Antoine Ayache
email: antoine.ayache@univ-lille.fr

Julien Hamonier
email: julien.hamonier@univ-lille.fr

Wavelet series representation for multifractional multistable Riemann-Liouville process

Keywords: 60G17, 60G22, 60G52 Stable and multistable distributions, fractional and multifractional processes, varying Hurst parameter, Haar basis, random series of functions

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Wavelet series representation for multifractional multistable Riemann-Liouville process

Introduction

The main idea behind multifractional processes is that Hurst parameter which governs path roughness is no longer a constant but a function whose values can change from point to point (see e.g. [START_REF] Ayache | Multifractional stochastic fields[END_REF]). Thus, such processes provide more flexible models than the classical fractional Brownian motion whose path roughness remains everywhere the same. In the same spirit, the articles [START_REF] Falconer | Multifractional multistable and other processes with prescribed local form[END_REF][START_REF] Le Guével | A Ferguson-Klass-Lepage series representation of multistable multifractional motions and related processes[END_REF][START_REF] Falconer | Multistable processes and localizability[END_REF] have proposed three different (non-equivalent) approaches allowing to generalize stable stochastic processes (see for instance [START_REF] Samorodnitsky | Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance[END_REF]) in such a way that the parameter α governing the heavy tail behaviour of their distributions becomes a function. Such generalizations are called multistable processes. The approach introduced in [START_REF] Falconer | Multistable processes and localizability[END_REF] relies on the construction of multistable stochastic integrals. Such an integral I depends on a functional parameter α(•); this deterministic Lebesgue measurable function α(•) is defined on the real line R and with values in some compact interval [α , α] included in (0, 2]. Throughout this article, we assume that α(•) belongs to the Hölder space C 1+ρ α ([0, 1]), for some ρ α ∈ (0, 1); in other words α(•) is continuously differentiable on [0, 1] and its derivative α (•) satisfies a uniform Hölder condition on [0, 1] of order ρ α , that is one has |α (s 1 )α (s 2 )| ≤ c|s 1s 2 | ρ α , for some constant c > 0 and for all (s 1 , s 2 ) ∈ [0, 1] 2 . Moreover, we assume that 1 < α ≤ α(s) ≤ α < 2, for all s ∈ R.

(1.1)

The integrands associated with the multistable stochastic integral I are the deterministic functions from R to R belonging to F α , the Lebesgue space of variable order defined as:

F α := f s.t. f is a Lebesgue measurable function from R to R and R |f (s)| α(s) ds < +∞ . (1.2)
Notice that, for any fixed f ∈ F * α := F α \ {0}, the function from (0, +∞) into itself λ → R λ -α(s) |f (s)| α(s) ds is continuous and strictly decreasing, and one has lim λ→0 + R λ -α(s) |f (s)| α(s) ds = +∞ and lim λ→+∞ R λ -α(s) |f (s)| α(s) ds = 0. Therefore, there exists a unique positive real number denoted by

f α such that R f -α(s) α |f (s)| α(s) ds = 1. (1.
3)

The map • α defined on F α in this way and by using the convention that 0 α = 0 is a quasi-norm on F α ; notice that the only difference between a norm and a quasi-norm is that in the latter case the triangular inequality holds up to a multiplicative constant, namely there exists c ∈ [1, +∞), such that f + g α ≤ c f α + g α , for all f , g ∈ F α . Also notice that one can derive from (1.2) and the inequality

R |f (s)| α(s) ds ≤ R |f (s)| α ds + R |f (s)| α ds,
which is satisfied by any Lebesgue measurable function f from R to R, that

L α (R) ∩ L α (R) ⊆ F α , (1.4)
where, for all p ∈ (0, +∞], one denotes L p (R) the classical Lebesgue space of order p of real-valued functions over R. Moreover, there is a finite constant κ 1 only depending on α and α, such that, for all f ∈ L α (R) ∩ L α (R), one has

f α ≤ κ 1 f α + f α = κ 1 R |f (s)| α ds 1/ α + κ 1 R |f (s)| α ds 1/α . (1.5)
The latter inequality simply results from the fact that

R 2 1/ α f α + 2 1/ α f α -α(s) |f (s)| α(s) ds ≤ R 2 1/ α f α + 2 1/ α f α -α |f (s)| α ds + R 2 1/ α f α + 2 1/ α f α -α |f (s)| α ds ≤ 2 -1 R f -α α |f (s)| α ds + 2 -1 R f -α α |f (s)| α ds = 1.
Let us now recall some fundamental properties of the multistable stochastic integral I which was introduced in [START_REF] Falconer | Multistable processes and localizability[END_REF]. Denote by L γ (Ω, A, P) the space of the real-valued random variables on a given probability space (Ω, A, P) whose absolute moment of order γ is finite, where γ ∈ (0, α) is arbitrary and fixed. The integral I is a linear map from F α into L γ (Ω, A, P) such that, for all f ∈ F α , the characteristic function Φ I (f ) of the random variable I (f ) satisfies

Φ I (f ) (ξ) := E e iξI (f ) = exp - R ξf (s) α(s) ds , for every ξ ∈ R. (1.6)
Observe that (1.6) implies that I (f ) has a symmetric distribution. Similarly to stable stochastic integrals (see for instance [START_REF] Samorodnitsky | Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance[END_REF]), one can in a natural way associate to the multistable stochastic integral I an independently scattered random measure denoted by M α (see [START_REF] Falconer | Multistable processes and localizability[END_REF]). Thus, I (f ) is frequently denoted by R f (s)M α (ds). It is worth mentioning that an upper bound of the asymptotic behavior at +∞ of the tail of the distribution of the random variable R f (s)M α (ds) is provided by Proposition 2.3 of [START_REF] Falconer | Multistable processes and localizability[END_REF]:

P R f (s)M α (ds) ≥ λ ≤ κ 2 R λ -α(s) |f (s)| α(s) ds, for all λ ∈ (0, +∞), (1.7)
where κ 2 is a constant only depending on α and α. The same proposition also provides, thanks to (1.7), an estimate for the absolute moment of any order γ ∈ (0, α) of R f (s)M α (ds):

E R f (s)M α (ds) γ ≤ κ 3 (γ) f γ α , for each fixed γ ∈ (0, α), ( 1.8)
where κ 3 (γ) is a finite constant only depending on γ, α and α. We mention in passing that the paper [START_REF] Ayache | Sharp estimates on the tail behavior of a multistable distribution[END_REF] has shown that the inequality (1.7) is sharp: the reverse inequality also holds.

Main result and simulations

Let us now give the main motivation behind our present article. The paper [START_REF] Hamonier | Multifractional Stable Motion: representation via Haar basis[END_REF] has introduced via Haar basis an almost surely uniformly convergent wavelet-type random series representation for the stable stochastic field which generates linear multifractional stable motions [START_REF] Stoev | Stochastic properties of the linear multifractional stable motion[END_REF][START_REF] Stoev | Path properties of the linear multifractional stable motion[END_REF]. In our present article, we intend to generalize this result to the framework of the multistable stochastic field which generates linear multifractional multistable motions of Riemann-Liouville type. The latter field is denoted by X(u, v) : (u, v) ∈ [0, 1] × (1/α , 1) , and defined, for all (u, v) ∈ [0, 1] × (1/α , 1), as:

X(u, v) := R K u,v (s)M α (ds) , ( 2.1)
where, for every (u, v, s)

∈ [0, 1] × (1/α , 1) × R, K u,v (s) := (u -s) v-1 α(s) + 1 [0,1] (s) =      0 if s [0, u), (u -s) v-1 α(s)
otherwise.

(2.

2)

It can easily be seen that, for each fixed (u, v) ∈ [0, 1] × (1/α , 1), one has

0 ≤ K u,v (s) ≤ 1 [0,1] (s), for all s ∈ R. (2.3)
Thus, the function K u,v belongs to the space F α (see (1.2)) which guarantees the existence of the multistable stochastic integral in (2.1). Also, one can derive from (2.3) that the function K u,v belongs to all the Lebesgue spaces L p ([0, 1]), p ∈ (0, +∞], and in particular to the Hilbert space L 2 ([0, 1]). A well-known orthonormal basis of the latter space was introduced by Haar in [START_REF] Haar | Zur theorie der orthogonalen fuctionnensysteme[END_REF]; it consists in the following collection of functions:

         1 [0,1) (•) , 2 j/2 h(2 j • -k) = 2 j/2 1 2 -j k,2 -j (k+ 1 2 ) (•) -1 2 -j (k+ 1 2 ),2 -j (k+1) (•) , j ∈ Z + and k ∈ {0, . . . , 2 j -1}, (2.4)
where

h := 1 [0,1/2) -1 [1/2,1)
. By expanding, for each fixed (u, v) ∈ [0, 1] × (1/α , 1), the function K u,v on the latter basis, one gets that

K u,v (•) = K u,v 1 1 [0,1) + +∞ j=0 2 j -1 k=0 w j,k (u, v)h(2 j • -k) , (2.5)
where K u,v 1 := 1 0 K u,v (s)ds and

w j,k (u, v) := 2 j 1 0 K u,v (s)h(2 j s -k)
ds, for all j ∈ Z + and k ∈ {0, . . . , 2 j -1}.

(2.6)

A priori, the series in (2.5) is convergent for the L 2 ([0, 1]) norm; yet (1.1), (1.5) and the Hölder inequality imply that this series is also convergent for the quasi-norm • α . Thus, using (2.1) and (1.8) one gets that

X(u, v) = K u,v 1 η + +∞ j=0 2 j -1 k=0 w j,k (u, v)ε j,k , (2.7)
where

η := R 1 [0,1) (s)M α (ds) = M α [0, 1) and ε j,k := R h(2 j s -k)M α (ds) , for all j ∈ Z + and k ∈ {0, . . . , 2 j -1}. ( 2

.8)

A priori, the series in (2.7) is convergent in the sense of the L γ (Ω, A, P) (quasi)-norm, for each fixed (u, v) ∈ [0, 1] × (1/α , 1) and γ ∈ (0, α). The main goal of our article is to show that it is also convergent in a much stronger sense, namely:

Theorem 2.1. For all integer J ≥ 1 and (u, v) ∈ [0, 1] × (1/α, 1)
, let X J (u, v) be the partial sum of the series in (2.7) defined as:

X J (u, v) = K u,v 1 η + J-1 j=0 2 j -1 k=0 w j,k (u, v)ε j,k .
(2.9)

Then, there exists an event Ω * of probability 1, such that, for all ω ∈ Ω * and for every real numbers a and b Thus, it is convergent in this space. Moreover, the multistable stochastic field

satisfying 1/α < a < b < 1, X J (•, •, ω) J∈N is a Cauchy sequence in C [0, 1]×[a
X(u, v) : (u, v) ∈ [0, 1] × [a, b]
with continuous paths, defined as:

X(•, •, ω) := lim J→+∞ X J (•, •, ω), if ω ∈ Ω * , and X(•, •, ω) := 0, else, (2.10) is a modification of X(u, v) : (u, v) ∈ [0, 1] × [a, b] ,

and one has, for any fixed

ζ > 1/α and ω ∈ Ω * , sup J -ζ 2 J min{ρ α , a-1/α } X(u, v, ω) -X J (u, v, ω) : (J, u, v) ∈ N × [0, 1] × [a, b] < +∞. (2.11)
Remark 2.1. In view of (2.9) and of the fact that X is a modification of X, the inequality (2.11) provides an almost sure estimate of the rate of convergence for the uniform norm • C of the random series of functions in (2.7). Notice that in the particular case where X is an α-stable field (that is α(s) = α, for all s ∈ [0, 1], where α ∈ (1, 2) is a constant parameter), this estimate of the rate of convergence becomes O(2 -J(a-1/α) J 1/α+η ), where η is an arbitrarily small fixed positive real number. Thus, it improves the estimate O(2 -J(a-1/α) J

(u, v) ∈ [0, 1] × (1/α, 1
), the random variable X J (u, v), defined in (2.9), can be expressed as:

X J (u, v) = 2 J -1 l=0 K J,l u,v M α 2 -J l, 2 -J (l + 1) , (2.13)
where, for all J ∈ N and l ∈ {0, . . . , 2 J -1}, K J,l u,v is the average value of the function K u,v on the dyadic interval 2 -J l, 2 -J (l + 1) , that is

K J,l u,v := 2 J 2 -J (l+1) 2 -J l K u,v (s)ds.
(2.14)

The equalities (2.13), (2.10) and (2.12) provide an efficient method for simulating paths of the mmRL process Y . To this end, when J is large enough, one uses the approximation:

M α 2 -J l, 2 -J (l + 1) ≈ Z α(2 -J l) 2 -J l, 2 -J (l + 1) , (2.15)
where the Z α(2 -J l) , l = 0, . . . , 2 J -1, are independent usual symmetric stable random measures with stability parameters α(2 -J l), l = 0, . . . , 2 J -1. Notice that the approximation (2.15) is justified by [START_REF] Falconer | Multistable processes and localizability[END_REF]Theorem 2.6].

Here are some simulations: The main two ideas of the proof of Theorem 2.1 are:

1. The use of the Abel's summation rule in order to express L j (u, v) in a convenient way (see Remark 3.1).

2. The use of the Doob's maximal inequality for discrete submartingales in order to derive, for each j ∈ N, a suitable upper for the supremum of the absolute values of the partial sums τ j,k , k ∈ {0, . . . , 2 j -1}, of the multistable random variables j,0 , . . . , j,2 j -1 (see Remark 3.2, Lemma 3.1 and its proof).

The first idea is borrowed from [START_REF] Hamonier | Multifractional Stable Motion: representation via Haar basis[END_REF] while the second one is completely new.

Remark 3.1. For all j ∈ Z + and (u, v) ∈ [0, 1] × (1/α , 1), one sets

L j (u, v) := 2 j -1 k=0 w j,k (u, v)ε j,k = [2 j u] k=0 w j,k (u, v)ε j,k . (3.1)
One mentions in passing that the last equality in (3.1), in which [2 j u] denotes the integer part of 2 j u, follows from (2.4) and (2.6). Using the Abel's summation rule one has that:

L j (u, v) = τ j,[2 j u] w j,[2 j u] (u, v) + [2 j u]-1 k=0 τ j,k w j,k (u, v) -w j,k+1 (u, v) , ( 3.2)
where τ j,k : j ∈ Z + and k ∈ {0, . . . , 2 j -1} is the sequence of the multistable random variables defined, for all j ∈ Z + and k ∈ {0, . . . , 2 j -1}, as:

τ j,k := k m=0 ε j,m = R k m=0 h(2 j s -m) M α (ds) . (3.3)
Notice that the last equality in (3.3) follows from (2.8).

Remark 3.2. One knows from (2.8) and (1.8) that the multistable random variables ε j,k , j ∈ N and k ∈ {0, . . . , 2 j -1}, belong to L γ (Ω, A, P), for all γ ∈ (0, α); which in particular means that they are in L 1 (Ω, A, P) since α > 1; notice that they are centered since their distributions are symmetric. Moreover, for each fixed j ∈ N, the random variables ε j,k , k ∈ {0, . . . , 2 j -1}, are independent since the random measure M α is independently scattered and the supports of the functions h(2 j • -k), k ∈ {0, . . . , 2 j -1}, are pairwise disjoints (up to Lebesgue negligible sets). Thus, in view of the first equality in (3.3), it turns out that, for each fixed j ∈ N, the sequence of random variables {τ j,k } 0≤k<2 j is a discrete martingale with respect to the filtration (A j,k ) 0≤k<2 j -1 such that, for any k ∈ {0, . . . , 2 j -2}, A j,k denotes the smallest σ -algebra for which the random variables ε j,0 , . . . , ε j,k are measurable.

Lemma 3.1. There exists an event Ω * of probability 1 such that on Ω * , one has, for all fixed ζ > 1/α , , that, for each fixed j ∈ N, the sequence of random variables |τ j,k | γ 0≤k<2 j is a discrete submartingale with respect to the filtration (A j,k ) 0≤k<2 j -1 . Hence, using the Doob's maximal inequality (see [START_REF] Dudley | Real analysis and probability[END_REF]Theorem 10.4.2 on page 360]) one has, for all positive real number M,

sup j∈Z + (1 + j) -ζ sup 0≤k<2 j |τ j,k | < +∞. ( 3
P sup 0≤k<2 j |τ j,k | γ > M ≤ M -1 E |τ j,2 j -1 | γ . (3.6)
Observe that it follows from the last equality in (3.3), (1.8) and the fact that

R 2 j -1 m=0 h(2 j s -m) α(s) ds = 1 , that E |τ j,2 j -1 | γ ≤ κ 3 (γ), for all j ∈ Z + , (3.7)
where κ 3 (γ) is the same finite constant as in (1.8). Next, taking in (3.6) M = (1 + j) γζ and using (3.7) and (3.5), one obtains that

+∞ j=1 P sup 0≤k<2 j |τ j,k | > (1 + j) ζ ≤ κ 3 (γ) +∞ j=1 (1 + j) -γζ < +∞.
Thus, it follows from the Borel-Cantelli's Lemma that the probability of the event

Ω * ζ := ω ∈ Ω : sup j∈Z + (1 + j) -ζ sup 0≤k<2 j |τ j,k (ω)| < +∞
is equal to 1. For finishing the proof, one sets

Ω * := ζ∈Q∩(1/α ,+∞) Ω * ζ
where Q denotes the countable set of the rational numbers.

In order to derive Theorem 2.1, one also needs the following five lemmas whose proofs are given in the Appendix A. From now on, for the sake of simplicity one denotes by I the interval [0, 1]. Lemma 3.2. There exists a positive and finite constant c 1 such that, for any j ∈ N, for each (u, v) ∈ I × [a, b] satisfying u ≥ 4.2 -j-1 , and for all s ∈ [0, u -4.2 -j-1 ], the following inequality holds

K u,v (s) -K u,v (s + 2 -j-1 ) -K u,v (s + 2.2 -j-1 ) + K u,v (s + 3.2 -j-1 ) ≤ c 1 2 -j 2 -jρ α + 2 -j |u -s -3.2 -j-1 | a-1/α-2 .
(3.8) Lemma 3.3. There exists a positive and finite constant c 2 such that, for any j ∈ Z + and (u, v) ∈ I × [a, b], the following inequality, in which [2 j u] denotes the integer part of 2 j u, is satisfied

w j,[2 j u] (u, v) ≤ c 2 2 -j a-1 α . ( 3
I 1 j (u, v) := 2 j u-2 -(j+1) u-2•2 -(j+1) (u -s) v-1 α(s) -(u -s -2 -j-1) ) v- 1 α(s+2 -j-1 ) ds ≤ c 3 2 -j a-1 α , (3.10)
with the convention that I 1 j (u, v) := 0 when u ≤ 2 -(j+1) .

Lemma 3.5. There exists a positive and finite constant c 4 such that, for any j ∈ Z + and (u, v) ∈ I × [a, b], one has

I 2 j (u, v) := 2 j u-2•2 -(j+1) u-3•2 -(j+1) (u -s) v-1 α(s) -(u -s -2 -(j+1) ) v- 1 α(s+2 -j-1 ) -(u -s -2 • 2 -(j+1) ) v- 1 α(s+2•2 -j-1 ) ds ≤ c 4 2 -j a-1 α , (3.11)
with the convention that I 2 j (u, v) := 0 when u ≤ 2 • 2 -(j+1) . Lemma 3.6. There exists a positive and finite constant c 5 such that, for any j ∈ Z + and (u, v) ∈ I × [a, b], one has

I 3 j (u, v) := 2 j u-3•2 -(j+1) u-4•2 -(j+1) (u -s) v-1 α(s) -(u -s -2 -(j+1) ) v- 1 α(s+2 -j-1 ) -(u -s -2 • 2 -(j+1) ) v- 1 α(s+2•2 -j-1 ) + (u -s -3 • 2 -(j+1) ) v- 1 α(s+3•2 -j-1 ) ds ≤ c 5 2 -j a-1 α , (3.12)
with the convention that j+1) . We are now in position to prove Theorem 2.1.

I 3 j (u, v) := 0 when u ≤ 3 • 2 -(
Proof of Theorem 2.1. Let J ∈ N, Q ∈ N, (u, v) ∈ I × [a, b]
and ω ∈ Ω * be arbitrary and fixed. Using (2.9), (3.1), (3.2), the triangular inequality, (2.6) and (2.4), one gets that

X J+Q (u, v, ω) -X J (u, v, ω) = J+Q-1 j=J L j (u, v, ω) ≤ J+Q-1 j=J sup 0≤k<2 j |τ j,k (ω)| |w j,[2 j u] (u, v)| + [2 j u]-1 k=0 |w j,k (u, v) -w j,k+1 (u, v)| = J+Q-1 j=J sup 0≤k<2 j |τ j,k (ω)|       |w j,[2 j u] (u, v)| + [2 j u]-1 k=0 2 j (k+1/2)2 -j k2 -j K u,v (s) -K u,v (s + 2 -j-1 ) -K u,v (s + 2 • 2 -j-1 ) + K u,v (s + 3 • 2 -j-1 ) ds       ≤ J+Q-1 j=J sup 0≤k<2 j |τ j,k (ω)| |w j,[2 j u] (u, v)| + 2 j u-2 -j-1 0 K u,v (s) -K u,v (s + 2 -j-1 ) -K u,v (s + 2 • 2 -j-1 ) + K u,v (s + 3 • 2 -j-1 ) ds . (3.13)
Next, putting together (3.13), (2.2) and Lemmas 3.1 to 3.6, one obtains, for any fixed ζ > 1/α, that:

X J+Q (u, v, ω) -X J (u, v, ω) ≤ C (ω) J+Q-1 j=J (1 + j) ζ 2 -j min{a-1 α ,ρ α } + 2 -j u-4•2 -j-1 0 (u -s -3 • 2 -j-1 ) a-1/α-2 ds ≤ C"(ω) J+Q-1 j=J (1 + j) ζ 2 -j min{a-1 α ,ρ α } ≤ C"(ω) +∞ j=J (1 + j) ζ 2 -j min{a-1 α ,ρ α } , ( 3.14)
where C and C" are two positive and finite random variables not depending on J, Q and (u, v). Thus, one can derive from (3.14) that (X J (•, •, ω)) J∈N is a Cauchy sequence in the Banach space C(I × [a, b]); its limit in this space is denoted by X(•, •, ω).

Let us now prove that (2.11) is satisfied. When Q goes to +∞, it follows from (3.14) that

X(u, v, ω) -X J (u, v, ω) ≤ C"(ω) +∞ j=J (1 + j) ζ 2 -j min{a-1 α ,ρ α } ≤ C"(ω)(1 + J) ζ 2 -J min{a-1 α ,ρ α } +∞ j=0 (1 + j) ζ 2 -j min{a-1 α ,ρ α } ≤ C (ω) J ζ 2 -J min{a-1 α ,ρ α } , (3.15)
where C is a positive and finite random variable not depending on J and (u, v). Thus, (3.15) implies that (2.11) holds.

A Appendix

Proof of Lemma 3.2. One assumes that j ∈ N and (u, v) ∈ I × [a, b] are arbitrary and such that u ≥ 4.2 -j-1 . Then, one denotes by L u,v the infinitely differentiable function on the open set (-∞, u) × (1/v, +∞) ⊂ R 2 defined as:

L u,v (x, y) := (u -x) v-1/y , for all (x, y) ∈ (-∞, u) × (1/v, +∞). (A.1)
Thus, using (2.2), one has

K u,v (z) = L u,v z, α(z) for all z ∈ [0, u -2 -j-1 ]. (A.2)
One can derive from (A.2) and the triangular inequality that, for every s

∈ [0, u -4.2 -j-1 ], K u,v (s) -K u,v (s + 2 -j-1 ) -K u,v (s + 2.2 -j-1 ) + K u,v (s + 3.2 -j-1 ) = L u,v (s, α(s)) -L u,v (s + 2 -j-1 , α(s + 2 -j-1 )) -L u,v (s + 2.2 -j-1 , α(s + 2.2 -j-1 )) + L u,v (s + 3.2 -j-1 , α(s + 3.2 -j-1 )) ≤ A j u,v (s) + B j u,v (s), (A.3)
where

A j u,v (s) := L u,v (s, α(s)) -L u,v (s + 2 -j-1 , α(s + 2 -j-1 )) -L u,v (s + 2.2 -j-1 , α(s)) + L u,v (s + 3.2 -j-1 , α(s + 2 -j-1 )) (A.4) and B j u,v (s) := L u,v (s + 2.2 -j-1 , α(s)) -L u,v (s + 3.2 -j-1 , α(s + 2 -j-1 ))
-L u,v (s + 2.2 -j-1 , α(s + 2.2 -j-1 )) + L u,v (s + 3.2 -j-1 , α(s + 3.2 -j-1 )) . (A.5)

First step: The goal of this step is to provide a suitable upper bound for the quantity A j u,v (s). For any fixed s ∈ [0, u -4.2 -j-1 ], one denotes by g 1,s the infinitely differentiable function defined as:

g 1,s :        [0, 2 -j ] -→ R x -→ L u,v (s + x, α(s)) -L u,v (s + 2 -j-1 + x, α(s + 2 -j-1 )). (A.6)
Thus, it follows from (A.4) that A j u,v (s) = g 1,s (2 -j )g 1,s (0) . Then using the mean value theorem, one obtains that A j u,v (s) = 2 -j |g 1,s (x * )|, for some x * ∈ (0, 2 -j-1 ). Therefore, one can derive from (A.6), (A.1), the triangular inequality and the inequalities |v -1/α(s + 2 -j-1 )| < 1 and |v -1/α(s)| < 1 that

A j u,v (s) ≤ 2 -j (u -s -2 -j-1 -x * ) v-1-1/α(s+2 -j-1 ) -(u -s -2 -j-1 -x * ) v-1-1/α(s) (A.7) + 1 α(s + 2 -j-1 ) - 1 α(s) (u -s -2 -j-1 -x * ) v-1-1/α(s) + (u -s -2 -j-1 -x * ) v-1-1/α(s) -(u -s -x * ) v-1-1/α(s) .
Next, notice that it follows from the assumption α ∈ C 1+ρ α (I), (1.1), x * ∈ (0, 2 -j-1 ) and v ∈ [a, b], that

1 α(s + 2 -j-1 ) - 1 α(s) (u -s -2 -j-1 -x * ) v-1-1/α(s) ≤ c 1 2 -j-1 (u -s -2 -j-1 -x * ) v-1-1/α(s) ≤ c 1 2 -j-1 (u -s -3.2 -j-1 ) a-1-1/α , (A.8)
where c 1 is a constant not depending on j, u, s, v. Thus, using (A.7) and (A.8), and applying the mean value theorem to the functions:

g 2,s,x * :        [0, 2 -j-1 ] → R w -→ (u -s -x * -w) v-1-1 α(s) g 3,s,x * :        [α(s) ∧ α(s + 2 -j-1 ), α(s) ∨ α(s + 2 -j-1 )] → R z -→ (u -s -2 -j-1 -x * ) v-1-1 z , one obtains that A j u,v (s) ≤ c 2 2 -2j (u -s -3.2 -j-1 ) a-1-1/α + (u -s -3.2 -j-1 ) a-2-1/α + | log(u -s -2 -j-1 )|(u -s -3.2 -j-1 ) a-1-1/α ≤ c 2 2 -2j 2(u -s -3.2 -j-1 ) a-2-1/α + | log(u -s -2 -j-1 )|(u -s -3.2 -j-1 ) a-1-1/α , (A.9)
where c 2 is a constant not depending on j, u, s, v. Finally, combining (A.9) and the inequality | log(x)| ≤ x -1 , for all x ∈ (0, 1], one gets that

A j u,v (s) ≤ c 3 2 -2j (u -s -3.2 -j-1 ) a-2-1/α , (A.10)
where c 3 is a constant not depending on j, u, s, v.

Second step: The goal of this step is to provide a suitable upper bound for the quantity B j u,v (s). In view of (A.5), the quantity B j u,v (s) can be rewritten as:

B j u,v (s) = L u,v (s + 2.2 -j-1 , α(s)) -L u,v (s + 2.2 -j-1 , α(s + 2.2 -j-1 )) -L u,v (s + 3.2 -j-1 , α(s + 2 -j-1 )) -L u,v (s + 3.2 -j-1 , α(s + 3.2 -j-1 )) .
Thus applying the mean value theorem to the functions g 4,s :

       [α(s) ∧ α(s + 2.2 -j-1 ), α(s) ∨ α(s + 2.2 -j-1 )] -→ R y -→ L u,v (s + 2.2 -j-1 , y), g 5,s :        [α(s + 2 -j-1 ) ∧ α(s + 3.2 -j-1 ), α(s + 2 -j-1 ) ∨ α(s + 3.2 -j-1 )] -→ R y -→ L u,v (s + 3.2 -j-1 , y).
* ∈ α(s) ∧ α(s + 2.2 -j-1 ), α(s) ∨ α(s + 2.2 -j-1 ) (A.11) and y * * ∈ α(s + 2 -j-1 ) ∧ α(s + 3.2 -j-1 ), α(s + 2 -j-1 ) ∨ α(s + 3.2 -j-1 ) , (A.12) that B j u,v (s) ≤ c 4 2 -j(1+ρ α ) 1 y 2 * log(u -s -2.2 -j-1 ) (u -s -2.2 -j-1 ) a-1/α + c 4 2 -j 1 y 2
where c 5 is a constant not depending on j, u, s, v, y * , y * * . Also, notice that applying the mean value theorem to the function g 6,s :

       [y * ∧ y * * , y * ∨ y * * ] -→ R x -→ (u -s -2.2 -j-1 ) v-1/x
and using v ∈ [a, b], y * , y * * ∈ (α, α), the second and the third inequality in (A.15), one obtains that

(u -s -2.2 -j-1 ) v-1/y * -(u -s -2.2 -j-1 ) v-1/y * * ≤ c 6 2 -j (u -s -2.2 -j-1 ) a-1/α log(u -s -2.2 -j-1 ) , (A.16)
where c 6 is a constant not depending on j, u, s, v, y * , y * * . Moreover, notice that applying the mean value theorem to the function g 7,s,y * * :

       [0, 2 -j-1 ] -→ R x -→ (u -s -2.2 -j-1 -x) v-1/y * * log(u -s -2.2 -j-1 -x)
and making use of v ∈ [a, b] and y * , y * * ∈ (α, α), it follows that (us -2.2 -j-1 ) v-1/y * * log(us -2.2 -j-1 ) -(us -3.2 -j-1 ) v-1/y * * log(us -3.2 -j-1 ) ≤ c 7 2 -j (us -3.2 -j-1 ) a-1/α-1 1 + log(us -3.2 -j-1 ) , (A.17) where c 7 is a constant not depending on j, u, s, v, y * , y * * . Next putting together (A.13) to (A.17), one gets that B j u,v (s) ≤ c 4 2 -j(1+ρ α ) log(us -2.2 -j-1 ) (us -2.2 -j-1 ) a-1/α + c 8 2 -2j (us -2.2 -j-1 ) a-1/α log(us -2.2 -j-1 ) + (us -2.2 -j-1 ) a-1/α log 2 (us -2.2 -j-1 ) + (us -3.2 -j-1 ) a-1/α-1 1 + log(us -3.2 -j-1 ) , (A.18)

where c 8 is a constant not depending on j, u, s, v, y * , y * * . Then, one can derive from (A.18) and the inequalities sup z∈(0,1] | log(z)|z a-1/α < +∞, sup z∈(0,1] | log 2 (z)|z a-1/α < +∞ and | log(x)| ≤ |x| -1 , for all x ∈ (0, 1], that B j u,v (s) ≤ c 9 2 -j(1+ρ α ) + 2 -2j |us -3.2 -j-1 | a-

(u -s) v-1 α(s) ds ≤ 2 j u 2 -j [2 j u]
2 -j(a-1/α) ds ≤ 2 -j(a-1/α) .

The proofs of Lemmas 3.4, 3.5 and 3.6 are very similar so we only give that of Lemma 3.6.

Proof of Lemma 3.6. Let j ∈ Z + and (u, v) ∈ I × [a, b] be arbitrary and such that u > 3 • 2 -(j+1) . In view of the assumptions on (u, v) and (1.1), it can easily be seen that, for all s ∈ [u -4 • 2 -(j+1) , u -3 • 2 -(j+1) ] and for any q ∈ {0, 1, 2, 3}, one has

0 ≤ u -s -q • 2 -(j+1) v- 1 α(s+q•2 -j-1 ) ≤ 7 • 2 -(j+1) v- 1 α(s+q•2 -j-1 ) ≤ 7 b-1/α • 2 -(j+1)(a-1/α) .
Thus, using the triangular inequality one gets that

I 3 j (u, v) := 2 j u-3•2 -(j+1) u-4•2 -(j+1) (u -s) v-1 α(s) -u -s -2 -(j+1) v- 1 α(s+2 -j-1 ) -u -s -2 • 2 -(j+1) v- 1 α(s+2•2 -j-1 ) + u -s -3 • 2 -(j+1) v- 1 α(s+3•2 -j-1 ) ds ≤ 2 j u-3•2 -(j+1) u-4•2 -(j+1) 4 • 7 b-1/α • 2 -(j+1)(a-1/α) ds = 2 -1 • 4 • 7 b-1/α • 2 -(j+1)(a-1/α) ,
which shows that (3.12) is satisfied.

  , b] the Banach space of the realvalued continuous functions over the rectangle [0, 1] × [a, b] equipped with the uniform norm denoted by • C .

Figure 1 :Figure 2 :

 12 Figure 1: Multifractional Multistable Riemann-Liouville process

. 9 ) 3 . 4 .

 934 Lemma There exists a positive and finite constant c 3 such that, for any j ∈ Z + and (u, v) ∈ I × [a, b], one has

  and putting together the triangular inequality, the assumption α ∈ C 1+ρ α (I), (1.1), v ∈ [a, b], and the equality dfgh = d(fh) + h(dg) , for all d, f , g, h ∈ R, one obtains, for some y

  Observe z → |z| γ is a convex function from R to R + , and one has E |τ j,k | γ < +∞, for all j and k (see the last equality in (3.3), (1.8) and (2.4)). Thus, it follows Remark 3.2 and from[START_REF] Dudley | Real analysis and probability[END_REF] Theorem 10.3.3 on page 354]

	(3.5)

.4) Proof of Lemma 3.1. Let ζ > 1/α and γ ∈ [1, α) be fixed and such that ζ > 1/γ > 1/α .

Acknowledgements

This work has been partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and the GDR 3475 (Analyse Multifractale).

where c 4 is a constant not depending on j, u, s, v, y * , y * * . Next, notice that using (1.1), (A.11), (A.12) and the assumption α ∈ C 1+ρ α (I), one gets that


Antoine Ayache
email: antoine.ayache@math.univ-lille1.fr

Céline Esser
email: celine.esser@uliege.be

Julien Hamonier
email: julien.hamonier@univ-lille2.fr

A new Multifractional Process with Random Exponent

Keywords: 60G17, 60G22 Fractional Brownian Motion, varying Hurst parameter, Haar basis, Hölder regularity, Itô integral

 by replacing in a wavelet series representation of Fractional Brownian Motion (FBM) the Hurst parameter by a random variable depending on the time variable. In the present article, we propose another approach for constructing another type of MPRE. It consists in substituting to the Hurst parameter, in a stochastic integral representation of the high-frequency part of FBM, a random variable depending on the integration variable. The MPRE obtained in this way offers, among other things, the advantages to have a representation through classical Itô integral and to be less difficult to simulate than the first type of MPRE, previously introduced in [2]. Yet, the study of Hölder regularity of this new MPRE is a significantly more challenging problem than in the case of the previous one. Actually, it requires to develop a new methodology relying on an extensive use of the Haar basis.

Introduction

Fractional Brownian Motion (FBM), which was introduced by Kolmogorov [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen raum[END_REF] and made popular by Mandelbrot and Van Ness [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF], is one of the most important stochastic process in both theory and applications (see for instance [START_REF] Samorodnitsky | Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance[END_REF][START_REF] Doukhan | Theory and Applications of Long-range Dependence[END_REF]). This continuous centred Gaussian process {B H (t) : t ∈ I}, where I denotes the closed interval [0, 1], depends on a deterministic constant parameter, denoted by H, belonging to the open interval (0, 1) and called the Hurst parameter. Let {B(s) : s ∈ R} be a standard Brownian Motion on a complete probability space (Ω, F , P) and let (•)dB be the associated Wiener integral. Then the FBM {B H (t) : t ∈ I} can be defined, for every t ∈ I, as:

B H (t) := 0 -∞ (t -s) H-1 2 -(-s) H-1 2 dB(s) + t 0 (t -s) H-1 2 dB(s). ( 1
)
The first one of these two Wiener integrals is called the low-frequency part of FBM and the other one its high-frequency part. Roughness of paths of FBM is mainly due to its high-frequency part, which is also called the Riemann-Liouville process and denoted by {R H (t) : t ∈ I}. For the sake or clarity, let us point out that, for all t ∈ I, one has R H (t) := t 0 (ts) H-1 2 dB(s).

(

) 2
Roughness of paths of {R H (t) : t ∈ I}, and consequently that of {B H (t) : t ∈ I}, is governed by the Hurst parameter H. More precisely, on any arbitrary non-degenerate compact interval included in I, the critical Hölder regularity of paths of these two processes is equal to H. Thus, in contrast with many real-life signals, roughness of paths of FBM is not allowed to change from one period of time to another which somehow restricts its areas of applicability.

In spite of importance and usefulness of FBM as a random model in signal processing, the constancy and non-randomness of its Hurst parameter H are serious limitations of it. This is the main motivation behind construction and study of several classes of Multifractional Processes since the mid-1990s (see for instance [START_REF] Ayache | Multifractional processes with random exponent[END_REF][START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF][START_REF] Bianchi | Pathwise identification of the memory function of multifractional Brownian motion with application to finance[END_REF][START_REF] Benassi | Elliptic Gaussian random processes[END_REF][START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF][START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Process[END_REF][START_REF] Surgailis | Nonhomogeneous fractional integration and multifractional processes Stochastic Process[END_REF] to mention just a few references). The paradigmatic example of such processes is the Gaussian Multifractional Brownian Motion (MBM) of Benassi, Jaffard, Lévy Véhel, Peltier and Roux [START_REF] Benassi | Elliptic Gaussian random processes[END_REF][START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF], which is obtained simply by replacing in [START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF] the constant Hurst parameter H by a deterministic function H(t), depending on the time variable t in a continuous way. Observe that the assumption that H(t) is deterministic, or more generally that the processes {H(t) : t ∈ I} and {B(t) : t ∈ I} are independent, is crucial. Indeed, as explained in [START_REF] Ayache | Multifractional processes with random exponent[END_REF], the stochastic integrals in [START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF] fail to be well-defined, at least in the usual Itô sense, when H is replaced by a stochastic process {S(t) : t ∈ I} which is not independent on the Brownian Motion {B(s) : s ∈ R}. In order to overcome this difficulty, the article [START_REF] Ayache | Multifractional processes with random exponent[END_REF] proposed to replace H by {S(t) : t ∈ I} in an almost surely uniformly convergent random wavelet series representation of FBM, due to Meyer, Sellan and Taqqu [START_REF] Meyer | Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion[END_REF], which is of a rather different nature from its stochastic integral representation [START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF]. Thus, the article [START_REF] Ayache | Multifractional processes with random exponent[END_REF] was able to construct a first type of Multifractional Process with Random Exponent (MPRE) which, among other things, turned out to be useful in stock prices modelling, thanks to some papers by Bianchi and his co-authors (see for instance [START_REF] Bianchi | Modeling and simulation of currency exchange rates using MPRE[END_REF][START_REF] Bianchi | Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity[END_REF][START_REF] Bianchi | Multifractional processes in finance[END_REF]).

The main goal of our present article is to propose another approach for constructing a new type of MPRE which, among other things, offers the advantages to have a representation through classical Itô integral and to be less difficult to simulate than the first type of MPRE, previously introduced in [START_REF] Ayache | Multifractional processes with random exponent[END_REF]. Yet, as we will see it in our article, the study of Hölder regularity of this new MPRE is a significantly more challenging problem than in the case of the previous one. Actually, it requires to develop a new methodology relying on an extensive use of the Haar basis [START_REF] Haar | Zur theorie der orthogonalen fuctionnensysteme[END_REF][START_REF] Daubechies | Ten lectures on wavelets[END_REF][START_REF] Meyer | Wavelets and operators[END_REF][START_REF] Meyer | Ondelettes et Opérateurs[END_REF][START_REF] Wojtaszczyk | A mathematical introduction to wavelets[END_REF], whose definition is recalled in [START_REF] Bianchi | Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity[END_REF] in the next section, and which is sometimes called in French in a humorous way: "l'ondelette du pauvre" (the wavelet of the poor). The approach we propose in our present article is to a certain extent inspired by the one used by Surgailis in [START_REF] Surgailis | Nonhomogeneous fractional integration and multifractional processes Stochastic Process[END_REF] which, roughly speaking, consists in replacing the constant Hurst parameter H by a deterministic function H(s) depending on the integration variable s (and not on the time variable t). Yet, in our case, H(s) is not only a deterministic function but more generally a stochastic process with continuous paths, denoted by {A(s) : s ∈ I}, which is assumed to be adapted to the natural filtration (F s ) s∈I associated with the Brownian Motion {B(s) : s ∈ I}; recall that F s := σ (B(u); 0 ≤ u ≤ s), for all s ∈ I. Another important assumption on {A(s) : s ∈ I} is that it takes its values in [a, a] where a and a are two deterministic arbitrary constants satisfying the inequalities:

1 2 < a ≤ a < 1 . ( 3
)
The MPRE we study in our present article is denoted by {X(t) : t ∈ I} and obtained by substituting in (2) the process {A(s) : s ∈ I} to the Hurst parameter H. More precisely, {X(t) : t ∈ I} is defined, for all t ∈ I, as the Itô integral:

X(t) = 1 0 K t (s)dB(s), (4)
where, for any (t, s) ∈ I 2 ,

K t (s) := (t -s) A(s)-1 2 + = 0 if t ≤ s (t -s) A(s)-1 2 otherwise. ( 5
)
One clearly has, for any s ∈ I, that K 0 (s) = 0 and therefore X(0) = 0. Let us show that, for any fixed t ∈ (0, 1], the Itô integral in (4), makes sense; that is the stochastic process {K t (s) : s ∈ I} belongs to usual class of integrands for the Itô integral over I.

1. From Relation (5), for any fixed ω ∈ Ω, the function s → K t (s, ω) is continuous on I. Therefore (s, ω)

→ K t (s, ω) is a measurable function from (I × Ω, B(I) ⊗ F ) to (R, B(R)).
2. For any s ∈ I, the random variable K t (s) is F s -measurable; this is obvious in view of (5), since A(s) is a F s -measurable random variable.

3. One has K t ∈ L 2 (I ×Ω, B(I)⊗F ); indeed, it follows from (5) and the fact

A(s) ∈ [a, a] ⊂ (1/2, 1), that Ω 1 0 |K t (s, ω)| 2 ds d P = Ω t 0 (t -s) 2A(s)-1 ds d P ≤ Ω t 0 (t -s) 2a-1 ds d P < +∞ . ( 6
)
From now on, we assume, in addition, that there exist a constant ρ ∈ (0, 1] and a positive constant c such that, for any x, y ∈ I, one has

E |A(x) -A(y)| 2 ≤ c|x -y| 2ρ . ( 7
)
Under this additional assumption, the stochastic process {X(t) : t ∈ I} has a modification whose paths are Hölder continuous function on I. More precisely, the following proposition holds. Let us emphasize that throughout our article the process {X(t) : t ∈ I} is systematically identified with its modification with Hölder continuous paths introduced in Proposition 1.1.

Also, it is worth mentioning that the proof of Proposition 1.1, which is given in the Appendix A, mainly relies on the classical Kolmogorov-Čentsov's continuity theorem (see [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] for instance). It is well-known that the latter theorem is of simple use and of very great utility. However, unfortunately, it can hardly allow to obtain very precise results on Hölder regularity of non-Gaussian processes such as the MPRE {X(t) : t ∈ I}. Thus, one the main goals of our present article is to derive, thanks to a different methodology relying on the use of the Haar basis, a much more precise result than Proposition 1.1 on Hölder regularity of {X(t) : t ∈ I}, namely Theorem 3.1.

The rest of our article is organized in the following way. In Section 2, we introduce via the Haar basis a random series representation of the MPRE {X(t) : t ∈ I}, and we derive the almost sure convergence of the series uniformly in t ∈ I. In Section 3, we assume in addition that the paths of {A(s) : s ∈ I} satisfy a uniform Hölder condition of an arbitrary order γ > 1/2, and under this additional assumption, we show that the pathwise uniform Hölder exponent of {X(t) : t ∈ I} on any arbitrary interval [ν 1 , ν 2 ] ⊆ I is almost surely bounded from below by min s∈[ν 1 ,ν 2 ] A(s). This result somehow means that roughness of paths of {X(t) : t ∈ I} is governed by {A(s) : s ∈ I} and thus is allowed to change from one period of time to another; in order to derive it, we make an essential use of the series representation of {X(t) : t ∈ I} via the Haar basis. In Section 4, thanks to the latter representation of {X(t) : t ∈ I}, we give two simulation methods for this MPRE and we test them; our simulations tend to confirm the fact that roughness of paths of {X(t) : t ∈ I} does not remain everywhere the same and is closely connected to the values of {A(s) : s ∈ I}. In the Appendix A the proofs of some auxiliary results are given.

Series representation via the Haar basis

First, we recall that the Haar basis of L 2 ([0, 1]) is the collection of functions:

         U := 1 [0,1) h j,k := 2 j 2 1 2 -j k,2 -j (k+ 1 2 ) - 1
2 -j (k+ 1 2 ),2 -j (k+1) , j ∈ Z + and k ∈ {0, . . . , 2 j -1}. ( 8
)
Let us point out that, for each j ∈ Z + and k ∈ {0, . . . , 2 j }, the dyadic number k2 -j is frequently denoted by δ j,k . Moreover, when k < 2 j , we frequently denote by ∆B j,k the increment of the Brownian motion {B(s) : s ∈ I} on the dyadic interval [δ j,k , δ j,k+1 ), that is one has

∆B j,k := B(δ j,k+1 ) -B(δ j,k ). ( 9
)
The main result of the present section is the following theorem which gives a random series representation of the MPRE {X(t) : t ∈ I}. We mention that, roughly speaking, this representation of {X(t) : t ∈ I} is obtained via the decomposition of the associated kernel function K t (•, ω) in the Haar basis.

Theorem 2.1. Assume that the exponent ρ (see [START_REF] Bianchi | Multifractional processes in finance[END_REF]) satisfies

1 2 < ρ ≤ 1 . ( 10
)
Then, there is an event Ω * * ⊆ Ω of probability 1 such that, for every ω ∈ Ω * * , one has

X(t, ω) = K t (•, ω), U η 0 (ω) + +∞ j=0 2 j -1 k=0 K t (•, ω), h j,k ε j,k (ω) , ( 11
)
where the convergence holds uniformly in t ∈ I, and where the N (0, 1) Gaussian random variables η 0 and ε j,k , j ∈ Z + , k ∈ {0, . . . , 2 j -1} are defined by

η 0 := 1 0 U (s)dB(s) = B(1) -B(0) = ∆B 0,0 (12)
and

ε j,k := 1 0 h j,k (s)dB(s) = 2 j 2       2B 2 -(j+1) (2k + 1) -B 2 -j k -B 2 -j (k + 1)       = 2 j 2 ∆B j+1,2k -∆B j+1,2k+1 . ( 13
)
Our first goal is to show that the convergence in [START_REF] Haar | Zur theorie der orthogonalen fuctionnensysteme[END_REF] holds, for each fixed t ∈ I, in L 1 (Ω). To this end, we need some preliminary results.

Lemma 2.1. There exists a deterministic constant c 0 > 0 such that, for every real numbers s , s , t

satisfying 0 ≤ s ≤ s < t ≤ 1, ( 14
)
one has K t (s ) -K t (s ) ≤ c 0 (t -s ) a-3 2 (s -s ) + A(s ) -A(s ) . ( 15
)
The proof of Lemma 2.1 is given in the Appendix A.

Remark 2.1. Let us mention that, for all t ∈ I, for every j ∈ Z + , and for each k ∈ {0, . . . , 2 j -1}, one has

K t , h j,k =2 j 2 1 0 K t (s) 1 2 -j k,2 -j (k+ 1 2 )
-1

2 -j (k+ 1 2 ),2 -j (k+1) ds =2 j 2        2 -j (k+ 1 2 ) 2 -j k K t (s)ds - 2 -j (k+1) 2 -j (k+ 1 2 ) K t (s) ds        =2 j 2 2 -j (k+ 1 2 ) 2 -j k K t (s) -K t (s + 2 -j-1 ) ds . ( 16
)
Therefore, using the inequalities 0 ≤ K t (s, ω) ≤ 1 one gets that

K t , h j,k ≤ 2 -j 2 . ( 17
)
The partial sums of the series in [START_REF] Haar | Zur theorie der orthogonalen fuctionnensysteme[END_REF] are defined in the following way:

Definition 2.1. For all t ∈ I and for each J ∈ Z + , one sets

X J (t) := K t , U η 0 + J-1 j=0 2 j -1 k=0 K t , h j,k ε j,k , ( 18
)
with the convention that when J = 0 one has

X 0 (t) := K t , U η 0 .
The following proposition provides an alternative expression of the process {X J (t) : t ∈ I}.

Proposition 2.1. For every J ∈ Z + and for all l ∈ {0, . . . , 2 J -1}, let K J,l t be the mean value of the function s → K t (s) on the dyadic interval [δ J,l , δ J,l+1 ] defined as:

K J,l t := 2 J δ J,l+1 δ J,l K t (s)ds. ( 19
)
Then, for every J ∈ Z + and for all l ∈ {0, . . . , 2 J -1}, one has

0 ≤ K J,l t ≤ 1 and K J,l t = K J+1,2l t + K J+1,2l+1 t 2 . ( 20
)
Moreover, for all t ∈ I and for each J ∈ Z + , X J (t) can be expressed as:

X J (t) = 2 J -1 l=0 K J,l t ∆B J,l , ( 21
)
where ∆B J,l is the increment of the Brownian motion B defined through [START_REF] Daubechies | Ten lectures on wavelets[END_REF] with j = J and k = l.

Proof. The proof of ( 20) is skipped since it is very easy. For proving [START_REF] Surgailis | Nonhomogeneous fractional integration and multifractional processes Stochastic Process[END_REF] one proceeds by induction on J. It is clearly satisfied when J = 0. Let us assume that it holds for an arbitrary J ∈ Z + and show that it remains true when J is replaced by J + 1. Thus, in view of the induction hypothesis and of [START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF], it suffices to prove that

2 J -1 k=0 K t , h J,k ε J,k = 2 J+1 -1 l=0 K J+1,l t ∆B J+1,l - 2 J -1 l=0 K J,l t ∆B J,l . (22)
First, let us note that

K t , h J,k = 2 J 2       δ J+1,2k+1 δ J+1,2k K t (s)ds - δ J+1,2k+2 δ J+1,2k+1 K t (s)ds       = 2 -J 2 -1 K J+1,2k t -K J+1,2k+1 t .
Thus, one can derive from (13) that

2 J -1 k=0 K t , h J,k ε J,k = 2 J -1 k=0        K J+1,2k t -K J+1,2k+1 t 2        ∆B J+1,2k -∆B J+1,2k+1 . (23)
On the other hand, using the equality ∆B J,l = ∆B J+1,2l + ∆B J+1,2l+1 and the equality in [START_REF] Stoev | How rich is the class of multifractional Brownian motions? Stochastic Process[END_REF] one obtains that

2 J+1 -1 l=0 K J+1,l t ∆B J+1,l - 2 J -1 l=0 K J,l t ∆B J,l = 2 J -1 k=0 K J+1,2k t ∆B J+1,2k + K J+1,2k+1 t ∆B J+1,2k+1 - 2 J -1 k=0        K J+1,2k t + K J+1,2k+1 t 2        ∆B J+1,2k + ∆B J+1,2k+1 = 2 J -1 k=0               K J+1,2k t -K J+1,2k+1 t 2        ∆B J+1,2k -        K J+1,2k t -K J+1,2k+1 t 2        ∆B J+1,2k+1        = 2 J -1 k=0        K J+1,2k t -K J+1,2k+1 t 2        ∆B J+1,2k -∆B J+1,2k+1 . ( 24
)
Thus combining ( 24) and (23) one gets [START_REF] Wojtaszczyk | A mathematical introduction to wavelets[END_REF].

In order to show that, for each fixed t ∈ I, X J (t) converges to X(t) in L 1 (Ω) when J goes to +∞, let us introduce the stochastic process X J (t) : t ∈ I defined through Itô integral in the following way: Definition 2.2. For all fixed J ∈ Z + and t ∈ I, let K J t (s) : s ∈ I be the elementary stochastic process defined, for every s ∈ I, as:

K J t (s) := 2 J -1 l=0 K t (δ J,l )1 [δ J,l ,δ J,l+1 ) (s). ( 25
)
One sets

X J (t) := 1 0 K J t (s)dB(s) = 2 J -1 l=0 K t (δ J,l )∆B J,l . ( 26
)
Remark 2.2. For each fixed t ∈ I, the function the function L t is defined, for all (u, v) ∈ [0, 1] × [a, a], as:

L t (u, v) := (t -u) v-1 2 + = 0 if t ≤ u , (t -u) v-1 2 otherwise. ( 27
)
This function is continuous on [0, 1] × [a, a] and satisfies, for every

(u, v) ∈ [0, 1] × [a, a], 0 ≤ L t (u, v) ≤ 1 . (28)
Moreover, in view of ( 5) and ( 27) one has

K t (s, ω) = L t (s, A(s, ω)) , for all (t, s, ω) ∈ I 2 × Ω, (29)
Remark 2.2, the continuity of the paths of the process {A(s) : s ∈ I}, the dominated convergence theorem, and the isometry property of Itô integral easily imply that the following lemma holds.

Lemma 2.2. For any fixed (t, s, ω) ∈ I 2 × Ω, one has lim J→+∞ K J t (s, ω) = K t (s, ω). ( 30
)
Hence, using the dominated convergence theorem and the isometry property of Itô integral, one gets:

1. the sequence ( K J t ) J∈Z + converges to K t in L 2 (I × Ω) when J goes to +∞; 2. the sequence X J (t) J∈Z + converges to X(t) = 1 0 K t (s)dB(s) in L 2 (Ω) when J goes to +∞.
Lemma 2.3. Assume that the exponent ρ (see [START_REF] Bianchi | Multifractional processes in finance[END_REF]) satisfies [START_REF] Doukhan | Theory and Applications of Long-range Dependence[END_REF]. Then, one has

lim J→+∞ sup t∈I E X J (t) -X J (t) = 0. ( 31
)
The proof of Lemma 2.3 is given in the Appendix A. The following proposition is a straightforward consequence of Lemma 2.2, Lemma 2.3 and (18). Proposition 2.2. Assume that the exponent ρ (see [START_REF] Bianchi | Multifractional processes in finance[END_REF]) satisfies [START_REF] Doukhan | Theory and Applications of Long-range Dependence[END_REF]. Then, for every fixed t ∈ I, one has

X(t) = K t , U η 0 + +∞ j=0 2 j -1 k=0 K t , h j,k ε j,k , ( 32
)
where the convergence holds in L 1 (Ω).

Proposition 2.3. Assume that the exponent ρ (see [START_REF] Bianchi | Multifractional processes in finance[END_REF]) satisfies [START_REF] Doukhan | Theory and Applications of Long-range Dependence[END_REF]. Then one has

E         +∞ j=0 sup t∈I         2 j -1 k=0 K t , h j,k |ε j,k |                 < +∞. ( 33
)
One of the main ingredients of the proof of Proposition 2.3 is the following lemma which concerns the N (0, 1) Gaussian random variables ε j,k and whose proof can be found in [START_REF] Ayache | Rate optimality of wavelet series approximations of fractional Brownian motion[END_REF].

Lemma 2.4. There are an event Ω * ⊆ Ω of probability 1 and a non-negative random variable C * with finite moment of any order, such that the inequality

|ε j,k (ω)| ≤ C * (ω) j + 1. ( 34
)
holds, for all ω ∈ Ω * , for every j ∈ Z + and for each k ∈ {0, . . . , 2 j -1}.

Proof of Proposition 2.3. For all t ∈ I and all j ∈ Z + , one has

2 j -1 k=0 K t , h j,k = K t , h j,[2 j t] + [2 j t]-1 k=0 K t , h j,k , (35)
with the conventions that K t , h j,2 j = 0 and -1 k=0 • • • = 0, which means that the second term in the right-hand side of (35) vanishes when [2 j t] = 0. In the other case [2 j t] ≥ 1, using ( 16) and Lemma 2.1, one gets

[2 j t]-1 k=0 K t , h j,k ≤ c 0 2 j 2 [2 j t]-1 k=0 2 -j (k+ 1 2 ) 2 -j k (t -2 -j-1 -s) a-3 2 2 -j-1 + A(s) -A(s + 2 -j-1 ) ds ≤ c 0 2 -j 2 -1 t-2 -j-1 0 (t -2 -j-1 -s) a-3 2 ds + c 0 2 j 2 t-2 -j-1 0 A(s) -A(s + 2 -j-1 ) ds ≤ c 0 2 -j 2 -1 (t -2 -j-1 ) a-1 2 a -1 2 + c 0 2 j 2 t-2 -j-1 0 A(s) -A(s + 2 -j-1 ) ds ≤ c 3 2 -j 2 + c 0 2 j 2 t-2 -j-1 0 A(s) -A(s + 2 -j-1 )
ds
where c 3 = c 0 2a-1 . Moreover, from ( 17), we know that, for any t ∈ I,

K t , h j,[2 j t] ≤ 2 -j 2 .
Thus, setting c 4 := c 3 + 1, one has, for each t ∈ I,

2 j -1 k=0 K t , h j,k ≤ c 4 2 -j 2 + c 0 2 j 2 1-2 -j-1 0 A(s) -A(s + 2 -j-1 ) ds . ( 36
)
Then using Lemma 2.4 and (36), one obtains, for every ω ∈ Ω * and for all j ∈ Z + , that

sup t∈I         2 j -1 k=0 K t , h j,k |ε j,k |         ≤ C * 1 + j        c 4 2 -j 2 + c 0 2 j 2 1-2 -j-1 0 A(s) -A(s + 2 -j-1 ) ds        ,
where the non-negative random variable C * has finite moments of any order. Thus, in order to derive (33), it is enough to prove that +∞ j=0

2 j 2 1 + j E        C * 1-2 -j-1 0 A(s) -A(s + 2 -j-1 ) ds        < +∞ . ( 37
)
For every j ∈ Z + , it follows from the Cauchy-Schwarz inequality and ( 7) that

E        C * 1-2 -j-1 0 A(s) -A(s + 2 -j-1 ) ds        ≤ E C 2 * 1 2         E                1-2 -j-1 0 A(s) -A(s + 2 -j-1 ) ds        2                 1 2 ≤ E C 2 * 1 2        E        1-2 -j-1 0 A(s) -A(s + 2 -j-1 ) 2 ds               1 2 = E C 2 * 1 2        1-2 -j-1 0 E A(s) -A(s + 2 -j-1 ) 2 ds        1 2 ≤ c 5 2 -ρj (38)
where c 5 > 0 is a deterministic finite constant not depending on j and t. Finally, combining (38) and the assumption [START_REF] Doukhan | Theory and Applications of Long-range Dependence[END_REF], one gets (37).

Proof of Theorem

β f [ν 1 , ν 2 ] := sup        γ ∈ [0, 1] : sup t ,t ∈[ν 1 ,ν 2 ] f (t ) -f (t ) |t -t | γ < +∞        . ( 39
)
Throughout this section, we assume that there are a deterministic constant γ ∈ ( 1 2 , 1) and Ω 0 an event of probability 1, such that, for each ω ∈ Ω 0 , the path A(•, ω) of the process {A(s) : s ∈ I} satisfies, on a I, a uniform Hölder condition of order γ. That is there exists a finite constant C 1 (ω) such that, for all s , s ∈ I, the following inequality holds :

A(s , ω) -A(s , ω) ≤ C 1 (ω) s -s γ . ( 40
)
Theorem 3.1. Assume that the condition (40) is satisfied. Then, there exists an event Ω 1 ⊆ Ω of probability 1 such that, for every ω ∈ Ω 1 and for all non-degenerate compact interval

[ν 1 , ν 2 ] ⊆ I, one has β X(•,ω) [ν 1 , ν 2 ] ≥ A ν 1 ,ν 2 (ω) , ( 41
)
where

A ν 1 ,ν 2 (ω) := min A(s, ω) : s ∈ [ν 1 , ν 2 ] . ( 42
)
X(t ) -X(t ) ≤ K t , U -K t , U |η 0 | + +∞ j=0 2 j -1 k=0 K t , h j,k -K t , h j,k |ε j,k | , ( 43
)
Observe that it can easily be seen that, for all fixed ω ∈ Ω (the probability space), t → K t (ω), U is a Lipschitz function on I. So, we only have to focus on the second term in the right-hand side of (43). Using Lemma 2.4, one has +∞ j=0

2 j -1 k=0 K t , h j,k -K t , h j,k |ε j,k | ≤ C * +∞ j=0         j + 1 2 j -1 k=0 K t , h j,k -K t , h j,k         . ( 44
)
Moreover, in view of ( 16) and (29), for each j ∈ Z + , one has

2 j -1 k=0 K t , h j,k -K t , h j,k ≤ 2 j 2 2 j -1 k=0
2 -j (k+ 1 2 ) 2 -j k L t s, A(s) -L t s + 2 -j-1 , A(s + 2 -j-1 ) -L t s, A(s) + L t s + 2 -j-1 , A(s + 2 -j-1 ) ds ≤ 2 j 2 1-2 -j-1 0 L t s, A(s) -L t s + 2 -j-1 , A(s + 2 -j-1 ) -L t s, A(s) + L t s + 2 -j-1 , A(s + 2 -j-1 ) ds . ( 45
)
Therefore, using the triangular inequality, one gets that

2 j -1 k=0 K t , h j,k -K t , h j,k ≤ λ 1 j (t , t ) + λ 2 j (t , t ) , ( 46
)
where

λ 1 j (t , t ) := 2 j 2 1-2 -j-1 0 L t s, A(s) -L t s, A(s + 2 -j-1 ) -L t s, A(s) + L t s, A(s + 2 -j-1 ) ds (47)
and

λ 2 j (t , t ) := 2 j 2 1-2 -j-1 0 L t s, A(s + 2 -j-1 ) -L t s + 2 -j-1 , A(s + 2 -j-1 ) -L t s, A(s + 2 -j-1 ) + L t s + 2 -j-1 , A(s + 2 -j-1 ) ds . ( 48
)
Let us set

t j := min{t , 1 -2 -j-1 } and t j := min{t , 1 -2 -j-1 } , ( 49
)
and let j 0 denote the unique nonnegative integer such that

2 -j 0 -1 < t -t ≤ 2 -j 0 . ( 50
)
From now on and till the end of the proof, we work on the event Ω 1 := Ω * * ∩ Ω 0 of probability 1.

• Step 1 : Upper bound for λ 1 j (t , t ) . From ( 27), ( 47) and (49), we know that

λ 1 j (t , t ) =2 j 2 t j 0 t -s A(s)-1 2 -t -s A(s+2 -j-1 )-1 2 -t -s A(s)-1 2 + t -s A(s+2 -j-1 )-1 2 ds + 2 j 2 t j t j t -s A(s)-1 2 -t -s A(s+2 -j-1 )-1 2 ds . ( 51
)
Let us first conveniently bound from above the first integral in (51). Observe that this integral vanishes when t j = 0, so there is no restriction to assume that t j > 0. Let us then fix s ∈ (0, t j ). By applying the Mean Value Theorem to the function x

→ (t -s) x-1 2 -(t -s) x-1 2 , one gets t -s A(s)-1 2 -t -s A(s+2 -j-1 )-1 2 -t -s A(s)-1 2 + t -s A(s+2 -j-1 )-1 2 = (t -s) e 1 -1 2 log(t -s) -(t -s) e 1 -1 2 log(t -s) A(s) -A(s + 2 -j-1 ) (52)
where e 1 depends on t , t , s, j and is between A(s) and A(s + 2 -j-1 ). This implies that

e 1 ∈ [a, a] ⊆ 1 2 , 1 . (53)
Let us apply once again the Mean Value Theorem to the function x → x e 1 -1 2 log(x); it follows that log(e -1 2 ) + 1

(t -s) e 1 -1 2 log(t -s) -(t -s) e 1 -1 2 log(t -s) ≤ e e 1 -3 2 2 e 1 - 1 2 | log(e 2 )| + 1 t -t , (54)
≤ (t -s) a-3 2 log (t -s) -1 + 1 , ( 56
)
where the last inequality comes from the fact that the function x → x a-3 2 log(x -1 ) is decreasing on (0, 1). By combining (52), ( 54) and (56), it follows that 2

j 2 t j 0 t -s A(s)-1 2 -t -s A(s+2 -j-1 )-1 2 -t -s A(s)-1 2 + t -s A(s+2 -j-1 )-1 2 ds ≤2 j 2 t -t t j 0 (t -s) a-3 2 log (t -s) -1 + 1 A(s) -A(s + 2 -j-1 ) ds ≤C 1 2 -j(γ-1 2 ) t -t t j 0 (t -s) a-3 2 log (t -s) -1 + 1 ds ≤C 1 2 -j(γ-1 2 ) t -t 1 0 u a-3 2 log u -1 + 1 du =C 2 2 -j(γ-1 2 ) t -t , ( 57
)
where we have used the assumption (40) on the paths of A, and where

C 2 = C 1 (a -1/2) -1 + (a -1/2) -2 .
Let us now conveniently bound from above the second integral in (51). There is no restriction to assume that t j = t < 1 -2 -j-1 since this integral vanishes when t j = 1 -2 -j-1 . Therefore, one has that

2 j 2 t j t j t -s A(s)-1 2 -t -s A(s+2 -j-1 )-1 2 ds = 2 j 2 t j t t -s A(s)-1 2 -t -s A(s+2 -j-1 )-1 2 ds . ( 58
)
Let us fix s ∈ (t , t j ) and apply the Mean Value Theorem to the function x → (ts) x-1 2 . One gets

t -s A(s)-1 2 -t -s A(s+2 -j-1 )-1 2 =(t -s) e 3 -1 2 log(t -s) A(s) -A(s + 2 -j-1 ) ≤(t -s) a-1 2 log(t -s) A(s) -A(s + 2 -j-1 ) , ( 59
)
since e 3 is between A(s) and A(s + 2 -j-1 ). Next, using (59) and the assumption (40), one obtains

2 j 2 t j t t -s A(s)-1 2 -t -s A(s+2 -j-1 )-1 2 ds ≤2 j 2 t j t (t -s) a-1 2 log(t -s) A(s) -A(s + 2 -j-1 ) ds ≤C 1 2 -j(γ-1 2 ) t t (t -s) a-1 2 log(t -s) ds ≤C 3 2 -j(γ-1 2 ) (t -t ) , ( 60
)
where

C 3 = C 1 sup x∈(0,1] x a-1 2 log(x) < +∞.
Next, combining (51), ( 57) and (60), it follows that

λ 1 j (t , t ) ≤ C 4 2 -j(γ-1 2 ) (t -t ) , ( 61
)
where

C 4 = C 2 + C 3 .
• Step 2 : Upper bound for λ 2 j (t , t ) . First, observe that one can derive from ( 27), ( 48) and ( 49), one has

λ 2 j (t , t ) = 2 j 2 t j 0 L t s, A(s + 2 -j-1 ) -L t s + 2 -j-1 , A(s + 2 -j-1 ) -L t s, A(s + 2 -j-1 ) + L t s + 2 -j-1 , A(s + 2 -j-1 ) ds . ( 62
)
Let us define ν 0 in the following way: if ν 1 = 0 then ν 0 := 0, otherwise ν 0 is an arbitrary fixed strictly positive real number belonging to the open interval (0, ν 1 ). Then, one sets

µ 1 j (t , t , ν 0 ) := 2 j 2 max{0,ν 0 -2 -j-1 } 0 L t s, A(s + 2 -j-1 ) -L t s + 2 -j-1 , A(s + 2 -j-1 ) -L t s, A(s + 2 -j-1 ) + L t s + 2 -j-1 , A(s + 2 -j-1 ) ds , ( 63
)
µ 2 j (t , t , ν 0 ) := 2 j 2 max{0,t -2 -j-1 } max{0,ν 0 -2 -j-1 } L t s, A(s + 2 -j-1 ) -L t s + 2 -j-1 , A(s + 2 -j-1 ) -L t s, A(s + 2 -j-1 ) + L t s + 2 -j-1 , A(s + 2 -j-1 ) ds , ( 64
)
and

µ 3 j (t , t ) := 2 j 2 t j max{0,t -2 -j-1 } L t s, A(s + 2 -j-1 ) -L t s + 2 -j-1 , A(s + 2 -j-1 ) -L t s, A(s + 2 -j-1 ) + L t s + 2 -j-1 , A(s + 2 -j-1 ) ds . ( 65
)
Notice that one has

λ 2 j (t , t ) = µ 1 j (t , t , ν 0 ) + µ 2 j (t , t , ν 0 ) + µ 3 j (t , t ) . ( 66
)
1. Upper bound for µ 1 j (t , t , ν 0 ) . Observe that µ 1 j (t , t , ν 0 ) = 0 when ν 0 -2 -j-1 ≤ 0, so there is no restriction to assume that j is such that ν 0 -2 -j-1 > 0. Then, using (27), (63) and the inequalities

ν 0 < ν 1 ≤ t ≤ t ≤ ν 2 , ( 67
)
one gets that

µ 1 j (t , t , ν 0 ) = 2 j 2 ν 0 -2 -j-1 0 (t -s) A(s+2 -j-1 )-1 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 + (t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 ds . ( 68
)
Let us fix s ∈ (0, ν 0 -2 -j-1 ) and apply the Mean Value Theorem to the function x → x A(s+2 -j-1 )-1 2 -(x -2 -j-1 ) A(s+2 -j-1 )-1 2 . One gets that

(t -s) A(s+2 -j-1 )-1 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 + (t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 = A(s + 2 -j-1 ) - 1 2 (t -t ) e A(s+2 -j-1 )-3 2 4
-(e 4 -2 -j-1 ) A(s+2 -j-1 )-3 2

≤(tt ) e

A(s+2 -j-1 )-3 24

-(e 4 -2 -j-1 ) A(s+2 -j-1 )- Then, the Mean Value Theorem applied to the function x → x A(s+2 -j-1 )-3 2 implies that there exists

e 5 ∈ (e 4 -2 -j-1 , e 4 ) (71)
such that e A(s+2 -j-1 )-3 24

-

(e 4 -2 -j-1 ) A(s+2 -j-1 )-1 2 =2 -j-1 A(s + 2 -j-1 ) - 3 2 e A(s+2 -j-1 )-5 2 5 ≤2 -j-1 (ν 1 -s -2 -j-1 ) a-5 2 , ( 72
)
where we have used ( 3), ( 67), ( 70) and (71). Putting together (68), ( 69) and ( 72), it follows that

µ 1 j (t , t , ν 0 ) ≤ 2 -j 2 -1 (t -t ) ν 0 -2 -j-1 0 (ν 1 -s -2 -j-1 ) a-5 2 ds ≤ 2 -j 2 (t -t ) (ν 1 -ν 0 ) a-3 2 3 -2a . ( 73
)
2. Upper bound for µ 2 j (t , t , ν 0 ) . Observe that µ 2 j (t , t , ν 0 ) = 0 when t -2 -j-1 ≤ 0, so there is no restriction to assume that j is such that t -2 -j-1 > 0. Then, using ( 27) and (64), we have

µ 2 j (t , t , ν 0 ) = 2 j 2 t -2 -j-1 max{0,ν 0 -2 -j-1 } (t -s) A(s+2 -j-1 )-1 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 + (t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 ds . ( 74
)
Assume first that j ∈ {0, . . . , j 0 -1}. Combining (74) and (42) in which ν 1 is replaced by ν 0 , we get that

µ 2 j (t , t , ν 0 ) = 2 -j 2 -1 2 j+1 t -1 max{0,2 j+1 ν 0 -1} (t -2 -j-1 u) A(2 -j-1 u+2 -j-1 )-1 2 -(t -2 -j-1 u -2 -j-1 ) A(2 -j-1 u+2 -j-1 )-1 2 -(t -2 -j-1 u) A(2 -j-1 u+2 -j-1 )-1 2 + (t -2 -j-1 u -2 -j-1 ) A(2 -j-1 u+2 -j-1 )-1 2 du ≤ 2 -(j+1)A ν 0 ,ν 2 2 j+1 t -1 max{0,2 j+1 ν 0 -1} (2 j+1 t -u) A(2 -j-1 (u+1))-1 2 -(2 j+1 t -(u + 1)) A(2 -j-1 (u+1))-1 2 -(2 j+1 t -u) A(2 -j-1 (u+1))-1 2 + (2 j+1 t -(u + 1)) A(2 -j-1 (u+1))-1 2 du = 2 -(j+1)A ν 0 ,ν 2 min{2 j+1 t -1,2 j+1 (t -ν 0 )} 0 (2 j+1 (t -t ) + v + 1) A(t -2 -j-1 v)-1 2 -(2 j+1 (t -t ) + v) A(t -2 -j-1 v)-1 2 -(v + 1) A(t -2 -j-1 v)-1 2 + v A(t -2 -j-1 v)-1 2 dv . (75)
Next, applying on the interval [0, 2 j+1 (tt )] the Mean Value Theorem to the function

x → (x + v + 1) A(t -2 -j-1 v)-1 2 -(x + v) A(t -2 -j-1 v)-1 2
, and using (3), it follows that (75) can be bounded from above by

2 (j+1)(1-A ν 0 ,ν 2 ) (t -t ) +∞ 0 v A(t -2 -j-1 v)-3 2 -(v + 1) A(t -2 -j-1 v)-3 2 dv .
Observe that, thanks to (3), this last integral can be bounded from above by a deterministic constant c > 0 not depending on t , t , j. Thus, in view of the definition (50) of j 0 and the fact that j ≤ j 0 -1, one gets that

µ 2 j (t , t , ν 0 ) ≤ c2 (j+1)(1-A ν 0 ,ν 2 ) (t -t ) ≤ c(t -t ) A ν 0 ,ν 2 , for all j ∈ {0, . . . , j 0 -1}. ( 76
)
Assume now that j ≥ j 0 . Let us fix s ∈ (max{0, ν 0 -2 -j-1 }, t -2 -j-1 ). Applying the Mean Value Theorem to the function x → (tsx) A(s+2 -j-1 )-1 2 -(tsx) A(s+2 -j-1 )-1 2 , we obtain, for some e 6 ∈ (0, 2 -j-1 ), that

(t -s) A(s+2 -j-1 )-1 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 + (t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 =2 -j-1 A(s + 2 -j-1 ) - 1 2 (t -s -e 6 ) A(s+2 -j-1 )-3 2 -(t -s -e 6 ) A(s+2 -j-1 )-3 2 ≤2 -j-1 (t -s -2 -j-1 ) A(s+2 -j-1 )-3 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-3 2 , ( 77
)
where we have used (3). Consequently, (74) can be bounded from above by

2 -j 2 -1 t -2 -j-1 max{0,ν 0 -2 -j-1 } (t -s -2 -j-1 ) A(s+2 -j-1 )-3 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-3 2 ds ≤2 -j 2 -1 t ν 0 (t -v) A(v)-3 2 -(t -v) A(v)-3 2 dv =2 -j 2 -1 (t -t ) -1 2 t -ν 0 t -t 1 (t -t ) A(t -(t -t )u) (u -1) A(t -(t -t )u)-3 2 -u A(t -(t -t )u)-3 2 du ≤2 -j 2 -1 (t -t ) A ν 0 ,ν 2 -1 2 +∞ 1 (u -1) A(t -(t -t )u)-3 2 -u A(t -(t -t )u)-3 2 du , (78)
where we have used the changes of variables v = s + 2 -j-1 and u = t -v t -t . Moreover, thanks to (3), the last integral can be bounded from above by a deterministic constant c > 0 not depending on t , t , j. Then (74), (77), and (78) imply

µ 2 j (t , t , ν 0 ) ≤ c 2 -j 2 -1 (t -t ) A ν 0 ,ν 2 -1 2
, for all j ≥ j 0 .

(79)

3. Upper bound for µ 3 j (t , t ) . Assume first that j ∈ {0, . . . , j 0 -1}. Let us then decompose the integral (65) defining µ 3 j (t , t ) into two parts: the integral on the interval [max{0, t -2 -j-1 }, max{0, t -2 -j-1 }] denoted by µ 3,1 j (t , t ), and the integral on the interval [max{0, t -2 -j-1 }, t j ] denoted by µ 3,2 j (t ). Let us now provide an appropriate upper bound for µ 3,1 j (t , t ). One can assume that t -2 -j-1 ≥ 0 since µ 3,1 j (t , t ) = 0 in the other case. Observe that one can derive from the latter inequality and from (50) that t ≥ t -2 -j-1 . Thus, one has

µ 3,1 j (t , t ) = 2 j 2 t -2 -j-1 max{0,t -2 -j-1 } (t -s) A(s+2 -j-1 )-1 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 ds ≤ 2 j 2 t -2 -j-1 t -2 -j-1 (t -s) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 ds + 2 j 2 t -2 -j-1 t -2 -j-1 (t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 ds . ( 80
)
Using ( 42), (50) and the assumption j ≤ j 0 -1, one gets that 2

j 2 t -2 -j-1 t -2 -j-1 (t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 ds ≤2 j 2 t -2 -j-1 t -2 -j-1 (t -s -2 -j-1 ) A ν 1 ,ν 2 -1 2 ds = 2 j 2 A ν 1 ,ν 2 + 1 2 (t -t ) A ν 1 ,ν 2 + 1 2 ≤C 5 (t -t ) A ν 1 ,ν 2 , ( 81
)
where

C 5 = 2 -1 2 (A ν 1 ,ν 2 + 1 2 ) -1 .
Using again (42) and the change of variable v = 2 j+1 (ts) -1, one obtains that 2

j 2 t -2 -j-1 max{0,t -2 -j-1 } (t -s) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 ds ≤2 -j 2 -1 2 j+1 (t -t ) 0 (2 -j-1 (v + 1)) A(t -2 -j-1 v)-1 2 -(t -t + 2 -j-1 (v + 1)) A(t -2 -j-1 v)-1 2 dv ≤2 -(j+1)A ν 1 ,ν 2 2 j+1 (t -t ) 0 (v + 1) A(t -2 -j-1 v)-1 2 -(2 j+1 (t -t ) + v + 1) A(t -2 -j-1 v)-1 2 dv ≤2 -(j+1)A ν 1 ,ν 2 2 j+1 (t -t ) 0 (v + 1) A(t -2 -j-1 v)-1 2 -(2 j+1 (t -t ) + v + 1) A(t -2 -j-1 v)-1 2 dv ≤2 (j+1)(1-A ν 1 ,ν 2 ) (t -t ) 1 0 v A(t -2 -j-1 v)-3 2 dv ( 82
)
where we have used in the last equality the Mean Value Theorem applied to the function x → (x + v + 1) A(t -2 -j-1 v)-1 2 , (50) and the assumption j ≤ j 0 -1. Observe that, thanks to (3), the last integral can be bounded from above by a deterministic constant c > 0 not depending on t , t , j. Thus, one can derive from (82), (50) and the inequality j ≤ j 0 -1, that 2

j 2 t -2 -j-1 max{0,t -2 -j-1 } (t -s) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 ds ≤ c (t -t ) A ν 1 ,ν 2 . ( 83
)
Let us now bound from above the integral µ 3,2 j (t ). One has

µ 3,2 j (t ) = 2 j 2 t j max{0,t -2 -j-1 } L t s, A(s + 2 -j-1 ) -L t s + 2 -j-1 , A(s + 2 -j-1 ) -L t s, A(s + 2 -j-1 ) + L t s + 2 -j-1 , A(s + 2 -j-1 ) ds = 2 j 2 t j max{0,t -2 -j-1 } (t -s) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 + 2 j 2 t j t j (t -s) A(s+2 -j-1 )-1 2 ds . ( 84
)
Observe that, one knows from (40) and (42) that

min A(x) : x ∈ [ν 1 -2 -j-1 , ν 2 + 2 -j-1 ] ∩ [0, 1] ≥ A ν 1 ,ν 2 -C 1 2 -(j+1)γ . ( 85
)
Hence, one has

2 -(j+1) min A(x) : x∈[ν 1 -2 -j-1 ,ν 2 +2 -j-1 ]∩[0,1] ≤ C 2 -(j+1)A ν 1 ,ν 2 , ( 86
)
where C is a positive almost surely finite random constant not depending on t , t , j. It follows from the change of variable u = 2 j+1 (ts), (86), and the Mean Value Theorem that 2

j 2 t j max{0,t -2 -j-1 } (t -s) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 ds ≤2 -j 2 -1 1-(t -t )2 j+1 (t -t j )2 j+1 (t -t + 2 -j-1 u) A(t +(1-u)2 -j-1 )-1 2 -(2 -j-1 u) A(t +(1-u)2 -j-1 )-1 2 du =C 2 -1 2 2 -(j+1)A ν 1 ,ν 2 1-(t -t )2 j+1 (t -t j )2 j+1 2 j+1 (t -t ) + u A(t +(1-u)2 -j-1 )-1 2 -u A(t +(1-u)2 -j-1 )-1 2 du ≤C 2 -1 2 2 (j+1)(1-A ν 1 ,ν 2 ) (t -t ) 1-(t -t )2 j+1 (t -t j )2 j+1 u A(t +(1-u)2 -j-1 )-3 2 du ≤C 2 -1 2 2 (j+1)(1-A ν 1 ,ν 2 ) (t -t ) 1 0 u A(t +(1-u)2 -j-1 )-3 2 du . ( 87
)
Observe that, thanks to (3), the last integral can be bounded from above by a deterministic constant c > 0 not depending on t , t , j. Then, (50) and the assumption j ≤ j 0 -1 entail that 2

j 2 t j max{0,t -2 -j-1 } (t -s) A(s+2 -j-1 )-1 2 -(t -s) A(s+2 -j-1 )-1 2 ds ≤ C 6 (t -t ) A ν 1 ,ν 2 , ( 88
)
where C 6 = C 2 -1 2 c . Let us now provide an appropriate upper bound for the second integral in the right-hand side of (84). There is no restriction to assume that t j = t since the integral vanishes in the other case. Using the triangular inequality, the Mean Value Theorem, (3), (40), (42), (50) and the assumption j ≤ j 0 -1, one gets

2 j 2 t j t (t -s) A(s+2 -j-1 )-1 2 ds ≤2 j 2 t j t (t -s) A(s+2 -j-1 )-1 2 -(t -s) A(s)-1 2 ds + 2 j 2 t j t (t -s) A(s)-1 2 ds ≤2 j 2 t j t A(s + 2 -j-1 ) -A(s) (t -s) a-1 2 log(t -s) ds + 2 j 2 t t (t -s) A ν 1 ,ν 2 -1 2 ds ≤C 3 2 -j(γ-1 2 ) (t -t ) + 2 j 2 (t -t ) A ν 1 ,ν 2 + 1 2 ≤C 3 2 -j(γ-1 2 ) (t -t ) + 2 -1 2 (t -t ) A ν 1 ,ν 2 (89)
where C 3 is the same random constant as in (60). Putting together, (81), ( 83), ( 88) and ( 89), one gets that

µ 3 j (t , t ) ≤ C 7 (t -t ) A ν 1 ,ν 2 + C 3 2 -j(γ-1 2 ) (t -t ) , for all j ∈ {0, . . . , j 0 -1}, (90)
where the finite random constants C 3 and

C 7 = C 5 + c + C 6 + 2 -1 2
do not depend on t , t , j, j 0 .

It remains for us to provide an appropriate upper bound for µ 3 j (t , t ) in the case where j ≥ j 0 . One knows from the latter inequality and from (50) that t -2 -j-1 < t < t -2 -j-1 ≤ t j . Thus, it results from (65), ( 27) and the triangular inequality that

µ 3 j (t , t ) =2 j 2 t j max{0,t -2 -j-1 } (t -s) A(s+2 -j-1 )-1 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 + -(t -s) A(s+2 -j-1 )-1 2 + ds ≤2 j 2 t j t -2 -j-1 (t -s) A(s+2 -j-1 )-1 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 + ds + 2 j 2 t t -2 -j-1 (t -s) A(s+2 -j-1 )-1 2 ds ≤2 j 2 t -2 -j-1 t -2 -j-1 (t -s) A(s+2 -j-1 )-1 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 ds + 2 j 2 t j t -2 -j-1 (t -s) A(s+2 -j-1 )-1 2 ds + 2 j 2 t t -2 -j-1 (t -s) A(s+2 -j-1 )-1 2 ds . ( 91
)
Let now bound from above in a suitable way each of the three integrals appearing in (91). Using standard computations, (86) and (42), one gets that

2 j 2 t j t -2 -j-1 (t -s) A(s+2 -j-1 )-1 2 ds ≤ 2 j 2 t j t -2 -j-1 2 -(j+1)(A(s+2 -j-1 )-1 2 ) ds ≤ C 2 j+ 1 2 (t j -t + 2 -j-1 )2 -(j+1)A ν 1 ,ν 2 ≤ C 2 -1 2 2 -(j+1)A ν 1 ,ν 2 (92) and 2 j 2 t t -2 -j-1 (t -s) A(s+2 -j-1 )-1 2 ds ≤ 2 j 2 t t -2 -j-1 (t -s) A ν 1 ,ν 2 -1 2 ds ≤ 2 j 2 t t -2 -j-1 2 -(j+1)(A ν 1 ,ν 2 -1 2 ) ds = 2 -1 2 2 -(j+1)A ν 1 ,ν 2 . ( 93
)
Moreover, the Mean Value Theorem, standard computations, and (42) allow us to obtain that 2

j 2 t -2 -j-1 t -2 -j-1 (t -s) A(s+2 -j-1 )-1 2 -(t -s -2 -j-1 ) A(s+2 -j-1 )-1 2 ds ≤2 -j 2 -1 t -2 -j-1 t -2 -j-1 (t -s -2 -j-1 ) A(s+2 -j-1 )-3 2 ds =2 -j 2 -1 t t (t -s) A(s)-3 2 ds ≤2 -j 2 -1 t t (t -s) A ν 1 ,ν 2 -3 2 ds = 2 -j 2 -1 A ν 1 ,ν 2 -1 2 (t -t ) A ν 1 ,ν 2 -1 2 . ( 94
)
Combining ( 91), ( 92), ( 93) and ( 94), it follows that

µ 3 j (t , t ) ≤(C 5 + 1)2 -1 2 2 -(j+1)A ν 1 ,ν 2 + 2 -j 2 -1 A ν 1 ,ν 2 -1 2 (t -t ) A ν 1 ,ν 2 -1 2
, for all j ≥ j 0 .

(95)

Conclusion of the second step.

In the case where j ∈ {0, . . . , j 0 -1}, putting together (66), ( 73), ( 76) and (90), one obtains

λ 2 j (t , t ) ≤ C 8 2 -j(γ-1 2 ) (t -t ) + (t -t ) A ν 0 ,ν 2 , ( 96
)
where the finite random constant

C 8 = (ν 1 -ν 0 ) a-3 2 3-2a
+ c + C 7 + C 3 does not depend on t , t , j, j 0 . In the other case where j ≥ j 0 , combining (66), ( 73), ( 79) and (95), one gets that

λ 2 j (t , t ) ≤ C 9 2 -j 2 (t -t ) + 2 -j 2 +1 (t -t ) A ν 0 ,ν 2 -1 2 + 2 -(j+1)A ν 1 ,ν 2 , ( 97
)
where the finite random constant

C 9 = (ν 1 -ν 0 ) a-3 2 3-2a + c 2 + (C 5 + 1)2 -1 2 + 1 2A ν 1 ,ν 2 -1
does not depend on t , t , j, j 0 .

• Step 3 : Conclusion.

Putting together (44), ( 46), (61), ( 96) and (97), one obtains that +∞ j=0

2 j -1 k=0 K t , h j,k -K t , h j,k |ε j,k | ≤C * j 0 -1 j=0 j + 1 (C 4 + C 8 )2 -j(γ-1 2 ) (t -t ) + C 8 (t -t ) A ν 0 ,ν 2 + C * +∞ j=j 0 j + 1 C 4 2 -j(γ-1 2 ) (t -t ) + C 9 2 -j 2 (t -t ) + 2 -j 2 +1 (t -t ) A ν 0 ,ν 2 -1 2 + 2 -(j+1)A ν 1 ,ν 2 . ( 98
)
Moreover, setting C 10 = C 4 + 2C 8 and using (50), it follows that

j 0 -1 j=0 j + 1 (C 4 + C 8 )2 -j(γ-1 2 ) (t -t ) + C 8 (t -t ) A ν 0 ,ν 2 ≤ j 0 (t -t ) A ν 0 ,ν 2 j 0 -1 j=0 (C 4 + C 8 )2 -j(γ-1 2 ) (t -t ) 1-A ν 0 ,ν 2 + C 8 ≤ log 2 (t -t ) (t -t ) A ν 0 ,ν 2 C 10 j 0 ≤C 10 log 2 (t -t ) and +∞ j=j 0 j + 1 C 4 2 -j(γ-1 2 ) (t -t ) + C 9 2 -j 2 (t -t ) + 2 -j 2 +1 (t -t ) A ν 0 ,ν 2 -1 2 + 2 -(j+1)A ν 1 ,ν 2 ≤(C 4 + C 9 ) +∞ j=0 2 -j(γ-1 2 ) j + 1 (t -t ) + 2 ε+ 1 2 C 9 +∞ j=0 2 -εj j + 1 (t -t ) (A ν 0 ,ν 2 -ε) + C 9 +∞ j=0 2 -εj j + 1 (t -t ) (A ν 1 ,ν 2 -ε) , ( 100
)
where ε is an arbitrarily small fixed positive real number.

The inequalities (98), ( 99) and (100) show that the stochastic process {X(t) : t ∈ I} satisfies almost surely a uniform Hölder condition of order (A ν 0 ,ν 2ε) on the interval [ν 1 , ν 2 ]. Therefore, one has almost surely that β X ([ν 1 , ν 2 ]) ≥ (A ν 0 ,ν 2ε). This implies that (41) is satisfied, since (A ν 0 ,ν 2ε) goes to A ν 1 ,ν 2 when ν 0 approaches ν 1 and ε approaches 0.

Simulations

In order to simulate paths of the MPRE {X(t) : t ∈ I} defined in (4), it seems natural to use its approximation { X J (t) : t ∈ I} given by (26). But so far, we only know that, for each fixed t ∈ I, the random variable X J (t) converges in L 2 (Ω) to the random variable X(t), when J goes to +∞ (see Lemma 2.2). The first main goal of the present section is to show that this weak convergence result can be greatly improved. In fact, Proposition 4.1, stated below, shows that the convergence also holds almost surely and uniformly in t ∈ I. It is worth mentioning that the main ingredient of the proof of this proposition is Theorem 2.1 which has been obtained in Section 2 thanks to the Haar basis. Also, we mention that another ingredient of the proof of Proposition 4.1 is the following classical theorem.

Let J ∈ Z + be arbitrary and fixed. Using the triangular inequality, [START_REF] Surgailis | Nonhomogeneous fractional integration and multifractional processes Stochastic Process[END_REF] and Theorem 4.1, one gets

X J (t) -X J (t) = [2 J t] l=0 K t (δ J,l ) -K J,l t ∆B J,l ≤ [2 J t] l=0 K t (δ J,l ) -K J,l t ∆B J,l ≤ Ĉ2 -J 2 1 + J [2 J t] l=0 K t (δ J,l ) -K J,l t = Ĉ2 -J 2 1 + J          K t (δ J,[2 J t] ) -K J,[2 J t] t + [2 J t]-1 l=0 K t (δ J,l ) -K J,l t          , ( 104
)
where Ĉ := sup j∈Z + ,l∈{0,...2 j -1}

|∆B j,l |2 j 2 (1 + j) -1 2 < ∞. One knows from the inequality (124) in the

Appendix A that K t (δ J,[2 J t] ) -K J,[2 J t] t ≤ 2.
Let us provide a suitable upper bound for

[2 J t]-1 l=0 K t (δ J,l ) -K J,l t .
0bserve that, for any l ∈ 0, . . . , [2 J t] -1 , one has t ≥ δ J,l+1 . Thus, it results from Lemma 2.1, [START_REF] Samorodnitsky | Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance[END_REF] and Condition (40) that

[2 J t]-1 l=0 K t (δ J,l ) -K J,l t ≤ c 0 2 J [2 J t]-1 l=0       δ J,l+1 δ J,l |A(s) -A(δ J,l )|ds + δ J,l+1 δ J,l (t -s) a-3 2 (s -δ J,l )ds       ≤ C * 0 2 J [2 J t]-1 l=0 δ J,l+1 δ J,l (s -δ J,l ) γ ds + c 0 [2 J t]-1 l=0 δ J,l+1 δ J,l (t -s) a-3 2 ds = C * 0 γ + 1 2 J 2 -J(1+γ) [2 J t] + c 0 δ J,[2 J t] 0 (t -s) a-3 2 ds ≤ C * 0 γ + 1 2 J(1-γ) + c 0 a -1 2 t a-1 2 ≤ C * 0 γ + 1 2 J(1-γ) + c 0 a -1 2 ,
where C * 0 := c 0 C 1 . Therefore, one deduces from (104) and the last inequality that

| X J (t) -X J (t)| ≤ Ĉ2 -J 2 1 + J          K t (δ J,[2 J t] ) -K J,[2 J t] t + [2 J t]-1 l=0 K t (δ J,l ) -K J,l t          ≤ Ĉ2 -J 2 1 + J       2 + c * 0 γ + 1 2 J(1-γ) + c 0 a -1 2       .
Then using the inequality γ > 1 2 , one obtains (103). The second main goal of the present section is to show that simulating paths of the MPRE {X(t) : t ∈ I}, defined in (4), can also be done by approximating it by the stochastic process { X J (t) : t ∈ I} defined as follows: Definition 4.1. For each fixed J ∈ Z + , the stochastic process { X J (t) : t ∈ I} is defined, for all t ∈ I, by:

X J (t) := 2 J -1 l=0 K J,l t ∆B J,l ,
where, for every l ∈ {0, . . . , 2 J -1}, ∆B J,l is as in [START_REF] Daubechies | Ten lectures on wavelets[END_REF] and

K J,l t := 2 J δ J,l+1 δ J,l (t -s) A(δ J,l )-1 2 + ds (105) = 2 J A(δ J,l ) + 1 2 (t -δ J,l ) A(δ J,l )+ 1 2 + -(t -δ J,l+1 ) A(δ J,l )+ 1 2 + . ( 106
)
The following proposition, whose proof mainly relies on Proposition 4.1, shows that, when J goes to +∞, { X J (t) : t ∈ I} converges to {X(t) : t ∈ I} almost surely and uniformly in t ∈ I. Proposition 4.2. Let Ω 2 be the same event of probability 1 as in Proposition 4.1. Then, for any ω ∈ Ω 2 , one has

lim J→+∞ sup t∈I X J (t, ω) -X(t, ω) = 0. ( 107
)
Proof. In view of Proposition 4.1, it enough to prove that, on Ω 2 , one has

lim J→+∞ sup t∈I X J (t) -X J (t) = 0. ( 108
)
Let J ∈ Z + be arbitrary and fixed. Similarly to (104), it can be shown that the following inequality holds on Ω 2 :

X J (t) -X J (t) ≤ Ĉ2 -J 2 1 + J          K t (δ J,[2 J t] ) -K J,[2 J t] t + [2 J t]-1 l=0 K t (δ J,l ) -K J,l t          , ( 109
)
where Ĉ is the same as in (104). One can derive from Remark 2.2 and the triangular inequality that

K t (δ J,[2 J t] ) -K J,[2 J t] t ≤ 2 J δ J,[2 J t]+1 δ J,[2 J t] L t δ J,[2 J t] , A(δ J,[2 J t] ) - 1 2 -L t s, A(δ J,[2 J t] ) - 1 2 ds ≤ 2 . ( 110
)
Let us provide a suitable upper bound for

[2 J t]-1 l=0 K t (δ J,l ) -K J,l t .
Recall that for any l ∈ 0, . . . , [2 J t] -1 , one has t ≥ δ J,l+1 . Thus, it results from ( 5), (105) and the Mean Value Theorem that

[2 J t]-1 l=0 K t (δ J,l ) -K J,l t ≤ 2 J [2 J t]-1 l=0 δ J,l+1 δ J,l |(t -δ J,l ) A(δ J,l )-1 2 -(t -s) A(δ J,l )-1 2 |ds ≤ 2 J [2 J t]-1 l=0 δ J,l+1 δ J,l (s -δ J,l )(t -s) A(δ J,l )-3 2 ds ≤ [2 J t]-1 l=0 δ J,l+1 δ J,l (t -s) A(δ J,l )-3 2 ds ≤ δ J,[2 J t] 0 (t -s) a-3 2 ds = 1 a -1 2 t a-1 2 -(t -δ J,[2 J t] ) a-1 2 ≤ 1 a -1 2 .
Finally, combining the last inequality with (110) and ( 109), one gets that

| X J (t) -X J (t)| ≤ Ĉ2 -J 2 1 + J          K t (δ J,[2 J t] ) -K J,[2 J t] t + [2 J t]-1 l=0 K t (δ J,l ) -K J,l t          ≤ Ĉ2 -J 2 1 + J       2 + 1 a -1 2       ,
which shows that the (108) holds.

Before concluding this section let us provide some simulations of the processes { X J (t) : t ∈ I} and { X J (t) : t ∈ I} with a random parameter {A(s) : s ∈ I} chosen so that, for all s ∈ I, one has:

A(s) = R H (s) b a := a + (b -a) R H (s) -min x∈I R H (x) max x∈I R H (x) -min x∈I R H (x) ,
where a, b are two arbitrary fixed real numbers satisfying 1/2 < a ≤ b <

E |X(t") -X(t )| 2 = 1 0 E |K t" (s) -K t (s)| 2 ds = t" t E (t" -s) 2A(s)-1 ds + t 0 E (t" -s) A(s)-1 2 -(t -s) A(s)-
E (t" -t + u) A(t -u)-1 2 -u A(t -u)-1 2 2 du = t 0 E        |t" -t | 2A(t -u)-1 1 + u t" -t A(t -u)-1 2 - u t" -t A(t -u)-1 2 2        du ≤ |t" -t | 2a-1 t 0 E        1 + u t" -t A(t -u)-1 2 - u t" -t A(t -u)-1 2 2        du ≤ |t" -t | 2a t t"-t 0 E (1 + v) A((1+v)t -vt")-1 2 -v A((1+v)t -vt")-1 2 2 dv ≤ |t" -t | 2a 1 0 E (1 + v) A((1+v)t -vt")-1 2 -v A((1+v)t -vt")-1 2 2 dv + +∞ 1
E (1 + v) A((1+v)t -vt")-1 2v A((1+v)t -vt")-1 E (1 + v) A((1+v)t -vt")-1 2v A((1+v)t -vt")- Also observe that by applying, for any fixed v ∈ [1, +∞), the Mean Value Theorem to the function x → (x + v) A((1+v)t -vt")-1 2 , and by using (3), one obtains that (1 + v) A((1+v)t -vt")-1 2v A((1+v)t -vt")-1 2 ≤ v a-3 2 , and consequently that +∞ 1

E (1 + v) A((1+v)t -vt")-1 2 -v A((1+v)t -vt")-1 2 2 dv ≤ +∞ 1 v 2a-3 dv = 1 2 -2a . ( 115
)
Finally, it results from (111), (112), (113), ( 114) and ( 115) that

E |X(t") -X(t )| 2 ≤ c 1 |t" -t | 2a ,
where c 1 > 0 is a constant not depending on t , t . Thus, one can derive from the Kolmogorov-Čentsov's continuity theorem, given in [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF], that the stochastic process {X(t) : t ∈ I} admits a modification whose sample paths are, with probability 1, Hölder continuous function of any order ζ ∈ (0, a -1/2).

Proof of Lemma 2.1. Let us fix s , s , t three arbitrary real numbers satisfying 0 ≤ s ≤ s < t ≤ 1. It follows from the triangular inequality and (5) that

K t (s ) -K t (s ) = (t -s ) A(s )-1 2 -(t -s ) A(s )-1 2 ≤ (t -s ) A(s )-1 2 -(t -s ) A(s )-1 2 + (t -s ) A(s )-1 2 -(t -s ) A(s )-1 2 . ( 116
)
First, let us apply the Mean Value Theorem to the function x → (tx) A(s )-1 2 . We obtain, for some s 0 ∈ (s , s ), that (ts ) A(s )-1 2 -(ts ) A(s )-1 2 ≤ A(s ) -1 2 (ts 0 ) A(s )-3 2 (ss ) ≤ a -1 2 (ts ) a-3 2 (ss ) . (117)

Secondly, by applying the Mean Value Theorem to the function x → (ts ) x-1 2 , we obtain, for some d ∈ min{A(s ), A(s )}, max{A(s ), A(s )} ⊆ [a, a], that (ts ) A(s )-1 2 -(ts ) A(s )-1 2 = log(ts ) (ts ) d-1 2 A(s ) -A(s )

≤ log(ts ) (ts ) a-1 2 A(s ) -A(s ) .

(118)

Finally, using the fact that sup x∈(0,1] log(x) x a-1 2 < +∞ and using (116), ( 117) and ( 118), it follows that (15) is satisfied.

Proof of Lemma 2.3. One can derive from ( 21), (26), the triangular and the Cauchy-Schwarz inequalities, and the equality

E ∆B J,l 2 = 2 -J , ( 119
)
that, for all J ∈ Z + and all l ∈ {0, . . . , 2 J -1}, one has

E X J (t) -X J (t) ≤ 2 J -1 l=0 E K J,l t -K t (δ J,l ) ∆B J,l ≤ 2 -J 2 2 J -1 E K J,l t -K t (δ J,l ) 2 1 2 . ( 120
)
Moreover, [START_REF] Samorodnitsky | Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance[END_REF]

Proposition 1 . 1 .

 11 The process {X(t) : t ∈ I} has a modification whose paths are Hölder continuous functions on I of any order ζ ∈ (0, a -1/2).

where e 2 ∈

 2 (ts, ts) ⊆ (0

Theorem 4 . 1 .Proposition 4 . 1 .

 4141 (Lévy modulus of continuity for Brownian Motion)[START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF] Theorem 9.25] Let B be the Brownian Motion in[START_REF] Benassi | Elliptic Gaussian random processes[END_REF]. There exists an event Ω ⊆ Ω of probability 1 such that, for any ω ∈ Ω, one has |B(s , ω) -B(s , ω)| : (s , s ) ∈ I 2 and |ss | ≤ δ 2δ log(1/δ) Let Ω 2 be the event of probability 1 defined as Ω 2 = Ω * * ∩ Ω, where Ω * * is the same event of probability 1 as in Theorem 2.1. Then, for any ω ∈ Ω 2 , one haslim J→+∞ sup t∈I X J (t, ω) -X(t, ω) = 0 . (102)Proof of Proposition 4.1. In view of Theorem 2.1 and (18), it is sufficient to show that, on the event Ω 2 , one has lim J→+∞ sup t∈I X J (t) -X J (t) = 0.

  that, one can derive from the inequality |a βb β | ≤ |a -b| β , where β ∈ (0, 1) and all a, b ∈ (0, +∞) are arbitrary, that 1 0

  2.1. The theorem is a straightforward consequence of Propositions 2.2 and 2.3.

	3 Study of Hölder regularity

Definition 3.1. Let f be a deterministic real-valued continuous function defined on I. The uniform Hölder exponent of f on an arbitrary non-degenerate compact interval

[ν 1 , ν 2 ] ⊆ I is denoted by β f [ν 1 , ν 2 ]

and defined as:

  Remark 3.1. We conjecture that the inequality in (41) can be replaced by an equality.Proof. Let us consider an arbitrary non-degenerate compact interval [ν 1 , ν 2 ] ⊆ I, and let us fix t , t ∈ [ν 1 , ν 2 ] such that t < t . One can derive from Theorem 2.1 and the triangular inequality that, on some event Ω * * of probability 1 included in Ω * (see Lemma 2.4) and not depending on t , t , ν 1 , ν 2 , the following inequality holds:

  1, and where R H is the Riemann-Liouville process of Hurst parameter H introduced in (2). In these simulations two significantly different values for Hurst parameter H of R H have be taken, namely H = 0.58 and H = 0.9. These simulations tend to confirm a phenomenon which somehow has already appeared in the statement of Theorem 3.1, namely that there are close connections between path behavior of the MPRE {X(t) : t ∈ I} and the random values taken by its parameter {A(s) : s ∈ I}: the paths of {X(t) : t ∈ I} are rather rough (resp. rather smooth) in the neighbourhood of the points where the values of {A(s) : s ∈ I} are close to 1/2 (resp. close to 1).Proof of Proposition 1.1. By combining (4) with the isometry property of Itô integral, for any real numbers t , t" satisfying 0 ≤ t ≤ t" ≤ 1, one has

	A Proofs of some auxiliary results
	1										0.2
	0.9 0.95										0.1										X X
	0.85										0
	0.8
	0.75										-0.1
	0.7
	0.65										-0.2
	0.6										-0.3
	0.55
	0.5										-0.4
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
				(a) A(•) with H = 0.58			(b) Superposition of realizations of X J and X J
	0.2										0.2
	0.1										0.1
	0										0
	-0.1										-0.1
	-0.2										-0.2
	-0.3										-0.3
	-0.4										-0.4
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
					(c) X J										(d) X J
	1										0.2
	0.9 0.95										0.1										X X
	0.85										0
	0.8
	0.75										-0.1
	0.7
	0.65										-0.2
	0.6										-0.3
	0.55
	0.5										-0.4
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
				(e) A(•) with H = 0.9			(f) Superposition of realizations of X J and X J
	0.2										0.2
	0.1										0.1
	0										0
	-0.1										-0.1
	-0.2										-0.2
	-0.3										-0.3
	-0.4										-0.4
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
					(a) X J										(b) X J

  Let us provide a suitable upper bound for each term in the previous sum. The process {A(s) : s ∈ I} being with values in the deterministic interval [a, a] ⊂ (1/2, 1), it follows that

										1 2	2	ds .	(111)
		t	t"	E (t" -s) 2A(s)-1 ds ≤	t	t"	(t" -s) 2a-1 ds =	1 2a	(t" -t ) 2a .	(112)
	Moreover, using the changes of variables u = t -s and v = u t"-t and standard computations, one gets that
	t	E (t" -s) A(s)-1 2 -(t -s) A(s)-1 2	2	ds =			t
	0							0

  Also, observe that in the case where l = [2 J t], using the triangular inequality and (28) one has Let us now study the last case where l < [2 J t]. One has [δ J,l , δ J,l+1 ) ⊆ [0, t). Thus, one can derive from Lemma 2.1 that

		implies that
		K	J,l
			δ J,l+1
					K t (s) -K t (δ J,l ) ds = 0 ;
			δ J,l
	therefore (121) implies that
					K	J,l t -K t (δ J,l ) = 0 .	(122)
			δ J,[2 J t]+1 δ J,[2 J t]	K t (s) -K t (δ J,[2 J t] ) ds ≤ 2 -J+1 ;	(123)
	therefore (121) entails that
				K	J,[2 J t] t	-K t (δ J,[2 J t] ) ≤ 2 .	(124)
	δ J,l+1 δ J,l	K t (s) -K t (δ J,l ) ds ≤ c 0	     	δ J,l+1 δ J,l	A(s) -A(δ J,l ) ds +	δ J,l+1 δ J,l	(t -s) a-3 2 |s -δ J,l |ds	     
			≤ c 0	     	δ J,l+1 δ J,l		δ J,l+1 δ J,l	(t -s) a-3 2 ds	      .	(125)

t -K t (δ J,l ) ≤ 2 J δ J,l+1 δ J,l K t (s) -K t (δ J,l ) ds . (

121

)

Observe that in the case where l ≥ [2 J t] + 1, [•] being the integer part function, one has that δ J,l > t and consequently that A(s) -A(δ J,l ) ds + 2 -J

(tt ) A ν 0 ,ν 2 (99)

Acknowledgements

This work has been partially supported by the Labex CEMPI (ANR-11-LABX-0007-01), the F.R.S.-FNRS and the GDR 3475 (Analyse Multifractale).