Alexis Fauth
email: alexis.fauth@invivoo.com

Ciprian A Tudor
email: tudor@math.univ-lille1.fr

Multifractal random walks with fractional Brownian motion via Malliavin calculus

Keywords: 2010 AMS Classification Numbers: 60C30, 60H07, 60H05 fractional Brownian motion, Malliavin calculus, multifractal random walk, scaling, infinitely divisible cascades, leverage effect, high frequency financial data

We introduce a Multifractal Random Walk (MRW) defined as a stochastic integral of an infinitely divisible noise with respect to a dependent fractional Brownian motion. Using the techniques of the Malliavin calculus, we study the existence of this object and its properties. We then propose a continuous time model in finance that captures the main properties observed in the empirical data, including the leverage effect. We illustrate our result by numerical simulations.

Introduction

Starting with the seminal work of Mandelbrot about cotton price [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF], several studies of financial stock prices times series, have allowed to exhibit some particularities of their 1 fluctuations. Without making a comprehensive list, we can mention the appearance in empirical data of the following properties: non Gaussian distributions due to now well known fat tails of financial returns, the so-called volatility clustering that means that the volatility fluctuations are of intermittent and correlated nature, scaling invariance, long run correlation in volatility, leverage effect and so on (see e.g [START_REF] Gopikrishnan | Scaling of the Distribution of Fluctuations of Financial Market Indices[END_REF], [START_REF] Chakraborti | Econophysics review: I. Empirical Facts[END_REF], [START_REF] Voit | The Statistical Mechanics of Financial Markets[END_REF] for an extensive review). Thus, constructing theoretical models for financial returns that include all the properties listed before appears as a very interesting challenge. Many scientific works, in economics or mathematics, proposed various models for asset returns. The ARCH model introduced by Engle [START_REF] Engle | Autoregressive conditional heteroskedasticity with estimates of the variance of the U.K. inflation[END_REF] offers an interesting base of work, and after the seminar work by Engle a vast literature on ARCH and related models has been developed. One of the first extensions of the ARCH model, called the GARCH model, has been introduced by Bollerslev in [9] and it also has been the object of various generalizations. While the GARCH models are capable of capturing volatility clustering, there are some drawbacks of the model. For example, the GARCH models are unable to represent volatility asymmetry. Due to the presence of the squared observed data in the conditional variance equation, the positive and negative values of the lagged innovations have the same effect on the conditional variance. In the finance literature, it has been noticed that volatility often is affected by negative and positive shocks in different ways. Another inconvenient of the GARCH model is the fact that, to ensure positiveness of the conditional variance, non-negative constraints on the coefficients in the variance equation must be imposed. To take into account the asymmetric effects on conditional second moments and to avoid the non-negativity constraints on the coefficients in the variance equation, in [START_REF] Nelson | Conditional heteroskedasticity in asset returns: A new approach[END_REF] the so-called EGARCH model has been proposed by D. Nelson. On the other hand, many recent empirical studies, based on huge data sets available nowadays, put in light new aspects of the financial returns. For example, they suggested that the fluctuations of the asset process displays multifractal properties (see e.g. [START_REF] Fisher | Multifractality of Deutschemark / US Dollar Exchange Rates[END_REF] or [START_REF] Ghashghaie | Turbulent Cascades in Foreign Exchange Markets[END_REF]). Taking into account these multifractal character, several authors proposed models based on the on "cascade" random processes and "multifractal random walk (MRW)". We refer, among others, to [START_REF] Calvet | Multifractal Volatility: Theory, Forecasting and Princing[END_REF], [START_REF] Bacry | Multifractal Random Walks[END_REF] or [START_REF] Bacry | Continuous cascade models for asset returns[END_REF]. Usually, the noise in these models are defined by

Z(t) = Y (X([0, t]), t ∈ R (1)
where X is a multifractal random measure and Y is a self-similar process with stationary increments, independent of X. Another construction defines the noise via a stochastic integral as (see e.g. [START_REF] Barral | Multifractal products of cylindrical pulses[END_REF], see [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF])

Z(t) = t 0 Q(u)dY (u) ( 2
)
where Q is a suitable fractal noise and Y is a self-similar process with stationary increments, independent of Q. A natural choice for the process Y , both in (1) and [START_REF] Bacry | Continuous-time skewed multifractal processes as a model for financial returns[END_REF], is the fractional Brownian motion (fBm). While these multifractal models (1), ( 2) are able to capture several properties observed on the empirical data (scaling, volatility clustering or long-range dependance), they reproduce poorly the leverage effect. The leverage effect is understood as the correlation between the log-return at a fixed time t and his volatility (that may be defined as the squared or absolute log-return) varying around t, for example t -100∆t ≤ t ≤ t + 100∆t. Empirical studies (see e.g. [8] or [START_REF] Bouchaud | The leverage Effect in Financial Markets: Retarded Volatility and Market Panic[END_REF]) have shown that this quantity is close to 0 for past volatility and follow an exponential law for future volatility. This stylized fact is interpreted as a panic effect, the two quantities are negatively correlated just after t and go back quickly to 0 as t increases.

We introduce here a generalization of the model ( 2), able to take into account the leverage effect and allows a more flexibility for long-range dependence in log-return. That is, in the model (2) we will consider the processes Q to be multifractal process called "infinitely divisible cascading noise" and the process Y to be a fractional Brownian motion with Hurst parameter H > 1 2 . The same situation has been treated in [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF], [START_REF] Ludena | L p variations for multifractal fractional random walks[END_REF], but in addition, we construct a model in which the processes Q and B H are not independent anymore. An alternative construction has been proposed in [START_REF] Bacry | Continuous-time skewed multifractal processes as a model for financial returns[END_REF] but in this reference the integral in ( 2) is not a true stochastic integral with respect to the fBm. We will define the integral (2) using the techniques of the Malliavin calculus and we will study the properties of the fractional Multifractal Random Walk.

Our paper is organized as follows. Section 2 contains some preliminaries on multifractal process and the basic elements of the Malliavin calculus that we will need in our paper. In Section 3 we analyze the stochastic integrals that are used to define the MRW while in Section 4 we study the existence of the fractional MRW as a limit of a family of stochastic integrals. The properties of the fractional MRW (scaling, moments etc) are discussed in Section 5. The last section (Section 6) contains a numerical analysis of the data for several financial indices and we compare the simulation of our theoretical model with the real data.

Preliminaries

We present here he basic facts related to the infinitely divisible cascading noises and we introduce the basic tools of the Malliavin calculus.

Infinitely divisible cascading noise

Let M denote an infinitely divisible, independently scattered random measure on the set

R × R + with generating infinitely divisible distribution G satisfying R e qx G(dx) = e -ρ(q)
for some function ρ and for every q ∈ R. We assume that M has control measure m on R × R + meaning that for every Borel set A ⊂ R × R + it holds Ee qM (A) = e -ρ(q)m(A) for every q ∈ R.

The fact that M is independently scattered means that the random variables

M (A 1 ), M (A 2 ), ..., M (A n )
are independent whenever the Borel sets A 1 , ..., A n ∈ R × R + are disjoint. We define the Infinitely Divisible Cascading noise (IRC) by

Q r (t) = e M (Cr(t))
Ee M (Cr(t))

(3)

for every r > 0 and t ∈ R.

Here C r (t) is the cone in R × R + defined by C r (t) = {(t , r ), r ≤ r ≤ 1, t - r 2 ≤ t ≤ t + r 2 }. (4)
We will use the following facts throughout our paper. We refer to [START_REF] Chainais | On Non-Scale Invariant infinitely Divisible Cascades[END_REF] or [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF] for the their proofs. First, let us note the scaling property of the moments of the IRC EQ r (t) q = e -ϕ(q)mr(0) and the expression of its covariance: for every r > 0 and t, s ∈ R EQ r (t)Q r (s) = e -ϕ(2)mr(|t-s|)

(

where we denoted by for u ≥ 0, r > 0.

m r (u) = m (C r (0) ∩ C r (u))
and by ϕ(q) = ρ(q) -qρ(1).

The scaling of the moment of Q can be extended to the following scaling property in distribution : for t ∈ (0, 1)

(Q rt (tu)) u∈R = (d) e Ωt (Q r (u)) u∈R (7)
where " = (d) " means equivalence of finite dimensional distributions. Here Ω t denotes a random variable independent by Q, which satisfies, if the measure m is given by [START_REF] Chakraborti | Econophysics review: I. Empirical Facts[END_REF],

Ee qΩt = t qϕ(q) . ( 8
)
Remark 1 As noticed in [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF], we have ϕ(2) < 0.

In [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF] (see also [START_REF] Ludena | L p variations for multifractal fractional random walks[END_REF] ) the Multifractal random walk (MRW) based on fractional Brownian motion is defined as limit when r → 0 (in some sense) of the family of stochastic integrals Z H r (t) r>0 defined by

Z H r (t) = t 0 Q r (u)dB H (u), t ∈ [0, T ] (9)
where (B H t ) t∈[0,T ] is a fractional Brownian motion with Hurst parameter H ∈ (0, 1). The fractional Brownian motion (B H t ) t∈[0,T ] with Hurst parameter H ∈ (0, 1) is a centered Gaussian process starting from zero with covariance function

R H (t, s) := 1 2 t 2H + s 2H -|t -s| 2H , s, t ∈ [0, T ].
In [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF] it is assumed that M and B H are independent. Therefore the stochastic integral with respect to B H in (9) behaves mainly as a Wiener integral since, because of the independence, the integrand Q r (u) can be viewed as deterministic function for the integrator B H . Another important fact in the development of this theory is that the IRC Q is a martingale with respect to the argument r. Let us recall the following result (see [START_REF] Chainais | On Non-Scale Invariant infinitely Divisible Cascades[END_REF]):

Lemma 1 For every u > 0 the stochastic process (Q r (u)) r>0 is a martingale with respect to its own filtration. As a consequence, for every u, v, r, r > 0 with r < r it holds

EQ r (u)Q r (v) = EQ r (u)Q r (v). ( 10
)
The property (10) plays an important role in the construction of the MRW process in [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF] or [START_REF] Chainais | On Non-Scale Invariant infinitely Divisible Cascades[END_REF].

Malliavin calculus

Let (W t ) t∈T be a classical Wiener process on a standard Wiener space (Ω, F, P). By W (ϕ) we denote the Wiener integral of the function ϕ ∈ L 2 (T ) with respect to the Brownian motion W . We denote by D the Malliavin derivative operator that acts on smooth functionals of the form F = g(W (ϕ 1 ), . . . , W (ϕ n )) (here g is a smooth function with compact support and

ϕ i ∈ L 2 (T ) for i = 1, .., n) DF = n i=1 ∂g ∂x i (W (ϕ 1 ), . . . , W (ϕ n ))ϕ i .
The operator D can be extended to the closure D p,2 of smooth functionals with respect to the norm

F 2 p,2 = EF 2 + p i=1 E D i F 2 L 2 T i
where the i th Malliavin derivative D (i) is defined iteratively. The adjoint of D is denoted by δ and is called the divergence (or Skorohod) integral. Its domain (Dom(δ) ) coincides with the class of stochastic processes u ∈ L 2 (Ω × T ) such that

|E DF, u | ≤ c F 2
for all F ∈ D 1,2 and δ(u) is the element of L 2 (Ω) characterized by the duality relationship

E(F δ(u)) = E DF, u .
For adapted integrands, the divergence integral coincides to the classical Itô integral. A subset of Dom(δ) is the space L 1,p of the stochastic processes such that u t is Malliavin differentiable for every t and

u p 1,p := u p L p (T ×Ω) + Du p L p (T 2 ×Ω) < ∞.
We will need Meyer's inequality that allows to estimate the L p moment of the Skorohod integral

E |δ(u)| p ≤ u p 1,p . (11)
For the L 2 moment of the Skorohod integral we have the explicit formula

Eδ(u) 2 = T u 2 s ds + T T D r u s D s u r drds (12) if u ∈ L 1,2 .
We also recall that the Malliavin derivative satisfies the chain rule

Df (F ) = f (F )DF (13) if f is a differentiable function and F ∈ D 1,2 .

The construction of the fractional Multifractal Random Walk with dependent noise

Our purpose is to give a meaning to the stochastic integral (9) in the situation when the IRC Q and the fBm B H are not independent. We will use techniques related to the Malliavin calculus. In order to apply these type of techniques we will restrict ourselves to the case when the measure M introduced in Section 2 is Gaussian.

The Gaussian isonormal noise

We introduce a MRW without independence between the measure M (denote by W in our settings) and the integrator B H . We will restrict to the case where M is a Gaussian measure. More precisely, we will consider (W (h), h ∈ H) an isonormal process, that is, a centered Gaussian family with

EW (h)W (g) = h, g H
for every g, h ∈ H. The Hilbert space H will be

H = L 2 (R × R + , B(R × R + ), m))
where m is the control measure. In this work we will consider

m(dt, dr) = dt cdr r 2 if 0 < r ≤ 1 ( 14
)
and m vanishes if r ≥ 1. Here c is a strictly positive constant. This is called in [START_REF] Bacry | Multifractal stationary random measures and multifractal walks with log-infinitely divisible scaling laws[END_REF] (see also [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF]) the exact invariant scaling case.

The following properties of the noise W are immediate: denote

W (A) = W (1 A ) for A ∈ B(R × R + ). • For every A ∈ B(R × R + ) it holds that W (A) ∼ N (0, m(A)). •We have EW (A)W (B) = 0 if the Borel sets A, B ⊂ R × R + are disjoint.
This implies that the random variables W (A) and W (B) are independent, so the random measure W is independently scattered.

• For every q ∈ R and

A ∈ B(R × R + ) we have Ee qW (A) = e -1 2 q 2 m(A)
which means that

ρ(q) = - 1 2 q 2 and ϕ(q) = - 1 2 q 2 + q 2 = - 1 2 q(q -1).
Let us use the following notation:

m 1 (dt) = dt and m 2 (dr) = c dr r 2 .
Above m 1 is the Lebesque measure on R and m 2 is a measure on R + . Clearly m = m 1 ⊗ m 2 , the product measure. For every t ≥ 0 and

A 2 ∈ B(R + ) such that m 2 (A 2 ) = 1 (take for example A 2 = (c, ∞)) we set W (1) (t) := W (1 [0,t]×A 2 ). ( 15
)
The following result is immediate.

Proposition 1 If A 2 ∈ B(R + ) is such that m 2 (A 2 ) = 1 then the process (W (1) (t)) t≥0
given by ( 15) is a standard Brownian motion on the same probability space as W .

Proof: It is clear that W (1) is a Gaussian process. Let us compute its covariance. For every s, t ≥ 0 it holds that

EW (1) (t)W (1) (s) = EW (1 [0,t]×A 2 )W (1 [0,s]×A 2 ) = 1 [0,t]×A 2 , 1 [0,s]×A 2 H = 1 [0,t] , 1 [0,s] L 2 (R) m 2 (A 2 ) = t ∧ s
and this implies that W (1) is a Wiener process with respect to its own filtration.

The approximating Multifractal Random Walk with dependent fractional Brownian motion

We introduce the Multifractal Random Walk with based on the fractional Brownian motion by the formula

Z H r (t) = t 0 Q r (u)dB H (u) (16)
where for every r > 0, u ≥ 0 the integrands Q r (u) is defined by (3) with W instead of M and the fractional Brownian motion B H is given by

B H (t) = t 0 K H (t, s)dW (1) (s) (17)
where K H is the usual kernel of the fractional Brownian motion and W (1) is the Brownian motion defined by [START_REF] Engle | Autoregressive conditional heteroskedasticity with estimates of the variance of the U.K. inflation[END_REF]. By Proposition 1, it is clear that B H is a fractional Brownian motion. Recall that, when H > 1 2 the kernel K H (t, s) has the expression

K H (t, s) = c H s 1/2-H t s (u -s) H-3/2 u H-1/2 du
(see [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]) where t > s and c H =

H(2H-1) β(2-2H,H-1/2) 1/2 and β(•, •) is the Beta function. For t > s, the kernel's derivative is ∂K H ∂t (t, s) = c H s t 1/2-H (t -s) H-3/2 .
In the sequel we will simply denote K H := K. The stochastic integral in ( 16) is a divergence (Skorohod) integral with respect to B H as defined in e.g. [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]. Actually, when H > 1 2 we can write

Z H r (t) = t 0 Q r (s)dB H (s) = t 0 dW (1) (s) t s da∂ 1 K(a, s)Q r (a) .
The last equality is due to the definition of the stochastic integral with respect to B H (see [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF], Chapter 5 for example). We can also express Z r (t) as a Skorohod integral with respect to the measure W by the formula

Z H r (t) = t 0 A 2 dW (s, r ) t s da∂ 1 K(a, s)Q r (a) ( 18
)
where A 2 is the Borel set satisfying m 2 (A 2 ) = 1 and that appears in the definition of W (1) .

Remark 2 Clearly the fBm B H and the noise W are dependent. Actually, it can be seen that B H and W are correlated in general. Indeed, for every Borel set A ⊂ R × R + and for every t ≥ 0

EB H (t)W (A) = E t 0 K(t, s)dW (1) (t)W (A) = E t 0 1 0 K(t, s)dW (s, r)W (A) = E ((0,t)×(1,∞))∩A K(t, s)ds dr r 2
and this is in general not zero.

Let us further discuss the dependence between the integrator B H and the integrand Q in (9). We need to distinguish two situations. It depends on the relations of the set A 2 and the unit interval (0, 1).

The disjoint case:

We have A 2 ∩ (0, 1) = ∅ (this happens when A 2 = (c, ∞) with c > 1 for example).

On the other hand, B H s is independent by W (C r (t)) for every s, t, r. Indeed, since C r (t) ⊂ R × (0, 1) (see ( 4)) we have

EB H s W (C r (t)) = 0
and since (B H (s), W (C r (t))) is a Gaussian vector, we obtain the independence.

The non-disjoint case: A 2 ∩ (0, 1) = ∅ (this happens when A 2 = (c, ∞) with c < 1). In this case EB H (s)W (C r (t)) = 0 in general and so B H (s) and W (C r (t)) are dependent.

We will refer throughout this work to the two situations above as the the disjoint case and the non-disjoint case. Basically, the results in the disjoint case can be obtained by following the arguments in [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF] while in the non-disjoint case the context is different because of the appearance of the Malliavin derivatives in the expression of square mean of (9).

Proposition 2 Suppose H > 1 2 . Then for every r > 0 the stochastic Skorohod integral in ( 16) is well-defined.

Proof: We will use the representation of (9) of Z H r (t) as a Skorohod integral with respect to the measure W . Next we use the bound [START_REF] Bouchaud | The leverage Effect in Financial Markets: Retarded Volatility and Market Panic[END_REF] with p = 2. Let us apply it to the process

(s, r ) → 1 [0,t] (s)1 A 2 (r) t s da∂ 1 K(a, s)Q r (a)
(which is a two-parameter process) and to H = L 2 (R × R + , m). The variables t, r and the set A 2 with m 2 (A 2 ) = 1 are fixed. Below D is the Malliavin derivative with respect to the isonormal process W (see Section 2). We will get, by [START_REF] Bouchaud | The leverage Effect in Financial Markets: Retarded Volatility and Market Panic[END_REF],

E(Z H r (t)) 2 ≤ E t 0 ds A 2 cdr (r ) 2 t s da∂ 1 K(a, s)Q r (a) 2 +E t 0 ds A 2 cdr (r ) 2 t 0 dα A 2 cdβ β 2 D α,β t s da∂ 1 K(a, s)Q r (a) 2 = E t 0 ds A 2 cdr (r ) 2 t s da∂ 1 K(a, s)Q r (a) 2 +E t 0 ds A 2 cdr (r ) 2 t 0 dα A 2 cdβ β 2 t s da∂ 1 K(a, s)D α,β Q r (a) 2 = E t 0 ds A 2 cdr (r ) 2 t s da∂ 1 K(a, s)Q r (a) 2 +E t 0 ds A 2 cdr (r ) 2 t 0 dα A 2 cdβ β 2 t s da∂ 1 K(a, s)Q r (a)1 Cr(a) (α, β) 2 := T 1 + T 2 .
since, by the chain rule of the Malliavin operator (13)

D α,β Q r (a) = Q r (a)1 Cr(a) (α, β). (19)
We need again to consider two cases.

The disjoint case:

We have A 2 ∩(0, 1) = ∅ (this happens when A 2 = (c, ∞), c > 1 for example). In this case the term denoted by T 2 vanishes because

1 Cr(a) (α, β)1 A 2 (β) = 0.
We need to show that

T 1 = m 2 (A 2 )E t 0 ds t s da∂ 1 K(a, s)Q r (a)
2 is finite under condition [START_REF] Ludena | L p variations for multifractal fractional random walks[END_REF] and this is exactly the computation in [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF], proof of Proposition 2.1.

The non-disjoint case: Suppose A 2 ∩ (0, 1) = ∅ (this happens when A 2 = (c, ∞) with c < 1). In this case we need also to show that T 2 < ∞ By bounding the indicator function 1 Cr(a) (α, β) by 1 (note that ∂ 1 K(a, s) ≥ 0 for every a, s)

E t s da∂ 1 K(a, s)Q r (a)1 Cr(a) (α, β) 2 ≤ E t s da∂ 1 K(a, s)Q r (a) 2
and by computing the integrals dr and dβ

T 2 ≤ m 2 (A 2 ) 2 E t 0 ds t 0 dα t s da∂ 1 K(a, s)Q r (a) 2 = m 2 (A 2 ) 2 E t 0 ds t s da∂ 1 K(a, s)Q r (a) 2 = tT 1
and this is finite under [START_REF] Ludena | L p variations for multifractal fractional random walks[END_REF].

The Multifractal Random Walk

The purpose of this section is to study the limit as r → 0 of the family of stochastic processes Z H r (t) with fixed t > 0. This limit will be called the the Multifractal Random Walk.

We will assume throughout this paragraph that Z H r (t) is defined by [START_REF] Fisher | Multifractality of Deutschemark / US Dollar Exchange Rates[END_REF] with W (1) given by [START_REF] Engle | Autoregressive conditional heteroskedasticity with estimates of the variance of the U.K. inflation[END_REF]. Moreover we will suppose

m 2 (A 2 ) = 1
and A 2 ∩ (0, 1) = ∅. The disjoint case A 2 ∩ (0, 1) = ∅ follows from [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF].

We will consider the following assumption:

cϕ(2) + 2H > 1. ( 20
)
Remark 3 Since ϕ(2) = -1, the condition [START_REF] Ludena | L p variations for multifractal fractional random walks[END_REF] means that c < 2H -1. Since H > 1 2 we can choose c > 0 in order to have [START_REF] Ludena | L p variations for multifractal fractional random walks[END_REF]. By assuming [START_REF] Ludena | L p variations for multifractal fractional random walks[END_REF], we are in the case (A) in [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF].

We have the following limit theorem.

Theorem 1 Assume [START_REF] Ludena | L p variations for multifractal fractional random walks[END_REF]. For every t > 0 the sequence of stochastic process Z H r (t) r>0 defined by [START_REF] Fisher | Multifractality of Deutschemark / US Dollar Exchange Rates[END_REF] converges in L 2 (Ω) to a random variable Z H (t).

Proof: Let us fix r, r ∈ (0, 1) with r < r . We will first compute the L 2 (Ω) norm of the increment Z H r (t) -Z H r (t) where t > 0 is fixed. We can write, with W (1) given by ( 15)

E Z H r (t) -Z H r (t) 2 = E t 0 (Q r (u) -Q r (u)) dB H u 2 = E t 0 t u (Q r (a) -Q r (a)) ∂ 1 K(a, u) dW (1) u 2 = E t 0 A 2 t u da (Q r (a) -Q r (a)) ∂ 1 K(a, u) dW (u, x) 2 ≤ E t 0 du A 2 cdx x 2 t u da (Q r (a) -Q r (a)) ∂ 1 K(a, u) 2 +E t 0 du A 2 cdx x 2 t 0 dv A 2 cdy y 2 D v,y t u da (Q r (a) -Q r (a)) ∂ 1 K(a, u) 2
where we used the bound [START_REF] Bouchaud | The leverage Effect in Financial Markets: Retarded Volatility and Market Panic[END_REF] for the L 2 norm of the divergence operator. Using the differentiation rule [START_REF] Lillo | The Long Memory of the Efficient Market[END_REF], we get

E Z H r (t) -Z H r (t) 2 ≤ m 2 (A 2 ) t 0 du t u da t u db∂ 1 K(a, u)∂ 1 K(b, u) ×E (Q r (a) -Q r (a)) (Q r (b) -Q r (b)) +m 2 (A 2 ) t 0 du t 0 dv A 2 cdy y 2 t u da∂ 1 K(a, u) Q r (a)1 Cr(a) (v, y) -Q r (a)1 C r (a) (v, y) 2 := A(r, r ) + B(r, r ).
Let us first treat the term denoted by A(r, r ). Using property (10),

A(r, r ) = m 2 (A 2 ) t 0 du t u da t u db∂ 1 K(a, u)∂ 1 K(b, u) ×E [Q r (a)Q r (b) -2Q r (a)Q r (b) + Q r (a)Q r (b)] = m 2 (A 2 ) t 0 du t u da t u db∂ 1 K(a, u)∂ 1 K(b, u)E [Q r (a)Q r (b) -Q r (a)Q r (b)] = E t 0 du t u da∂ 1 K(a, u)Q r (a) 2 du -E t 0 du t u da∂ 1 K(a, u)Q r (a) 2 du = E t 0 Q r (u)d BH u 2 -E t 0 Q r (u)d BH u 2
where BH denotes a fractional Brownian motion independent by W . The convergence of this term is similar to the study in [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF] we are in the case (A) in their paper (see Remark 3).

The summand B(r, r ) will be handled as follows. First, note that

B(r, r ) = m 2 (A 2 ) t 0 du t 0 dv A 2 cdy y 2 t u da t u db∂ 1 K(a, u)∂ 1 K(b, u) ×E Q r (a)1 Cr(a) (v, y) -Q r (a)1 C r (a) (v, y) Q r (b)1 Cr(b) (v, y) -Q r (b)1 C r (b) (v, y) = m 2 (A 2 ) t 0 du t 0 dv A 2 cdy y 2 t u da t u db∂ 1 K(a, u)∂ 1 K(b, u) ×E Q r (a)Q r (b)1 Cr(a) (v, y)1 Cr(b) (v, y) -Q r (a)Q r (b)1 Cr(a) (v, y)1 C r (b) (v, y) -Q r (a)Q r (b)1 C r (a) (v, y)1 Cr(b) (v, y) + Q r (a)Q r (b)1 C r (a) (v, y)1 C r (b) (v, y)
where we used again (10). By decomposing

C r (a) = C r (a) ∪ (C r (a) \ C r (a))
we will have

B(r, r ) = m 2 (A 2 ) t 0 du t 0 dv A 2 cdy y 2 t u da t u db∂ 1 K(a, u)∂ 1 K(b, u) ×E [Q r (a)Q r (b) -Q r (a)Q r (b)] 1 C r (a) (v, y)1 C r (b) (v, y) +m 2 (A 2 ) t 0 du t 0 dv A 2 cdy y 2 t u da t u db∂ 1 K(a, u)∂ 1 K(b, u) ×EQ r (a)Q r (b)1 Cr(a)\C r (a) (v, y)1 Cr(b)\C r (b) (v, y) := B 1 (r, r ) + B 2 (r, r ).
Obviously, by bounding the indicator functions by 1,

B 1 (r, r ) ≤ t 0 du t 0 dv A 2 cdy y 2 t u da t u db∂ 1 K(a, u)∂ 1 K(b, u) ×E [Q r (a)Q r (b) -Q r (a)Q r (b)]
and therefore it converges to zero as r, r → 0 by using exactly the same argument as in the case of the term A(r, r ). Concerning B 2 (r, r ), since for 0 < r < r < 1 and

y ∈ A 2 1 Cr(a)\C r (a) (v, y) ≤ 1 (r,r )×A 2 (v, y)
we can bound it in the following way

B 2 (r, r ) ≤ m 2 (A 2 ) t 0 du t 0 dv A 2 cdy y 2 t u da t u db∂ 1 K(a, u)∂ 1 K(b, u) ×EQ r (a)Q r (b)1 (r,r )×A 2 (v, y) = m 2 (A 2 ) 2 (r -r)E t 0 du t u da t u db∂ 1 K(a, u)∂ 1 K(b, u)EQ r (a)Q r (b)
and by interchanging the order of integration and using

a∧b 0 du∂ 1 K(a, u)∂ 1 K(b, u) = c H |a -b| 2H-2 . ( 21
)
and since EQ r (a)Q r (b) = e -ϕ( 2)mr(|a-b|) ≤ e cϕ(2) log |a-b| = |a -b| cϕ (2) we get

B 2 (r, r ) ≤ m 2 (A 2 ) 2 (r -r) t 0 t 0 dadb|a -b| cϕ(2)+2H-2
we clearly obtain than B 2 (r, r ) goes to zero as r, r → 0 under condition [START_REF] Ludena | L p variations for multifractal fractional random walks[END_REF].

Definition 1 The process (Z H ) t>0 from Theorem 1 will be called as fractional Multifractal Random Walk.

Properties of the MRW

We will discuss here some immediate properties of the fractional MRW from Definition 1.

Basically, as we mentioned before, in the disjoint case the fractional MRW Z H has the same properties as in the situation when Q and B H are independent: self-similarity, stationarity of increments and long-range dependence (see [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF]). But in the non-disjoint case, we will show that even the L 2 moment of the fractional MRW does not scale exactly. We provide an exact calculation in order to show this phenomenon. On the other hand, we can control the L p norm of the increment of the process and we find some kind of asymptotic scaling even in the non-disjoint case.

Scaling of the second moment

We first analyze the second moment of the increments of Z H .

The disjoint case: Following [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF], we have

E Z H (t) 2 = t 0 ds t s t s dadbE(Q 0 (a)Q 0 (b))∂ 1 K(a, u)∂ 1 K(b, u) = t 0 ds t s t s dadbe -ρ 2 m 0 (|a-b|) ∂ 1 K(a, u)∂ 1 K(b, u) = c H t 0 t 0 db|a -b| 2H-2 E(Q 0 (a)Q 0 (b))
where the meaning of the quantity EQ 0 (a)Q 0 (b) is given by ( 5) with r = 0 and where we used the identity [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF]. Then, for every h > 0

E Z H (ht) 2 = c H ht 0 ht 0 db|a -b| 2H-2 E(Q 0 (a)Q 0 (b)) = h 2H c H t 0 t 0 db|a -b| 2H-2 E(Q 0 (ha)Q 0 (hb)) = h 2H Ee 2Ω h c H t 0 t 0 db|a -b| 2H-2 E(Q 0 (a)Q 0 (b)) = h 2H Ee 2Ω h E Z H (t) 2 = h 2H+cϕ(2) E Z H (t) 2 .
where we used the scaling property [START_REF] Barral | Multifractal products of cylindrical pulses[END_REF]. Actually, it is not difficult to see that for every

p > 1 E Z H (ht) p = h 2Hq Ee pΩt = h 2Hp+qϕ(p) .
Moreover, we have the self-similarity (Z H (at)) t∈[0,1] = (d) a H+Ωa (Z H (t)) t∈[0,1] , the stationarity of the increments and the long-range dependence in the sense that EX k X 0 ∼ τ 2 Hk 2H where X k = Z H ((k + 1)τ ) -Z H (kτ ) with k ≥ 0 integer and τ > 0.

The non-disjoint case:

The situation is different in the non-disjoint case and we will see that even the second moment of the fractional MRW does not scale. We can compute exactly the L 2 norm of Z H (t).

E Z H (t) 2 = t 0 ds t s daQ 0 (a)∂ 1 K(a, s)da 2 + t 0 ds A 2 cdr r 2 t 0 dα A 2 cdβ β 2 × D α,β t s dbQ 0 (b)∂ 1 K(b, s)db D s,r t α daQ 0 (a)∂ 1 K(a, α)da = t 0 ds t s daQ 0 (a)∂ 1 K(a, s)da 2 + t 0 ds A 2 cdr r 2 t 0 dα A 2 cdβ β 2 × t α daQ 0 (a)∂ 1 K(a, α)1 C 0 (a) (s, r) t s dbQ 0 (b)1 C 0 (b) (α, β)∂ 1 K(b, s)db = I(t) + J(t)
Consider h > 0. Then, applying the above formula to t = ht and making several changes of variables, we will get

E Z H (ht) 2 = h 2H+cϕ(2) I(t) +h 2H+1+cϕ(2) t 0 ds A 2 cdr r 2 t 0 dα A 2 cdβ β 2 × t α daQ 0 (a)∂ 1 K(a, α)1 C 0 (ha) (hs, r) t s dbQ 0 (b)1 C 0 (hb) (hα, β)∂ 1 K(b, s)db .
The integral with respect to dβ and dr can be computed explicitly. For example, when

A 2 = (c, ∞), c < 1, we find A 2 cdβ β 2 1 C 0 (hb) (hα, β) = c 1 c ∨ 2h|b -α| -1
and it is clear that the term J(t) does not scale exactly as I(t).

The control of the increments

Let us estimate the L p norm of the increment Z H (t) -Z H (s) of the fractional Multifractal Random Walk introduced in Definition 1. Fix s, t ∈ [0, 1] with t > s. We will not insist on the disjoint case because the calculations in [START_REF] Abry | Multifractal random walks as fractional Wiener integrals[END_REF] still hold, so we will have

E Z H (t) -Z H (s) p ∼ C p |t -s| 2Hp+cϕ(p)
with ϕ given by [START_REF] Bardet | Generators of long-range dependent processes: A survey[END_REF].

Let us consider the non-disjoint case. In this case Z H (t), which can formally be written as t 0 Q 0 (y)dB H (u) is an anticipating (Skorohod integral). We need to use Meyer's inequalities [START_REF] Bouchaud | The leverage Effect in Financial Markets: Retarded Volatility and Market Panic[END_REF] in order to estimate its L p norm.

Actually

Z H (t) -Z H (s) = 1 0 dW (1) (u)F s,t (u)
where we denoted by

F s,t (u) = 1 (0,t) (u) t u ∂ 1 K(a, u)Q 0 (a)da -1 (0,s) (u) s u ∂ 1 K(a, u)Q 0 (a)da.
By Meyer's inequality ( 11)

E Z H (t) -Z H (s) p ≤ E 1 0 du A 2 cdr r 2 |F s,t (u)| p + E 1 0 du A 2 cdr r 2 1 0 dα A 2 cdβ β 2 |D α,β F s,t (u)| 2 := A(t, s) + B(t, s).
The terms A is exactly the L p norm in the disjoint case, so

A(t, s) ≤ c p |t -s| 2Hp+cϕ(p) .
Concerning the summand denoted by B(t, s), we proceed as in the proof of Proposition 2: we use first the Malliavin differentiation D α,β Q r (a) = Q r (a)1 C 0 (α, β), then we bound the indicator function 1 C 0 (α, β) by 1, then we integrate dr and dβ and we obtain

B(t, s) ≤ c p 1 0 du |F s,t (u)| p ≤ c p |t -s| 2Hp+cϕ(p)
because the right hand side is equal, modulo a constant, to A(t, s). Taking into account the above estimates, we conclude that

E Z H (t) -Z H (s) p ≤ c p |t -s| 2Hp+cϕ(p) (22)
for every t, s.

Financial Statistics

As mentioned in the Introduction, the multifractal random walks appears nowadays as a serious candidate to model the financial time series. In order to compare its behavior with real data, one needs to simulate it. The main difficulty consists in the fact that, in our construction, the variables Q and B H are dependent. From the theoretical point of view, the Malliavin calculus offers convenient techniques but thinks are complicated in from the practical point of view.

Simulation Scheme

Recall that the fractional MRW is defined by

Z(t) = t 0 Q(s)dB H (s) = t 0 e M (C(t)) Ee M (C(t)) dB H (s), (23)
where the measure M and the fBm B H are dependent. As throughout our paper, we choose M to be the Gaussian measure. Consider a random variable w such that

e M (C(t)) Ee M (C(t)) = e w(t) .
The first step consists in generating a sequence of random variables W = (w(t i )) i such that w(t i+1 ) and w(t i ) are correlated and with Gaussian distribution N n (µ, Σ) where the covariance matrix Σ depending on the auto-correlation function γ(|t -(t + h)|) = Cov(w(t), w(t + h)). We consider the choice of γ from [START_REF] Bacry | Multifractal Random Walks[END_REF] with γ(k) decreasing with respect to k. Then, assuming that the matrix Σ is positively definite, there exists a triangular matrix C such that C t C = Σ. In this way, the components of

W = µ + C t N n (0, I),
are correlated and with normal law.

Let us explain the idea to simulate the integral [START_REF] Nelson | Conditional heteroskedasticity in asset returns: A new approach[END_REF]. This integral is a divergence integral and in principle its simulation is difficult. But in our particular case, we can use the following approach. Let {0 = t 0 < t 1 < • • • < t n = t} a partition of the interval [0, t]. in the dependent case (meaning when W and B H are independent), ( 23) can be naturally approximated by

n-1 i=0 e w(t i ) (B H (t i+1 ) -B H (t i )).
Since the simulation of B H is well-known (see e.g. [START_REF] Bardet | Generators of long-range dependent processes: A survey[END_REF] for an explicit algorithm), we can generate the above sum. In the dependent case, using the integration by parts formula δ(F u) = F δ(u) + DF, u , δ being the Skorohod integral with respect to B H (see [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]), the same sum can be expressed as

n-1 i=0 e w(t i ) (B H (t i+1 ) -B H (t i )) + De w(t i ) , 1 (t i ,t i+1 ) = n-1 i=0 e w(t i ) (B H (t i+1 ) -B H (t i )) + e w(t i ) Dw(t i ), 1 (t i ,t i+1 )
where the scalar product is in the Hilbert space associated to the fBm and, since w(t i ) is Gaussian, Dw(t i ) is deterministic and it depends on the set A 2 (more exactly on the intersection of A 2 and the interval (0,1) which is in principle very small). Thus, a possibility to simulate [START_REF] Nelson | Conditional heteroskedasticity in asset returns: A new approach[END_REF] is to approximate it by n-1 i=0 e w(t i ) (B H (t i+1 ) -B H (t i )) + a i e w(t i ) with suitable coefficients a i .

Stylized Facts

Let S(t) be the price at time t ≥ T of an financial asset, X(t) = ln S(t) the log price and then, log-returns at lag τ are given by

δ τ X(t) = X(t) -X(t -τ ) = ln S(t) S(t -τ ) ,
and, for sake of simplicity and with no loss of generality, we assume that log-returns are centered, E[δ τ X(t)] = 0, thus, we can interpret the volatility as the squared log-returns.

The construction of the proposed MRW allows us to verify several properties of stock market fluctuations, what we call stylized facts. The first two that we present are specific to the MRW, long memory in absolute log-return and in volatility; the others are achieved thanks to our construction, leverage effect and long memory in return.

The random walk that we introduce in our work allows the conditional variance and the random noise, which is a fBm with Hurst parameter H > 1 2 , to be dependent. This dependence should involve the so-called leverage effect, that is, the correlation between the log-return at time t and the squared log-return (the volatility) in the future. The case H > 1/2 implies that the noise B H exhibits long-range dependence. This should involve a long-range dependence of the fluctuations of the financial time series. These facts (leverage effect and long-range dependence) will be checked on the data. Actually the leverage effect is empirically observed on financial index such as the S&P 500 (US market), Nikkei 225 (Japan market), FTSE 100 (UK market), CAC 40 (french market) etc. But it does not appear on the Forex market and on the commodities, for example. We also notice the absence of auto-correlation in log-returns if we consider low frequency financial data but this auto-correlation structure clearly appears in the case of square or absolute log-returns. Nevertheless, at high frequency, we observe a dependence relation which is going to be formed even for usual (not squared and not absolute) auto-correlation function. (see e.g. [START_REF] Lillo | The Long Memory of the Efficient Market[END_REF] or [START_REF] Voit | The Statistical Mechanics of Financial Markets[END_REF]).

The data that we used in order to check the leverage effect and the long-range dependence come from S&P 500 index with a frequency of 15 seconds from 2012-02-28 to 2012-06-26, 131011 points. We compare the characteristics of the S&P 500 index with the simulation of our multifractal random walk. As we mention, we compare our simulations with the data from the S&P 500 index. In fact, we only present the comparison for the auto-correlation function since the multifractal structure of the asset return distribution are already well known and well describe by standard multifractal random walk (see e.g. [START_REF] Muzy | Modeling fluctuations of financial time series: from cascade process to stochastic volatility model[END_REF]). The numerical results are presented in Figures 1 and2. To compare, we also present the simulation of the fractional MRW with H = 0.62. This value has been obtained by a classical R/S test.

Since the Hurst parameter is bigger that 0.5, we should have a long-range dependence for the log-returns. The auto-correlation function of the log-returns is defined by

C(k) = Corr(δ τ X(t), δ τ X(t + k)).
Usually, we say that the log-return have long-range dependence if it satisfies,

C(k) ∼ k -α , α > 0,
We give in Figure 3 the empirical auto-correlation function of the log-returns, the best fit obtained in power law and the simulated auto-correlation function of the fractional MRW with H = 0.62.

We also notice a long-range dependence for the log-returns in the case of S&P 500 index at high frequency, but this less than for the squared or absolute long-returns. We could advance two main ideas to explain this small persistence in high frequency log-returns. First one is that all agents include information, good or bad news, in their investment strategies, but there may have some latencies in the response of investor and cause this auto-correlation. The second one can be cause by huge investors such as mutual fund who want to invest significant sums and place large order. They have to split their large order in a sequence of smaller orders to acquire the number of shares wanted, and then, could cause some persistence. A simple and current strategy to do that is called iceberg order, more sophisticate could be found in the theory of 'optimal trading'.

The persistence of dependence for the log-returns it not easily observed, but the presence of the cluster of log-returns indicates basically that strong variations are followed by other strong variations. Then, we should observe a persistence in absolute and squares log-returns. As before, we present in Figure 5 and Figure 4 the empirical results for S&P 500 with its fit in power law and the simulation of the MRW (H still equal to 0.62).

We now present the second result that we obtain via Malliavin calculus construction. We have assumed that the measure M and the integrator B H was dependent, hence, the MRW should verify the correlation between return at time t and squared return (volatility) around t, say t + k ≤ t ≤ t + k, this quantity correspond to the leverage effect,

L(t, t ) = E δ τ X(t)(δ τ X(t )) 2 , t + k ≤ t ≤ t + k, ( 24
)
This quantity is usually normalize by E[(δ τ X(t)) 2 ] 2 , we use this normalized formulation, see [START_REF] Bouchaud | The leverage Effect in Financial Markets: Retarded Volatility and Market Panic[END_REF] for details. In the literature, one can usually find studies of the stylized facts at low frequency, based on financial indices or on the average of financial indices (see [10] for a very interesting empirical study), that's why we have chosen to work with the S&P 500. We will follow the same approach, but we also present results for high frequency.

The empirical study of the leverage effect is given in Figure 6. To compare, we also plot the leverage effect for the fractional MRW in Figure 6 at low frequency. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 20 40 C(k) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 3: Auto-correlation of log-returns, k from 1 to 100. In black the S&P 500, in grey the confidence interval, in red the MRW and in blue to fit in power law (α = 2.1). Notice that the mean square error between the data and the numerical results is 3.868 × 10 -4 and if we take into account the fit, it is 0.404 × 10 -4 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 20 40 60 80 100 0.05 0.10 0.15 0.20 k (15 seconds) C(k) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 4: Auto-correlation of the absolute log-returns, k from 1 to 100. In black S&P 500, in grey the confidence interval, in red the MRW and in blue the fit in power law (α = 1.5).

The mean square error is 1.720 × 10 -4 and with the fit it is 0.223 × 10 -4

Figure 1 :

 1 Figure 1: S&P 500 index , at 15 seconds, from 2012-02-28 to 2012-06-26, 131011 points.

Figure 2 :

 2 Figure 2: Top: log-return of S&P 500, at 15 seconds, from 2012-02-28 to 2012-06-26, 131011 points. Down: simulation of the MRW with H = 0.62.

† Supported by the CNCS grant PN-II-ID-PCCE-2011-2-0015. Associate member of the team Samm, Université de Panthéon-Sorbonne Paris 1

q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 20 40 60 C(k) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 5: Auto-correlation of the squared log-returns, k from 1 to 100. In black S&P 500, in grey the confidence interval, in red the MRW and in blue the fit in power law (α = 2.4).

The mean square error is 2.834 × 10 -4 and with the fit 1.209 × 10 -2 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 20 40 60 80 100 -0.08 -0.06 -0.04 -0.02 0.00 k L(k) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 20 40 60 80 100 -0.10 -0.05 0.00 k L(k) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 6: First, the empirical leverage effect on S&P 500 at 5 seconds in black, best fit in blue, with α = 0.2, in red. Second, the leverage effect for daily data on S&P 500, from 1985-02-10 to 2012-06-26, the fit is obtained for α = 0.03, here H = 0.5.

[8] F. Black (1976): Studies of stock price volatility changes. Proceedings of the 1976 meetings of the American Statistical Association, Business and Economics Statistics Section, 177-181.

[9] T. Bollerslev (1986): Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307-327.

[10] T. Bollerslev, J. Litvinova, G. and Tauchen (2006): Leverage and Volatility Feedback Effects in High-Frequency Data. Journal of Financial Econometrics, 4, 353-384.


Franco Flandoli
email: flandoli@dma.unipi.it

Peter Imkeller
email: imkeller@mathematik.hu-berlin.de

Ciprian A Tudor
email: tudor@math.univ-lille1.fr

2D-stochastic currents over the Wiener sheet *

Keywords: 2000 AMS Classification Numbers: 60G15, 60G18, 60H05, 76M35, 60H05 currents, multiple stochastic integrals, Brownian sheet, two-parameter processes

By using stochastic calculus for two-parameter processes and chaos expansion into multiple Wiener-Itô integrals, we define a 2D-stochastic current over the Brownian sheet. This concept comes from geometric measure theory. We also study the regularity of the stochastic current with respect to the randomness variable in the Watanabe spaces and with respect to the spatial variable in the deterministic Sobolev spaces.

Introduction

The purpose of this paper is to study two-dimensional stochastic currents. The concept of current comes from geometric measure theory. Let us briefly recall a few definitions on deterministic currents used in the sequel. More informations can be found in [START_REF] Federer | Geometric Measure Theory[END_REF], [START_REF] Morgan | Geometric Measure Theory -A Beginners Guide[END_REF], [START_REF] Simon | Lectures on Geometric Measure Theory[END_REF], [START_REF] Giaquinta | Cartesian Currents in the Calculus of Variations I[END_REF].

We shall denote the Euclidean norm and scalar product in R d (d is fixed throughout the paper) by |•| and •, • respectively. We shall also need the space Λ 2 R d of 2-vectors, its dual space Λ 2 R d of 2-covectors, and the duality between them, still denoted by •, • . If we represent vectors and covectors in the standard basis (the summations are extented from 1 to

d) v = i<j v ij e i ∧ e j , w = i<j w ij e i ∧ e j then w, v = i<j w ij v ij .
Notationally, we prefer to write the covectors in the first argument of ., . since later on the notation for stochastic integrals is more natural. The norm of 2-vectors and 2-covectors is also defined as

|v| 2 = i<j v ij 2 , |w| 2 = i<j (w ij ) 2 .
Notice that for all v 1 , v 2 ∈ R d and w 1 , w 2 ∈ R d , the duality between the 2-vector

v 1 ∧ v 2 = i<j (v 1 ) i (v 2 ) j -(v 1 ) j (v 2 ) i e i ∧ e j
and the 2-covector w 1 ∧ w 2 similarly defined, is given by

w 1 ∧ w 2 , v 1 ∧ v 2 = det v i , w j .
Similarly we have

|v 1 ∧ v 2 | 2 = det ( v i , v j ) = |v 1 | 2 |v 2 | 2 -v 1 , v 2 2 .
Let D k be the space of all infinitely differentiable and compactly supported k-forms on R d . A k-dimensional current is a linear continuous functional on D k . We denote by D k the space of k-currents. In this paper we are only interested in the cases k = 1, 2, so we recall a few corresponding notations. Let ϕ (resp. ψ) be an element of D 1 (resp. D 2 ). We shall write them as

ϕ = d i=1 ϕ i dx i , ψ = i<j ψ ij dx i ∧ dx j
where ϕ i and ψ ij are infinitely differentiable and compactly supported functions on R d . Typical examples of 1-currents are those induced by regular curves (X t ) t∈[0,T ] in R d :

ϕ → S(ϕ) := T 0 ϕ (X t ) , Ẋt dt (1)
while typical examples of 2-currents are given by regular surfaces (f (t, s)) (t,s)∈A in R d , where A is a Borel set in R 2 :

ψ → T (ψ) := A ψ • f, ∂f ∂t ∧ ∂f ∂s dtds = A i<j (ψ ij • f ) dx i ∧ dx j , ∂f ∂t ∧ ∂f ∂s dtds = A i<j (ψ ij • f ) ∂f i ∂t ∂f j ∂s - ∂f j ∂t ∂f i ∂s dtds. (2)
The generalization of the previous example of 1-current to the stochastic case has been the object of previous research in the recent literature. We refer to [START_REF] Flandoli | Stochastic currents[END_REF], [START_REF] Flandoli | Random currents and probabilistic models of vortex filaments[END_REF], [START_REF] Flandoli | On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model[END_REF], [START_REF] Flandoli | Brownian and fractional Brownian stochastic currents via Malliavin calculus[END_REF] for the study of 1D-stochastic currents over the Brownian motion or over the fractional Brownian motion using various interpretation (Itô, Skorohod, Stratonovich etc) for the stochastic integral appearing in the definition of the stochastic current. We will investigate in our paper the 2D-current in the stochastic context using the theory of stochastic integration for two -parameter stochastic processes which has been developed in the eighties (see e.g. [START_REF] Dozzi | Stochastic processes with a multidimensional parameter[END_REF], [START_REF] Imkeller | Stochastic analysis and local time for (N:d) -Wiener process[END_REF], [START_REF] Nualart | Une formule d'Ito pour les martingales continues a deux indices et quelques applications[END_REF]). Somehow, our analysis related geometric aspects of stochastic processes with Malliavin calculus in the spirit of [START_REF] Malliavin | Stochastic Analysis[END_REF].

Let (W s,t ) (t,s)∈[0,1] 2 be a (2, d)-Brownian sheet, with d > 2. By this we mean a centered Gaussian d-dimensional process with 2-dimensional time parameter (t, s)

∈ [0, 1] 2 such that Cov W i s,t , W j s ′ ,t ′ = δ ij (s ∧ t) s ′ ∧ t ′
where δ ij denotes the Kronecker symbol. Note that the functional (1) defines a vector valued distribution

1 0 ∆(x -X t )dX t
with ∆ the Dirac distribution and the meaning of ∆(x -X t ) will be clarified later. Given (2) the correct multiparameter stochastic integrals to be studied for the theory of currents are formally written as

T ij (x) := [0,1] 2 ∆ (x -W s,t ) ∂ t W i s,t ∂ s W j s,t with 1 ≤ i = j ≤ d.
An important question is how to define the above stochastic integral.

At this level, there is no restriction to take i = 1, j = 2. Thus we need to define and to study study the mapping

T (x) := [0,1] 2 ∆ (x -W s,t ) ∂ t W 1 s,t ∂ s W 2 s,t . (3)
This is actually the aim of this work: to give a meaning to the integral (3) and to study its regularity in the deterministic Sobolev spaces H -r (R d ; R d ) as a function of x and in the Sobolev-Watanabe spaces D α,2 as a function of ω. As we mentioned above, the idea to define the stochastic integral (3) comes from the stochastic integration theory for twoparameter processes. We will explain it in details in Section 3. This integral also involves the divergence (Skorohod) integral with respect to the Brownian sheet and multiple Wiener-Itô integrals. Therefore, before defining (3) we will recall in the next section some elements of the Malliavin calculus for Gaussian processes. We will find the following answer: although the definition of the 2D -stochastic current is substantially different from the 1D-case, the 2D-current inherits the regularity of the one-dimensional case, with respect to ω as well as with respect to x. That means the following: the integral (3) belongs to the Sobolev -Watanabe space D -α,2 for every α > 1 2 and to the Sobolev space H -r (R; R) for r > 1 2 . We notice the same interesting phenomenon already remark in the one-dimensional case (see [START_REF] Flandoli | Brownian and fractional Brownian stochastic currents via Malliavin calculus[END_REF]): the order of regularity of (3) is the same with respect to ω and with respect to x.

Our paper is structured as follows: Section 2 contains some preliminaries on multiple Wiener-Itô integrals and the divergence integral. In Section 3 we explain how we arrive to the our definition of the 2D -stochastic currents, making links with the stochastic calculus for two-parameter processes. In Sections 4 and 5 we analyze the 2D-stochastic current over the Brownian sheet in Watanabe spaces and Sobolev spaces respectively. Finally in Section 6 we treat the 2D -stochastic current over the d -dimensional Brownian sheet.

Malliavin calculus, chaos expansion and Sobolev-Watanabe spaces

Here we describe the elements from stochastic analysis that we will need in the paper. Consider H a real separable Hilbert space and (B(ϕ), ϕ ∈ H) an isonormal Gaussian process on a probability space (Ω, A, P ), that is a centered Gaussian family of random variables such that E (B(ϕ)B(ψ)) = ϕ, ψ H . Denote by I n the multiple stochastic integral with respect to B (see [START_REF] Nualart | Malliavin calculus and related topics[END_REF]). This I n is actually an isometry between the Hilbert space H ⊙n (symmetric tensor product) equipped with the scaled norm 1 √ n! • H ⊗n and the Wiener chaos of order n which is defined as the closed linear span of the random variables H n (B(ϕ)) where ϕ ∈ H, ϕ H = 1 and H n is the Hermite polynomial of degree n ≥ 1

H n (x) = (-1) n n! exp x 2 2 d n dx n exp - x 2 2 , x ∈ R.
The isometry of multiple integrals can be written as: for m, n positive integers,

E (I n (f )I m (g)) = n! f, g H ⊗n if m = n, E (I n (f )I m (g)) = 0 if m = n. (4)
It also holds that

I n (f ) = I n f
where f denotes the symmetrization of f defined by f (x 1 , . . . , x x ) = 1 n! σ∈Sn f (x σ(1) , . . . , x σ(n) ). We recall that any square integrable random variable which is measurable with respect to the σ-algebra generated by B can be expanded into an orthogonal sum of multiple stochastic integrals

F = n≥0 I n (f n ) ( 5
)
where f n ∈ H ⊙n are (uniquely determined) symmetric functions and

I 0 (f 0 ) = E [F ].
Let L be the Ornstein-Uhlenbeck operator

LF = - n≥0 nI n (f n )
if F is given by [START_REF] Flandoli | Stochastic currents[END_REF].

For p > 1 and α ∈ R we introduce the Sobolev-Watanabe space D α,p as the closure of the set of polynomial random variables with respect to the norm

F α,p = (I -L) α 2 L p (Ω)
where I represents the identity. In this way, a random variable F as in (5) belongs D α,2 if and only if

n≥0 (1 + n) α I n (f n ) 2 L 2 (Ω) = n≥0 (1 + n) α n! f n 2 H ⊗n < ∞. (6)
We denote by D the Malliavin derivative operator that acts on smooth functions of the form F = g(B(ϕ 1 ), . . . , B(ϕ n )) (g is a smooth function with compact support and ϕ i ∈ H)

DF = n i=1 ∂g ∂x i (B(ϕ 1 ), . . . , B(ϕ n ))ϕ i .
The operator D is continuous from D α,p into D α,p (H) . The adjoint of D is denoted by δ and is called the divergence (or Skorohod) integral. It is a continuous operator from D α,p (H) into D α,p . For adapted integrands, the divergence integral coincides to the classical Itô integral. We will use the notation

δ(u) = T 0 u s dB s .
Let u be a stochastic process having the chaotic decomposition u s = n≥0 I n (f n (•, s)) where f n (•, s) ∈ H ⊗n for every s. One can prove that u ∈ Dom δ if and only if fn ∈ H ⊗(n+1) for every n ≥ 0, and ∞ n=0 I n+1 ( fn ) converges in L 2 (Ω). In this case,

δ(u) = ∞ n=0 I n+1 ( fn ) and E|δ(u)| 2 = ∞ n=0 (n + 1)! fn 2 H ⊗(n+1) . (7)
We will also recall the integration by parts formula

F δ(u) = δ(F u) + DF, u H (8)
which holds for F ∈ D α,2 , u ∈ D α,2 (H) and F u ∈ D α,2 (H). In the present work we will consider divergence integral with respect to a Brownian motion in R d . Throughout this paper we will denote by p s (x) the Gaussian kernel of variance s > 0 given by p s (x) =

1 √ 2πs e -x 2 2s , x ∈ R and for x = (x 1 , . . . , x d ) ∈ R d by p d s (x) = d i=1 p s (x i ).

Definition of the 2D-stochastic current

The idea to define a stochastic integral (3) comes from the stochastic calculus for twoparameter martingales developed in the eighties. Let us briefly recall the context. Consider a two-parameter Wiener process (W s,t ) s,t∈[0,1] and let us express the random variable W 2 s,t (here W s,t is one dimensional random variable and W 2 s,t is its square; we mention this in order to avoid the confusion with the components of the two-dimensional Brownian sheet that appears in Section 2) as a stochastic integral plus a deterministic integral in the spirit of the classical Itô formula. Let t, s ∈ [0, 1] be fixed and let 0 = t 0 < t 1 < . . . t n = t and 0 = s 0 < s 1 < . . . s m = s be two partitions of the intervals [0, t] and [0, s] respectively. For fixed t, we can write

W 2 s,t = 2 m-1 i=0 W s i ,t W s i+1 ,t -W s i ,t + m-1 i=0 W s i+1 ,t -W s i ,t 2 .
It is standard to prove that the last summand m-1 i=0

W s i+1 ,t -W s i ,t 2 converges in L 2 (Ω)
as m → ∞ to st which, in some sense, constitutes the quadratic variation of the twoparameter martingale W s,t . Let us analyze the first summand. Writing

g(t) = g(0) + n-1 j=0 (g(t j+1 ) -g(t j )) we get m-1 i=0 W s i ,t W s i+1 ,t -W s i ,t = m-1 i=0 m-1 j=0 W s i ,t j+1 W s i+1 ,t j+1 -W s i+1 ,t j -W s i ,t j+1 + W s i ,t j + m-1 i=0 m-1 j=0 W s i ,t j+1 -W s i ,t j W s i+1 ,t j W s i ,t j .
The first summand above usually converges to the stochastic integral with respect to the Wiener sheet [START_REF] Imkeller | Stochastic analysis and local time for (N:d) -Wiener process[END_REF] or [START_REF] Nualart | Une formule d'Ito pour les martingales continues a deux indices et quelques applications[END_REF] for the definition of the integral and the proof of the convergence; but the stochastic integral can be also understood as the Skorohod integral δ W

s 0 t 0 W u,v dW u,v (see
•,• 1 [0,s]×[0,t] (•, •) with respect to the Gaussian process W ). The summand m-1 i=0 m-1 j=0 W s i ,t j+1 -W s i ,t j W s i+1 ,t j W s i ,t j
is a specific two-parameter case term and it converges, for fixed s, t ∈ [0, 1] as m, n → ∞ to a random variable denoted Ms,t such that the stochastic process ( Ms,t ) s,t∈[0,1] is a two-parameter martingale (we refer again to [START_REF] Imkeller | Stochastic analysis and local time for (N:d) -Wiener process[END_REF] or [START_REF] Nualart | Une formule d'Ito pour les martingales continues a deux indices et quelques applications[END_REF] for details). A similar term M appears also in the stochastic calculus for the fractional Brownian sheet (see [START_REF] Tudor | Ito Formula and Local Time for the Fractional Brownian Sheet[END_REF]). The stochastic process M can be interpreted as

Ms,t = s 0 t 0 d u W u,v d v W u,v .
We refer to [START_REF] Tudor | Ito Formula and Local Time for the Fractional Brownian Sheet[END_REF] for a discussion in this sense. In terms of the divergence integral it can be written as

Ms,t = δ (2) (u 1 ,v 1 ),(u 2 ,v 2 ) 1 ⊗2 [0,s]×[0,t] ((u 1 , v 1 ), (u 2 , v 2 ))1 [0,u 1 ] (u 2 )1 [0,v 2 ] (v 1 )) .
Let us explain the above notation. The symbol δ (2) means the double (iterated) Skorohod integral with respect to the Wiener sheet W . The indices (u 1 , v 1 ), (u 2 , v 2 ) in the formula δ

(u 1 ,v 1 ),(u 2 ,v 2 ) means that the double Skorohod integral acts with respect to the two variables (u 1 , v 1 ) and (u 2 , v 2 ). More generally, if (X s,t ) s,t∈[0,1] is an adapted two-parameter process (let us be vague on this concept because several definitions of the adaptability exist in the two-parameter context; we refer again to [START_REF] Imkeller | Stochastic analysis and local time for (N:d) -Wiener process[END_REF], [START_REF] Nualart | Une formule d'Ito pour les martingales continues a deux indices et quelques applications[END_REF] or [START_REF] Dozzi | Stochastic processes with a multidimensional parameter[END_REF]), then the sequence

m-1 i=0 n-1 j=0 X s i ,t j W s i ,t j+1 -W s i ,t j W s i+1 ,t j -W s i ,t j converges to the integral t 0 s 0 X u,v dM u,v = t 0 s 0 X u,v d u W u,v d v W u,v
which can be also written as a double Skokorod integral in the following way (assuming that X is regular enough in the Malliavin calculus sense)

t 0 s 0 X u,v d u W u,v d v W u,v = δ (2) X u 1 ,v 2 1 [0,u 1 ] (u 2 )1 [0,v 2 ] (v 1 )
where δ (2) denotes the double Skorohod integrals which as above acts with respect to the variables (u

1 , v 1 ), (u 2 , v 2 ) ∈ [0, 1] 2 .
The above discussion motivate the following definition of the 2D stochastic current.

Definition 1 Let (W s,t ) s,t∈[0,1] be a Wiener sheet and for every x ∈ R, s, t ∈ [0, 1] consider the functional ∆(x -W s,t ) which is a distribution in the Watanabe sense. We define the stochastic current over the Wiener sheet W as the mapping

x → ξ(x) := δ (2) (u 1 ,v 1 ),(u 2 ,v 2 ) ∆(x -W u 1 ,v 2 )1 [0,u 1 ] (u 2 )1 [0,v 2 ] (v 1 ) . (9)
The meaning of this definition will be clarified in the sequel. The term ∆(x-W u 1 ,v 2 ) is a distribution in the Sobolev-Watanabe spaces. In the rest of our paper we will analyze the mapping ξ introduced above from different perspectives: first as a functional of the randomness ω and then as a function of x ∈ R d .

The 2D-stochastic current in Watanabe spaces

Let us first analyze the function ξ(x), for fixed x ∈ R, as a distribution in the Watanabe (or Sobolev-Watanabe) sense (see Section 2). Concretely, we will prove in this section that for every x the function ξ(x) belongs to the Watanabe space D α,2 for every α < -1 2 . We also prove that ξ(x) can be approximated, under the norm of the space D α,2 by some Riemann sums. In this way, we make the link with the discussion in the previous section.

Regularity of the 2D stochastic current in the Watanabe spaces

Our method to prove the regularity of the functional ( 9) is based on the Wiener-Itô chaos decomposition. Let (X s ) s∈S be an isonormal Gaussian process with covariance

EX s X t := R(s, t) s, t ∈ S.
The set S is a subset of R N . We will use the following decomposition of the delta Dirac function (see Nualart and Vives [START_REF] Nualart | Smoothness of Brownian local times and related functionals[END_REF], Imkeller et al. [START_REF] Imkeller | The asymptotic behavior of local times and occupation integrals of the N -parameter Wiener process in R d[END_REF], Eddahbi et al. [START_REF] Eddahbi | Regularity of the local time for the d-dimensional fractional Brownian motion with N -parameters[END_REF]) into orthogonal sum of multiple Wiener-Itô integrals

∆(x -X s ) = n≥0 R(s) -n 2 p R(s) (x)H n x R(s) 1 2 I n 1 ⊗n [0,s] (10)
:= n≥0 a x n (s)I n 1 ⊗n [0,s] , x ∈ R, s ∈ S where R(s) := R(s, s), p R(s)
is the Gaussian kernel of variance R(s), H n is the Hermite polynomial of degree n and I n represents the multiple Wiener-Itô integral of degree n with respect to the Gaussian process X as defined in the previous section. We also denoted by

a x n (s) = R(s) -n 2 p R(s) (x)H n x R(s) 1 2 (11)
for every s ∈ S, x ∈ R, n ≥ 0.

For X = W = (W s,t ) s,t∈[0,1] , the Wiener sheet, we have

∆(x -W u 1 ,v 2 ) = m≥0 a x m (u 1 , v 2 )I m 1 ⊗m [0,u 1 ]×[0,v 2 ] (12)
where for every u 1 , v 2 ∈ [0, 1],

a x m (u 1 , v 2 ) := (u 1 v 2 ) -m 2 p u 1 v 2 (x)H m x √ u 1 v 2 . ( 13
)
We will show in the next result that the 2D-stochastic current defined in Definition 1 is well-defined as a distribution in the Watanabe sense.

Theorem 1 For every x ∈ R the functional ξ(x) defined in Definition 1 belongs to the Watanabe space D -α,2 for any α > 1 2 .

Proof: Let x ∈ R be fixed throughout the proof. We will compute first the divergence integral δ

(2)

(u 1 ,v 1 ),(u 2 ,v 2 ) ∆(x -W u 1 ,v 2 )1 [0,u 1 ] (u 2 )1 [0,v 2 ] (v 1
) with respect to the variables (u 1 , v 1 ) and (u 2 , v 2 ) belonging to [0, 1] 2 . Using [START_REF] Imkeller | Chaos expansion of double intersection local time of Brownian motion in R d and renormalization[END_REF] and the chaotic form of the divergence integral ( 7), we will need to symmetrize the function ((x 1 , y 1 ), (x 2 , y 2 ), . . . , (x m+1 , y m+1 ), (x m+2 , y m+2 ))

→ a x m (x m+1 , y m+2 )1 [x m+2 ,1] (x m+1 )1 [0,y m+2 ] (y m+1 )1 ⊗m [0,x m+1 ]×[0,y m+2 ] ((x 1 , y 1
), (x 2 , y 2 ), . . . , (x m , y m )) with respect to the m + 2 variables (x 1 , y 1 ), (x 2 , y 2 ), . . . , (x m+ , y m+2 ).

The above function is already symmetric in its first m variables. Noting that for any function f (a 1 , a 2 , . . . , a m , x, y) which is symmetric with respect to a 1 , .., a m , its symmetrization with respect to its all m + 2 variables is given by

f (a 1 , .., a m+2 ) = 1 (m + 1)(m + 2) m+2 k,l=1;k =l f (a 1 , .., âk , .., âl , ..a m+2 , a k , a l ) (14)
(the notation â means that the variable a is missing above) we get, for every x ∈ R,

ξ(x) = m≥0 1 (m + 1)(m + 2) I m+2 m+2 k,l=1;k =l a x m (x k , y l )1 [0,x k ]×[0,y l ] (x l , y k ) ×1 ⊗m [0,x k ]×[0,y l ] (x 1 , y 1 ), (x 2 , y 2 ), . . . , (x k , y k ), (x l , y l ), . . . , (x m+2 , y m+2 ) = m≥0 I m+2 1 (m + 1)(m + 2) m+2 k,l=1;k =l a x m (x k , y l ) 1 ⊗m+1 [0,x k ] (x 1 , .., x k , .., x m+2 )1 ⊗m+1 [0,y l ] (y 1 , .., y l , .., y m+2 ) (15)
where (x k , y k ) means, as above, that the variable (x k , y k ) is missing. Using (6), the D α,2 norm of the random distribution ξ(x) can be written as

ξ(x) 2 α,2 = m≥0 (1 + m) α (m + 2)!   1 m + 1 1 m + 2 m+2 k,l=1;k =l a x m (x k , y l )1 [0,x k ]×[0,y l ] (x l , y k ) ×1 ⊗m [0,x k ]×[0,y l ] (x 1 , y 1 ), (x 2 , y 2 ), . . . , (x k , y k ), (x l , y l ), . . . , (x m+2 , y m+2 ) L 2 (([0,1] 2 ) m+2 ) = m≥0 (1 + m) α (m + 2)! 1 m + 1 1 m + 2 2 m+2 k,l=1;k =l m+2 k ′ ,l ′ =1;k ′ =l ′ ([0,1] 2 ) m+2 dx 1 dy 1 ..dx m+2 dy m+2 m+2 k,l=1;k =l a x m (x k , y l )1 ⊗m+1 [0,x k ] (x 1 , .., x k , .., x m+2 )1 ⊗m+1 [0,y l ] (y 1 , .., y l , .., y m+2 ) m+2 k ′ ,l ′ =0;k ′ =l ′ a x m (x k ′ , y l ′ )1 ⊗m+1 [0,x k ′ ] (x 1 , .., x k ′ , .., x m+2 )1 ⊗m+1 [0,y l ′ ] (y 1 , .., y l ′ , .., y m+2 ). Notice that if k = k ′ , the terms 1 [0,x k ] (x k ′ ) and 1 [0,x k ′ ] (x k
) appear in the above expression. Therefore, all the summand with k = k ′ will vanish. Similarly, the summands with l = l ′ will vanish. Consequently,

ξ(x) 2 α,2 = m≥0 (1 + m) α m! 1 m + 1 1 m + 2 m+2 k,l=1;k =l ([0,1] 2 ) m+2 dx 1 dy 1 ..dx m+2 dy m+2 ×(a x m (x k , y l )) 2 1 ⊗m+1 [0,x k ] (x 1 , .., x k , .., x m+2 )1 ⊗m+1 [0,y l ] (y 1 , .., y l , .., y m+2 ) = m≥0 (1 + m) α m! 1 m + 1 1 m + 2 m+2 k,l=1;k =l 1 0 dx k 1 0 dy l (a x m (x k , y l )) 2 × [0,x k ] m+1 dx 1 ...dx k ..dx m+2 [0,y l ] m+1 dy 1 ...dy l ..dy m+2 = m≥0 (1 + m) α m! 1 m + 1 1 m + 2 m+2 k,l=1;k =l 1 0 dx k 1 0 dy l (a x m (x k , y l )) 2 x m+1 k y m+1 l .
Using the expression (13) of the coefficient a x m we have

(a x m (x k , y l )) 2 x m+1 k y m+1 l = p x k y l (x)H m x √ x k y l 2 x k y l . Therefore ξ(x) 2 α,2 = m≥0 (1 + m) α m! 1 0 1 0 dudv p uv (x)H m x √ uv 2 uv.
We use the identity, for every y ∈ R,

H n (y)e -y 2 2 = (-1) [n/2] 2 n 2 2 n!π ∞ 0 u n e -u 2 g(uy √ 2)du (16)
where g(r) = cos(r) if n is even and g(r) = sin(r) if n is odd. Since |g(r)| ≤ 1, we have the bound

H n (y)e -y 2 2 ≤ 2 n 2 2 n!π Γ( n + 1 2 ) := c n . (17)
This implies that

ξ(x) 2 α,2 ≤ 1 2π m≥0 (1 + m) α m!c 2 m 1 0 1 0 dudv(uv) 1 2 = 1 2π 4 9 m≥0 (1 + m) α m!c 2 m
and this is finite for every α < -1 2 because, by Stirling's formula, m!c 2 m (this has been already used in [START_REF] Nualart | Smoothness of Brownian local times and related functionals[END_REF], [START_REF] Eddahbi | Regularity of the local time for the d-dimensional fractional Brownian motion with N -parameters[END_REF], [START_REF] Coutin | The Tanaka formula for the fractional Brownia motion[END_REF]) behaves as n -1 2 .

Remark 1 We remark that the regularity of the functional ξ(x) in the Watanabe sense is the same as in the one-parameter case (see [START_REF] Flandoli | Brownian and fractional Brownian stochastic currents via Malliavin calculus[END_REF]). This implies in particular that it is not a random variable, but a distribution.

Remark 2 Note that the integration interval for the stochastic integral dW in the definition of ξ(x) is the entire interval [0, 1]. In the one-parameter case (see [START_REF] Flandoli | Brownian and fractional Brownian stochastic currents via Malliavin calculus[END_REF]), we need to consider the quantity ξ(x) = 1 a ∆(x -B s )dB s with a > 0 in order to obtain that ξ(x) ∈ D α,2 with α < -1 2 .

Approximation of 2D-stochastic current in Watanabe spaces

Let us introduce the Riemann sum

T N,M (x) = N -1 i=0 M -1 j=0 ∆(x -W s i ,t j ) W s i+1 ,t j -W s i ,t j W s i ,t j+1 -W s i ,t j . ( 18
)
Remark 3 Following the lines of Theorem 1, we can prove that T N,M (x) belongs to the Sobolev-Watanabe space D α,2 for every α < -1 2 . This is also a consequence of the proof of Theorem 2 below.

Following the discussion at the beginning of Section 3 it is natural to expect that T N,M (x) converges as M, N → ∞ to ξ(x) for every x ∈ R. We will actually show that the sequence (18) approximates ξ(x) in the norm of the Watanabe space D -α,2 with α > 1 2 .

Theorem 2 For every x ∈ R, the sequence T N,M (x) given by ( 18) converges to ξ(x) as M, N → ∞ in the norm • -α,2 for any α > 1 2 .

Proof: Fix again x ∈ R. The first step is to express T N,M using multiple Wiener-Itô integrals. We have, by the integration by parts formula [START_REF] Flandoli | Brownian and fractional Brownian stochastic currents via Malliavin calculus[END_REF] and the symmetrization formula ( 14)

T N,M (x) = δ (2)   N -1 i=0 M -1 j=0 ∆(x -W s i ,t j )1 [0,s i ]×[t j ,t j+1 ] (u 1 , v 1 )1 [s i ,s i+1 ]×[0,t j ] (u 2 , v 2 )   = N -1 i=0 M -1 j=0 a x m (s i , t j ) × m≥0 I m+2 1 ⊗m [0,s i ]×[0,t j ] ((x 1 , y 1 ), ..., (x m , y m ))1 [0,s i ]×[t j ,t j+1 ] (u 1 , v 1 )1 [s i ,s i+1 ]×[0,t j ] (u 2 , v 2 ) = N -1 i=0 M -1 j=0 a x m (s i , t j ) m≥0 I m+2 1 (m + 1)(m + 2) m+2 k,l=1;k =l 1 ⊗m [0,s i ]×[0,t j ] ((x 1 , y 1 ), .., (x k , y k ), (x l , y l ), .., (x m+2 , y m+2 )) 1 [0,s i ]×[t j ,t j+1 ] (x k , y k ))1 [s i ,s i+1 ]×[0,t j ] (x l , y l ) . (19)
Taking into account relation [START_REF] Simon | Lectures on Geometric Measure Theory[END_REF] and its analogous for ξ(x) (see the proof of Theorem 1, formula 15) we obtain

T N,M (x) -ξ(x) 2 α,2 = m≥0 (3 + m) α m! (m + 1)(m + 2) m+2 k,l=1;k =l [0,1] 2m dx 1 ..dx m+2 dy 1 ..dy m+2   N -1 i=0 M -1 j=0 a x m (s i , t j )1 ⊗m+1 [0,s i ] (x 1 , .., x k , .., x m+2 )1 [s i ,s i+1 ] (x k )1 ⊗m+1 [0,t j ] (y 1 , .., y l , ..y m+2 ) -a x m (x k , y l )1 ⊗m+1 [0,x k ] (x 1 , .., x k , .., x m+2 )1 ⊗m+1 [0,y l ] (y 1 , .., y l , ..y m+2 ) 2
and as in the proof of Theorem 1

T N,M (x) -ξ(x) 2 α,2 = m≥0 (3 + m) α m! (m + 1)(m + 2) m+2 k,l=1;k =l   N -1 i=0 M -1 j=0 (a x m (s i , t j )) 2 s m+1 i (s i+1 -s i )t m+1 j (t j+1 -t j ) -2 N -1 i=0 M -1 j=0 a x m (s i , t j )s m+1 i t m+1 j s i+1 s i t j+1 t j dx k dy l a x m (x k , y l ) + 1 0 1 0 dx k dy l (a x m (x k , y l )) 2 x m+1 k y m+1 l = m≥0 (3 + m) α m!   N -1 i=0 M -1 j=0 (a x m (s i , t j )) 2 s m+1 i (s i+1 -s i )t m+1 j (t j+1 -t j ) -2 N -1 i=0 M -1 j=0 a x m (s i , t j )s m+1 i t m+1 j s i+1 s i t j+1 t j dudva x m (u, v) + 1 0 1 0 dudv(a x m (u, v)) 2 (uv) m+1 .
Now we claim that, for every fixed m ≥ 0, the sequence

  N -1 i=0 M -1 j=0 (a x m (s i , t j )) 2 s m+1 i (s i+1 -s i )t m+1 j (t j+1 -t j ) -2 N -1 i=0 M -1 j=0 a x m (s i , t j )s m+1 i t m+1 j s i+1 s i t j+1 t j dudva x m (u, v) + 2 1 0 1 0 dudv(a x m (u, v)) 2 (uv) m+1  
converges to zero as N, M → ∞. This is true because

N -1 i=0 M -1 j=0 (a x m (s i , t j )) 2 s m+1 i (s i+1 -s i )t m+1 j (t j+1 -t j ) → N,M →∞ 1 0 1 0 dudv(a x m (u, v)) 2 (uv) m+1 and N -1 i=0 M -1 j=0 a x m (s i , t j )s m+1 i t m+1 j s i+1 s i t j+1 t j dudva x m (u, v) → N,M →∞ 1 0 1 0 dudv(a x m (u, v)) 2 (uv) m+1
using Riemann sums convergence. To conclude the convergence of T N,M (x)ξ(x) 2 α,2 to zero for every α < -1 2 it suffices to check that

N -1 i=0 M -1 j=0 (a x m (s i , t j )) 2 s m+1 i (s i+1 -s i )t m+1 j (t j+1 -t j ) ≤ cm -1 2 , N -1 i=0 M -1 j=0 a x m (s i , t j )s m+1 i t m+1 j s i+1 s i t j+1 t j dudva x m (u, v) ≤ cm -1 2 ,
and

1 0 1 0 dudv(a x m (u, v)) 2 (uv) m+1 ≤ cm -1 2 .
with c > 0 a constant not depending on m. The last bound has been proved before (see the proof of Theorem 1) and the first two bounds can be obtained analogously using relations ( 16) and ( 17).

2D-stochastic currents in deterministic Sobolev spaces

The purpose is this paragraph in to study the mapping (9) as a function of the spatial variable x ∈ R.

Regularity in the deterministic Sobolev spaces

We study in this part the regularity with respect to the variable x ∈ R d of the mapping given by ( 9) in the (deterministic) Sobolev spaces H -r (R d ; R d ). This is the dual space to H r R d , R d , or equivalently the space of all vector valued distributions ϕ such that

R d 1 + |x| 2 -r | ϕ (x)| 2 dx < ∞, ϕ
being the Fourier transform of ϕ. Assume first that d = 1. The case d ≥ 2 will be treated later.

We will recall the following lemma (see [START_REF] Flandoli | Brownian and fractional Brownian stochastic currents via Malliavin calculus[END_REF], Lemma 1).

Lemma 1 Let s = (s 1 , .., s N ) ∈ [0, 1] N be fixed. The Fourier transform of the function

x → a x m (s) : R → R
is denoted by a x m (s) and it is given by

a x m (s) = e -x 2 2 |s| (-i) m x m m! where |s| = s 1 ...s N .
As a consequence, the Fourier transform with respect to x, denoted in the sequel

∆( x -W u 1 ,v 2 ), of the random distribution x ∈ R → ∆(x -W u 1 ,v 2 ) is given by ∆( x -W u 1 ,v 2 ) = m a x m (u 1 , v 2 )I m 1 ⊗m [0,u 1 ]×[0,v 2 ] = e -ixWu 1 ,v 2 .
Proposition 1 The Fourier transform of the function

x → ξ(x)
is given by ξ

(x) = δ (2) (u 1 ,v 1 ),(u 2 ,v 2 ) e -ixWu 1 ,v 2 1 [0,u 1 ]×[0,v 2 ] .
Proof: Using formula [START_REF] Morgan | Geometric Measure Theory -A Beginners Guide[END_REF] and the expression of the Fourier transform of x → a x m from Lemma 1, we get (to justify the interchange of the order of integration below we refer to Exercise 3.2.7 in [START_REF] Nualart | Malliavin calculus and related topics[END_REF])

ξ(x) = m≥0 I m+2 1 (m + 1)(m + 2) m+2 k,l=1;k =l a x m (x k , y l ) 1 ⊗m+1 [0,x k ] (x 1 , .., x k , .., x m+2 )1 ⊗m+1 [0,y l ] (y 1 , .., y k , .., y m+2 ) = δ (2)
(u 1 ,v 1 ),(u 2 ,v 2 ) e -x 2 2 u 1 v 2 m≥0 (-i) m x m m! I m 1 ⊗m [0,u 1 ]×[0,v 2 ] (•) = δ (2) (u 1 ,v 1 ),(u 2 ,v 2 ) e -ixWu 1 ,v 2 1 [0,u 1 ]×[0,v 2 ] .
Here we used the fact that

e -ixWu 1 ,v 2 = m≥0 (-i) m x m m! I m 1 ⊗m [0,u 1 ]×[0,v 2 ] (•)
which can be obtained using Stroock's formula (see [START_REF] Flandoli | Brownian and fractional Brownian stochastic currents via Malliavin calculus[END_REF] relation [START_REF] Giaquinta | Cartesian Currents in the Calculus of Variations I[END_REF]).

Consequently, we obtain the following result:

Theorem 3 For almost all ω, it holds that ξ ∈ H -r (R, R) if r > 1 2
Proof: It is easy to see that

E ξ(x) 2 = 1 0 du 1 u 1 0 du 2 1 0 dv 2 v 2 0 dv 1 e -ixWu 1 ,v 2 2 = 1 0 u 1 du 1 1 0 v 2 dv 2 = 1 4
and consequently

E ξ(x) H -r = R 1 (1 + x 2 ) r E ξ(x) 2 dx is finite if and only if r > 1 2 .
Remark 4 We note that, as in the case of 1D -currents, the regularity of ( 9) is the same with respect to the space variable x and with respect to the randomness variable ω.

Convergence of the Fourier transforms

Here we study the convergence of the sequence T N,M [START_REF] Nualart | Smoothness of Brownian local times and related functionals[END_REF] to the 2D-stochastic current [START_REF] Giaquinta | Cartesian Currents in the Calculus of Variations I[END_REF] in the Sobolev space. Precisely, we have the following:

Theorem 4 Let T N,M be given by [START_REF] Nualart | Smoothness of Brownian local times and related functionals[END_REF]. Then as M, N → ∞, the sequence T N,M converges to ξ in L 2 (Ω; H -r (R; R)) for every r > 1 2 .

Proof: From ( 18) it is immediate that

T N,M (x) = N -1 i=0 M -1 j=0 e i xWs i ,t j W s i+1 ,t j -W s i ,t j W s i ,t j+1 -W s i ,t j
where T N,M (x) denotes the Fourier transform of x → T N,M (x). We also know from Proposition 1 that ξ(x) = δ

(2)

(u 1 ,v 1 ),(u 2 ,v 2 ) e -xWu 1 ,v 2 1 [0,u 1 ] (u 2 )1 [0,v 2 ] (v 1 ) .
Then the L 2 (Ω; H -r (R; R))-norm of the difference T N,Mξ can be computed as follows

E |T N,M -ξ| 2 H -r (R;R) = E R |T N,M (x) -ξ(x)| 2 (1 + x 2 ) -r dx.
Let us first compute

E |T N,M (x) -ξ(x)| 2 = E N -1 i=0 M -1 j=0 cos(-xW s i ,t j ) W s i+1 ,t j -W s i ,t j W s i ,t j+1 -W s i ,t j -δ (2)
(u 1 ,v 1 ),(u 2 ,v 2 ) cos(-xW u 1 ,v 2 )1 [0,u 1 ] (u 2 )1 [0,v 2 ] (v 1 ) 2 +E N -1 i=0 M -1 j=0 sin(-xW s i ,t j W s i+1 ,t j -W s i ,t j W s i ,t j+1 -W s i ,t j -δ (2)
(u 1 ,v 1 ),(u 2 ,v 2 ) sin(-xW u 1 ,v 2 )1 [0,u 1 ] (u 2 )1 [0,v 2 ] (v 1 )
2 and from the independence of the increments of the Brownian sheet

E |T N,M (x) -ξ(x)| 2 = E N -1 i=0 M -1 j=0 cos(-xW s i ,t j ) 2 s i t j (s i+1 -s i )(t j+1 -t j ) +E 1 0 du 1 u 1 0 du 2 1 0 dv 2 v 2 0 dv 1 cos(-xW u 1 ,v 2 ) 2 -2E N -1 i=0 M -1 j=0 s i+1 s i du 1 s i 0 du 2 t j+1 t j dv 2 t j 0 dv 1 cos(-xW s i ,t j ) cos(-xW u 1 ,v 2 ) + E N -1 i=0 M -1 j=0 sin(-xW s i ,t j ) 2 s i t j (s i+1 -s i )(t j+1 -t j ) +E 1 0 du 1 u 1 0 du 2 1 0 dv 2 v 2 0 dv 1 sin(-xW u 1 ,v 2 ) 2 -2E N -1 i=0 M -1 j=0 s i+1 s i du 1 s i 0 du 2 t j+1 t j dv 2 t j 0 dv 1 sin(-xW s i ,t j ) sin(-xW u 1 ,v 2 ).
We will obtain, using the well-known trigonometric formulas sin 2 (x) + cos 2 (x) = 1 and sin(x) sin(y) + cos(x) cos(y) = cos(x + y)

E |T N,M (x) -ξ(x)| 2 = E N -1 i=0 M -1 j=0 s i t j (s i+1 -s i )(t j+1 -t j ) + 1 0 udu 2 -2E N -1 i=0 M -1 j=0 s i t j s i+1 s i du 1 t j+1 t j dv 2 cos x(W s i ,t j -W u 1 ,v 2 ) .
Since W s i ,t j -W u 1 ,v 2 with s i < u 1 and t j < v 2 is a centered Gaussian random variable with variance u 1 v 2s i t j we get

E cos x(W s i ,t j -W u 1 ,v 2 ) = e - x 2 (u 1 v 2 -s i t j ) 2
and therefore

E |T N,M (x) -ξ(x)| 2 = N -1 i=0 M -1 j=0 s i t j (s i+1 -s i )(t j+1 -t j ) + 1 0 udu 2 -2 N -1 i=0 M -1 j=0 s i t j s i+1 s i du 1 t j+1 t j dv 2 e - x 2 (u 1 v 2 -s i t j ) 2
.

We can write

E |T N,M -ξ| 2 H -r (R;R) = E R |T N,M (x) -ξ(x)| 2 (1 + x 2 ) -r dx = R dx(1 + x 2 ) -r   N -1 i=0 M -1 j=0 s i t j (s i+1 -s i )(t j+1 -t j ) + 1 0 udu 2   -2 R dx(1 + x 2 ) -r   N -1 i=0 M -1 j=0 s i t j s i+1 s i du 1 t j+1 t j dv 2 e - x 2 (u 1 v 2 -s i t j ) 2   By a Riemmann sum convergence, it is clear that N -1 i=0 s i (s i+1 -s 1 ) → N →∞ 1 0 udu, M -1 j=0 t j (t j+1 -t j ) → M →∞ 1 0 vdv and for every fixed x ∈ R N -1 i=0 M -1 j=0 s i t j e x 2 s i t j 2 s i+1 s i du 1 t j+1 t j dv 2 e -x 2 u 1 v 2 2 → N,M →∞ 1 0 1 0 uvdudve -x 2 uv 2 e x 2 uv 2 = 1 0 udu 2 and this implies that for every x ∈ R the sequence E |T N,M (x) -ξ(x)| 2 converges to zero as N, M → ∞. The convergence of E |T N,M -ξ| 2 H -r (R;R)
to zero follows from the dominated convergence theorem since

(1 + x 2 ) -r   N -1 i=0 M -1 j=0 s i t j s i+1 s i du 1 t j+1 t j dv 2 1 -e - x 2 (u 1 v 2 -s i t j ) 2   ≤ (1 + x 2 ) -r   N -1 i=0 M -1 j=0 s i t j s i+1 s i du 1 t j+1 t j dv 2   ≤ (1 + x 2 ) -r
which is integrable for every r > 1 2 .

Multidimensional 2D-currents

Newt we study in this parts d -dimensional stochastic currents over a Wiener sheet. Let W = W (1) , W (2) , ..., W (d) be a d-dimensional Brownian sheet. That means, for every i = 1, .., d, the component

W (i) = (W (i) s,t ) s,t∈[0,1]
is a Wiener sheet and the components of W are mutually independent.

We will defined the d -dimensional current over the Wiener sheet as follows.

Definition 2 The 2D-stochastic current over W in R d is given by the mapping

x ∈ R d → ξ(x) = (ξ i,j ) i,j=1,..,d := δ i (u 1 ,v 1 ) δ j (u 2 ,v 2 ) ∆(x -W u 1 ,v 2 )1 [0,u 1 ] (u 2 )1 [0,v 2 ] (v 1 )
i,j=1,..,d .

Consequently ξ(x) is, for every x ∈ R d , a d × d matrix whose components are distributions in Sobolev-Watanabe spaces. The notation δ i refers to the divergence integral with respect to the component W (i) for every i = 1, .., d.

At this point, we will introduce some notation. Let us denote, for every i, j = 1, .., d, by

∆ i,j (x -W u,v ) := d k=1;k =i,j ∆(x k -W (k) u,v ) ∀u, v ∈ [0, 1].
Therefore, we can formally write

∆ (x -W u,v ) = ∆ i,j (x -W u,v )∆(x i -W (i) u,v )∆(x j -W (j) u,v ). ( 20
)
We have the following result.

Theorem 5 The stochastic current x → ξ(x) introduced in Definition (2) belongs to the Sobolev space H -r (R d ; R d ) for every r > d 2 .

Proof: Let us consider i = j. Using (20) and the general chaos expansion formula [START_REF] Imkeller | Stochastic analysis and local time for (N:d) -Wiener process[END_REF] for the delta Dirac functional, the component ξ i,j can be decomposed as follows

ξ i,j (x) = δ i (u 1 ,v 1 ) δ j (u 2 ,v 2 ) ∆ i,j (x -W u 1 ,v 2 )1 [0,u 1 ] (u 2 )1 [0,v 2 ] (v 1 ) × m i ≥0 a x i m i (u 1 , v 2 )I i m i 1 ⊗m i [0,u 1 ]×[0,v 2 ] m j ≥0 a x j m j (u 1 , v 2 )I j m j 1 ⊗m j [0,u 1 ]×[0,v 2 ] .
Here I i m denotes the multiple integral of order m with respect to the Wiener sheet W (i) . We will get, after symmetrization, for every i, j = 1, .., d, i = j ξ i,j (x) = If i = j it follows from the proof of Theorem 3 that E ξ i,i (x) 2 = 1 4 for every i = 1, .., d. As a consequence

m i ≥0 m j ≥0 1 m i + 1 1 m j + 1 I i m i +1 I j m j +1   m i +1 k=1 m j +1 l=1 a x i m i (a k , d l )a
R d 1 (1 + |x| 2 ) r E ξ(x) 2 dx = d 2 4 R d 1 (1 + |x| 2 ) r dx is finite if and only if r > d 2 .
Remark 5 The 2D-stochastic current keeps the regularity of the one-dimensional counterpart (see [START_REF] Flandoli | Brownian and fractional Brownian stochastic currents via Malliavin calculus[END_REF]).

Remark 6 Let D R d be the space of smooth compact support functions on R d and let D ′ R d be its dual, the space of distributions, endowed with the usual topologies. We denote by S, ϕ the dual pairing between S ∈ D ′ R d and ϕ ∈ D R d . Then the quantity T d , ϕ because a well-defined random variable for every ϕ ∈ D ′ R d . Moreover, the functional ∆(x -W u 1 ,v 2 ) can be interpreted as a random distribution as described in Section 3 of [START_REF] Flandoli | Brownian and fractional Brownian stochastic currents via Malliavin calculus[END_REF].

e -x 2 i a k d l e -x 2 j 1 0i a k d l e -x 2 j a k d l = m i ,m j ≥0 1 +x 2 j+x 2 j )ab e x 2 i ab e x 2 j ab = 1 4

 2121224 x j m j (a k , d l )∆ i,j (x -W a k ,d l )1 ⊗m i [0,a k ]×[0,d l ] (a 1 , b 1 ), ..., (a k , b k ), ..., (a m i +1 , b m i +1 ) 1 [0,a k ] (c l )1 [0,d l ] (b k ) 1 ⊗m j [0,a k ]×[0,d l ] (c 1 , d 1 ), ..., (c l , d l ), ..., (c m j +1 , d m j +1 ) .Let us comment on the above expression. The multiple integral I i m i +1 acts with respect to the variables (a 1 , b 1 ), (a 2 , b 2 ), ...(a m i +1 , b m i +1 ). The multiple integral I j m j +1 acts with a k ] (a 1 , .., a k , .., a m i +1 )1⊗m j +1 [0,a k ] (c 1 , ...., c m j +1 ) 1 m i +1 [0,d l ] (b 1 , ..., b m i +1 )1 m j [0,d l ] (d 1 , .., d l , .., d m j +1 )and taking the square mean, we will obtainE ξ i,j (x) a k d l [0,1] m i +1 da 1 ...da m i +1 [0,1] m i +1 db 1 ...db m i +1 [0,1] m j +1 dc 1 ...dc m j+1 [0,1] m i +1 dd 1 ...dd m j +1 1 ⊗m i [0,a k ] (a 1 , .., âk , ..., a m i +1 )1 ⊗m i +1 [0,d l ] (b 1 , ...., b m i +1 ) 1 ⊗m j +1 [0,a k ] (c 1 , ...., c m j +1 )1 ⊗m j [0,d l ] (d 1 , .., dl , .., d m i +1 )dd l (a k d l ) m i +m j +1 e -x 2 )ab (ab) m i +m j +1 . .

* Dedicated to the memory of Constantin Tudor. † Supported by the CNCS grant PN-II-ID-PCCE-2011-2-0015. Associate member of the team Samos, Université de Panthéon-Sorbonne Paris 1. The author would like to acknowledge generous support from the Alexander von Humboldt Foundation which made possible several research visits at the Humboldt Universität zu Berlin. The support by the ANR grant "Masterie" BLAN 012103 is also acknowledged.

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