T γ -c,c 4 = E sinh cB T γ -1,1 4 c→0 ≃ c 4 E B T γ -1,1 4 = c 4 3E T γ -1,1 2 = 5c
E [X 2 t ] = t(2+3t)
3 for X t = C 1 and t = 1). This asymptotics may also be obtained by (74) by developing cos(4c) and cos(2c) into series up to the second order term and keeping the terms of the order c 4 . We replace x = T θ -c,c , and by the dominated convergence theorem [START_REF] Billingsley | Probability and Measure[END_REF], for c < π 8 , we obtain:

1 λ E T (λ) c - z 2 0 2D E sinh B T γ -c,c 2 λ→0
-→ -1 3

z 2 0 2D 2 E (T θ -c,c ) 2 = - 1 3
z 2 0 2D 2 E sinh B T γ -c,c 4
,

where E sinh B T γ -c,c 4
is given by (73), (74) and asymptotically, for c → 0 by (75).

  c = inf{t : |θ Z t | = c}, the corresponding time for an Ornstein-Uhlenbeck process with parameter λ. Secondly, we identify the distribution of T (λ) c . More specifically, we derive the asymptotics of E T (λ) c

Figure 1 :

 1 Figure 1: Exit times for a planar BM. This figure presents the exit times (a) T θ c (t 1 doesn't matter because the angle is negative) and (b) T θ -d,c for a planar BM starting from x 0 + i0.

2

 2

2 2. 5

 25 On the distributions of T θ c ≡ T θ -∞,c and T θ -c,c

Figure 2 :

 2 Figure 2: Graph of f 9,9 (t), with c = 2π.

Figure 3 :

 3 Figure 3: Graph of f K,N (t) for several values of K and N, with c = 2π.

2 3. 2

 22 On the distribution of T θ -c,c for an Ornstein-Uhlenbeck processNow we turn to the study of the density function of:

4 , 2 . 2 2 4 = 1 2 4 1 2 4

 42241414 which is finite if and only if c < π 8 . In order to prove this, it suffices to use the standard expressions: sinh(x) = e x -e -x 2 and cosh(x) = e x +e -x On the other hand (note T ≡ T γ -c,c ), we remark that -B T(law)= B T and[START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], ex.3.10,E e kB T = E e k T = 1 cos(kc) , for 0 ≤ k < π(2c) -1 , thus: E (sinh (B T )) 4 = 1 2 4 E e B Te -B T E e 4B T -4e 3B T -B T + 6e 2B T -2B T -4e B T -3B T + e -4B T = 2E e 4B T -8E e 2B T

29

 29

2 Remark 3 . 4 0 Z2D t 0 e 0 e 2 0

 2340002 If we slightly modify the above study for the Ornstein-Uhlenbeck process by inserting a diffusion coefficient D:Z t = z 0 + √ 2D Ztλ t s ds ,we obtain:Z t = e -λt z 0 + √ λs d Zs = e -λt (B αt ) ,(76)where in the second equation we used Dambis-Dubins-Schwarz Theorem withα t = 2D t 2λs ds = D e 2λt -(z 0 ) + E ln T θ (1) (cz)) + ln (2) + c E ,(78)where T θ (1) -c,c denotes the first hitting time of the symmetric conic boundary of angle c for a Brownian motion Z starting from 1. For λ small, we replace 2T θ -c,c by z D T θ -c,c in the proof of Proposition 3.3 (λ small case) and we have: By repeating the previous calculation, we make the elementary computation:

‡ When we simply write: Brownian motion, we always mean real-valued Brownian motion, starting from 0. For 2-dimensional Brownian motion, we indicate planar or complex BM.

Acknowledgements

I would like to thank Alain Comtet for his contribution in the proof of Proposition 2.8 and the Human Frontier Science Program for its support. This article represents the first part of my PhD thesis, under the supervision of Professors D. Holcman and M. Yor.