P Del Moral

S S Singh

Backward Itô-Ventzell and stochastic interpolation formulae

Keywords: Stochastic flows, variational equations, tangent and Hessian processes, perturbation semigroups, backward Itô-Ventzell formula, Alekseev-Gröbner lemma, Skorohod stochastic integral, two-sided stochastic integration, Malliavin differential, Bismut-Elworthy-Li formulae. Mathematics Subject Classification : 47D07, 93E15, 60H07

We present a novel backward Itô-Ventzell formula and an extension of the Alekseev-Gröbner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between the two stochastic flows when their drift and diffusion functions are not the same, yielding what seems to be the first results of this type for this class of anticipative models. We illustrate the impact of these results in the context of diffusion perturbation theory, interacting diffusions and discrete time approximations.

Introduction

Let b t pxq be a vector-valued function from R d into R d and σ t pxq " rσ t,1 pxq, . . . , σ t,r pxqs be a matrixvalued function from R d into R dˆr , for some parameters d, r ě 1. Both functions will be assumed to be differentiable. Let W t be an r-dimensional Brownian motion and denote by W s,t the σ-field generated by the increments pW u ´Wv q of the Brownian motion, with u, v P rs, ts.

For any time horizon s ě 0 we denote by X s,t pxq the stochastic flow defined for any t P rs, 8r and any starting point X s,s pxq " x P R d by the stochastic differential equation dX s,t pxq " b t pX s,t pxqq dt `σt pX s,t pxqq dW t (1.1)

We assume that x Þ Ñ b t pxq and x Þ Ñ σ t pxq have continuous and uniformly bounded derivatives up to the third order. This condition is clearly met for linear Gaussian models as well as for the geometric Brownian motion. This condition ensures that the stochastic flow x Þ Ñ X s,t pxq is a twice differentiable function of the initialisation x. In addition, all absolute moments of the flow and the ones of its first and second order derivatives exists for any time horizon. As it is well known, dynamical systems and hence stochastic models involving drift functions with quadratic growth require additional regularity conditions to ensure non explosion of the solution in finite time. It is also implicitly assumed that all functions pb t , σ t q are smooth functions w.r.t. the time parameter. The present article develop several constructive and stochastic analysis tools including Bismut-Elworthy-Li formulae, stochastic semigroup perturbation formulae, extended two-sided stochastic integration, Malliavin calculus, gradient and Hessian semigroup processes estimates. We are also looking for useful quantitative and time uniform estimates which are valid under a single set of easily checked conditions that only depend on the parameters of the model. Various techniques presented in the article and many results can be separately and readily extended to more general models with weaker and abstract custom assumptions that depend on the different quantities to handle. Let X s,t pxq be the stochastic flow associated with a stochastic differential equation defined as (1.1) by replacing pb t , σ t q by some drift and diffusion functions pb t , σ t q with the same regularity properties. Constant diffusion functions pσ t , σ t q are defined by σ t pxq " Σ t and σ t pxq " Σ t for some matrices Σ t and Σ t .

(1.2)

In this context, we will assume that Σ t and Σ t are uniformly bounded w.r.t. the time horizon.

The Markov transition semigroups associated with the flows X s,t pxq and X s,t pxq are defined for any measurable function f on R d by the formula P s,t pf qpxq :" E pf pX s,t pxqqq and P s,t pf qpxq :" E `f pX s,t pxqq Ȋn this paper we derive equations for the differences pX s,t ´Xs,t q and pP s,t ´P s,t q in terms of the difference of their corresponding drifts and diffusion functions, ∆a t :" a t ´at ∆b t :" b t ´bt and ∆σ t " σ t ´σt (

where a t pxq :" σ t pxq σ t pxq 1 and a t pxq :" σ t pxq σ 1 t pxq. In some applications the functions b t " b t ´∆b t and σ t " σ t ´∆σ t can be interpreted as a local perturbation of the drift and the diffusion of the stochastic flow X s,t .

We also address the problem of finding time-uniform estimates for the difference between the stochastic flows X s,t and X s,t and their corresponding Markov transition kernels P s,t and P s,t .

These important questions arise in a variety of domains including stochastic perturbation theory as well as in the stability and the qualitative theory of stochastic systems. Classical analytic estimates on the difference between the stochastic flows driven by different drift and diffusion functions are often much too large for most diffusion processes of practical interest. In some instances none of the diffusion flows are stable. In this context, any local perturbation of the stochastic model propagates so that any global error estimate eventually tends to 8 as the time horizon t Ñ 8.

Whenever one of the stochastic flows is stable, classical perturbation bounds combining Lipschitz type inequalities with Gronwall lemma [START_REF] Bellman | Stability Theory of Differential Equations[END_REF][START_REF] Gronwall | Note on the derivatives with respect to a parameter of the solutions of a system of differential equations[END_REF] yield exceedingly pessimistic global estimates that grows exponentially fast w.r.t. the time horizon. Notice that an exponential type estimate of the form e λt for some parameter λ ą 0 and some time horizon t s.t. λ t ě 199 would induce an error bound larger than the estimated number 10 86 of elementary particles of matters in the visible universe. As mentioned in [START_REF] Iserles | Global bounds on numerical error for ordinary differential equations[END_REF] in the context of Euler scheme type approximations of deterministic dynamical systems, one may encounter situations where λ " 10 8 and t " 10 2 and the resulting exponential bounds are clearly impractical from a numerical perspective.

The statement of the main results of the article are presented in section 1.1: i. Section 1.1.1 presents a novel generalized backward Itô-Ventzell formula (cf. theorem 1.1). The Itô-Ventzell is a very important formula, arguably as useful as the Itô's change of variable, but surprisingly the backward Itô-Ventzell presented in this work has never been studied before. Theorem 1.1 can be seen as a new generalized backward version of the generalized Itô-Ventzell formula presented in [START_REF] Ocone | A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations[END_REF].

ii. In section 1.1.2 we apply the backward Itô-Ventzell formula to derive a forward-backward stochastic perturbation formula that expresses the difference between the stochastic flows X s,t and X s,t in terms of first and second order derivatives of the flows, which we call the tangent and Hessian processes respectively, with respect to the space parameter (cf. theorem 1.2).

iii. Section 1.1.2 also provides a novel forward-backward Itô type differential formula for interpolating stochastic diffusion flows (cf. the change of variable formula (1.9)).

iv. In the beginning of section 1.1.2 we present a discrete time approach based on the pivotal interpolating telescoping sum formula (4.2). This interpolating stochastic semigroup technique can be seen as an extension to stochastic flows of the stochastic perturbation analysis developed in [START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF][START_REF] Moral | Feynman-Kac formulae. Genealogical and interacting particle systems with applications[END_REF][START_REF] Del Moral | On the stability of measure valued processes with applications to filtering[END_REF][START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF] and in [START_REF] Arnaudon | A variational approach to nonlinear and interacting diffusions[END_REF][START_REF] Arnaudon | A second order analysis of McKean-Vlasov semigroups[END_REF][START_REF] Bishop | A perturbation analysis of stochastic matrix Riccati diffusions[END_REF] in the context of discrete time models, matrix and nonlinear interacting processes (see also [START_REF] Arnaudon | A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces[END_REF][START_REF] Arnaudon | A second order analysis of McKean-Vlasov semigroups[END_REF]). For a more thorough discussion on these models, we refer to section 1.2. This approach allows to derive a stochastic interpolation formula (1.10) with a fluctuation term (1.12) defined by an extended two-sided stochastic integral.

v. Section 1.1.3 presents some natural spectral conditions on the gradients of b t pxq, σ t pxq, b t pxq and σ t pxq that allows us to derive in a direct way a series of realistic uniform estimates with respect to the time horizon.

The rest of the article is organized as follows: Section 3 provides some basic tools associated with the first and second variational equations associated with a diffusion flow. We also present some quantitative estimates of the tangent and the Hessian processes. For a more thorough discussion on stochastic flows and their differentiability properties we refer to [START_REF] Carverhill | Flows of Stochastic Dynamical Systems: The Functional Analytic Approach[END_REF][START_REF] Kunita | Lectures on stochastic flows and applications[END_REF][START_REF] Norris | Simplified Malliavin calculus[END_REF].

Section 4 is mainly concerned with the forward-backward stochastic interpolation formula (1.10) stated in theorem 1.2. Two approaches are presented: The first one discussed in section 4.1 is based on an extension of the two-sided stochastic calculus introduced by Pardoux and Protter in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] to stochastic interpolation flows. The second one discussed in section 4.2 is based on the generalized backward Itô-Ventzell formula. This section also discusses a multivariate Skorohod-Alekseev-Gröbner formula. Apart from more complex and sophisticated tensor notation, the quantitative stochastic analysis of these multivariate formulae follows the same arguments as the ones used in the proof of theorem 1.3. Thus, we have chosen to concentrate this introduction on stochastic flows. Some extensions of the stochastic interpolation formula (1.10) are discussed in section 4.4. Section 5 is dedicated to the analysis of the Skorohod fluctuation process introduced in (1.12). Section 6 is dedicated to the analysis of an extended version of two-sided stochastic integrals and a generalized backward Itô-Ventzell formula.

Section 7 presents some illustrations of the forward-backward interpolation formulae discussed in the present article in the context of diffusion perturbation theory, interacting diffusions and discrete time approximations.

The technical proofs of some results are housed in the appendix.

1.1 Statement of some main results

A backward Itô-Ventzell formula

We represent the gradient of a real valued function of several variables as a column vector while the gradient and the Hessian of a (column) vector valued function as tensors of type p1, 1q and p2, 1q, see for instance (2.2) and (2.3); in more layman terms a p1, 1q tensor is a matrix while the p2, 1q tensor can be visualized as a "row of matrices" rA 1 , . . . , A n s where the entries A i are matrices of a common dimension. We also use the tensor product and the transpose operator defined in (2.1), see also (2.4). We denote by D t the Malliavin derivative from some dense domain D 2,1 Ă L 2 pΩq into the space L 2 pΩ ˆR`; R r q. For multivariate d-column vector random variables F with entries F j , we use the same rules as for the gradient and D t F is the pr, pq-matrix with entries pD t F q i,j :" D i t F j . For pp ˆqq-matrices F with entries F j k we let D t F be the tensor with entries pD t F q i,j,k " D i t F j k . For a more thorough discussion on Malliavin derivatives and Skorohod integration we refer to section 2.3.

Let F be some function from R p into R q , and let y P R p be some given state, for some p, q ě 1. Suppose we are given a forward p-dimensional continuous semi-martingale Y s,t and a backward random field F s,t from R p into R q with a column-vector type canonical representation of the following form: pH 3 q The processes B s,u , Σ s,u as well as D v Σ s,u are continuous and have moments of any order.

$ ' ' & ' ' % Y s,t " y `ż t s B s,u du `ż t s Σ s,u dW u F s,t pxq " F pxq `ż t s G u,
In this notation, the first main result of this article is the following theorem.

Theorem 1.1. Assume conditions pH i q i"1,2,3 are satisfied. In this situation, for any s ď u ď v ď t we have the generalized backward Itô-Ventzell formula

F v,t pY s,v q ´Fu,t pY s,u q " ż v u p∇F r,t pY s,r q 1 B s,r `1 2 ∇ 2 F r,t pY s,r q 1 Σ s,r Σ 1 s,r ´Gr,t pY s,r qq dr `ż v u
`∇F r,t pY s,r q 1 Σ s,r ´Hr,t pY s,r q ˘dW r (1.5)

The stochastic anticipating integral in the r.h.s. of 1.5 is understood as a Skorohod stochastic integral.

The above theorem can be seen as the backward version of the generalized Itô-Ventzell formula presented in [START_REF] Ocone | A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations[END_REF][START_REF] Pardoux | Applications of anticipating stochastic calculus to stochastic differential equations[END_REF]. The proof of the above theorem is provided in section 6.2 (see theorem 6.3).

Conventional forward and backward Itô stochastic integrals are particular instances of the twosided stochastic integrals introduced by Pardoux and Protter in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF]. The terminology " two-sided " coined by the authors in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] comes from the fact that the integrand of the Skorohod integral depend on the past as well as on the future of the history generated by the Brownian motion.

The stochastic anticipating integral in the r.h.s. of (1.5) involves a backward random field and a forward semimartingale, thus it is tempting to interpret this integral as a two sided integral. Unfortunately, this class of integrands are not considered in the construction of the two-sided stochastic integrals defined in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF]. In section 4.1 and section 6.1 we shall present an extended version of the two-sided stochastic integrals introduced in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] that applies to integrands defined as a compositions of backward and forward stochastic flows. This extended version applies to backward stochastic flows but it doesn't encapsulate more general backward random fields. We believe more general extensions of the two-sided integrals can be developed but it is out of the scope of this article to develop a theory on generalized two-sided stochastic integrals. We finally mention that all two-sided stochastic integrals discussed in this article are particular instances of Skorohod integrals

A stochastic flow interpolation formula

The diffusion flow (1.1) is defined in term of a column vector with twice continuously differentiable entries. For h » 0 we use the backward approximation: X s,t pxq ´Xs´h,t pxq " X s,t pxq ´pX s,t ˝Xs´h,s qpxq » X s,t pxq ´Xs,t px `bs pxq h `σs pxq pW s ´Ws´h qq »

´"ˆ∇ X s,t pxq 1 b s pxq `1 2 ∇ 2 X s,t pxq 1 a s pxq ˙h `∇X s,t pxq 1 σ s pxq pW s ´Ws´h q

 (1.6)
In the above display, X s,t ˝Xs´h,s stands for the composition of the mappings X s,t and X s´h,s . The above approximations are rigorously justified in section 4.1 and lead to the backward stochastic flow evolution equation:

d s X s,t pxq "
´"`∇ X s,t pxq 1 b s pxq `1 2 ∇ 2 X s,t pxq 1 a s pxq ˘ds `∇X s,t pxq 1 σ s pxq dW s ‰ (1.7)

In the above display, d s X i s,t pxq represents the change in X i s,t pxq w.r.t. the variable s. In the same vein, for any s ă u ă t we have the interpolating semigroup decompositions X u`h,t ˝Xs,u`h ´Xu,t ˝Xs,u " pX u`h,t ´Xu,t q ˝Xs,u ``X u`h,t ˝Xs,u`h ´Xu`h,t ˝Xs,u ȃs well as the forward approximations X u`h,t `Xs,u pxq ``X s,u`h pxq ´Xs,u pxq ˘˘´X u`h,t pX s,u pxqq » p∇X u`h,t q pX s,u pxqq 1 pX s,u`h pxq ´Xs,u pxqq `1 2 `∇2 X u`h,t ˘pX s,u pxqq 1 a u pX s,u pxqq h

(1.8)
The above approximations are rigorously justified in section 4.1 and lead to the forward-backward stochastic interpolation equation d u `Xu,t ˝Xs,u ˘pxq " pd u X u,t q pX s,u pxqq `p∇X u,t q pX s,u pxqq 1 d u X s,u pxq `1 2 `∇2 X u,t ˘pX s,u pxqq 1 a u pX s,u pxqq du

(1.9)
The discrete time version of the forward-backward stochastic formula in the above display reduces to the telescoping sum formula (4.2) and the second order Taylor expansions discussed in section 4.1. We already mention that (4.2) can be interpreted as a discrete time version of the Alekseev-Gröbner lemma [START_REF] Alekseev | An estimate for the perturbations of the solution of ordinary differential equations[END_REF][START_REF] Gröbner | Die Lie-Reihen und ihre Anwendungen[END_REF]. The terminology forward-backward comes from the forward and backward nature of (1.9) and the telescoping sum formula (4.2).

Also notice that (1.7) can also be deduced formally from (1.9) by replacing X s,u by the stochastic flow X s,u in (1.9), and then letting s " u.

This yields the following interpolation theorem.

Theorem 1.2. We have the forward-backward stochastic interpolation formula X s,t pxq ´Xs,t pxq " T s,t p∆a, ∆bqpxq `Ss,t p∆σqpxq (1.10)

with the stochastic process T s,t p∆a, ∆bqpxq :"

ż t s " p∇X u,t q pX s,u pxqq 1 ∆b u pX s,u pxqq `1 2 `∇2 X u,t ˘pX s,u pxqq 1 ∆a u pX s,u pxqq  du (1.11)
and the fluctuation term given by the Skorohod stochastic integral S s,t p∆σqpxq :"

ż t s p∇X u,t q pX s,u pxqq 1 ∆σ u pX s,u pxqq dW u (1.12)
The fluctuation term in the above display can also be seen as the extended two-sided stochastic integral defined in (4.3) (see also proposition 6.2).

These interpolation formulae combine the backward evolution (1.7) with the conventional forward evolution of the perturbed flow.

The proof of the interpolation formula (1.10) is provided in section 4.

We will present two different approaches: The first one presented in section 4.1 is rather elementary and very intuitive. It combines the conventional Itô-type discrete time approximations of stochastic integrals discussed above with the two-sided stochastic integration calculus introduced in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF]. Using this approximation technique the fluctuation term is defined by the extended twosided stochastic integral defined in (4.3). In this interpretation, the equation (1.10) can be seen as an extended version of the Itô-type change rule formula stated in theorem 6.1 in the article [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] to the interpolating flow

Z s,t : u P rs, ts Þ Ñ Z s,t u :" X u,t ˝Xs,u ùñ Z s,t s ´Zs,t t " X s,t ´Xs,t (1.13)
Roughly speaking, the increments of the interpolating path are decomposed into two parts: One comes from the backward increments of the flow u Þ Ñ X u,t given the past values of the stochastic flow X s,u . The other one comes from the conventional Itô increments of u Þ Ñ X s,u given the future values of the stochastic flow X u,t .

The second approach discussed in section 4.2 is based on the generalized backward Itô-Ventzell formula stated in theorem 1.1. More precisely we also recover (1.10) from (1.5) by choosing pF s,t pxq, Y s,t pyqq " pX s,t pxq, X s,t pyqq pB s,t , Σ s,t q " `bt `Xs,t pxq ˘, σ t `Xs,t pxq ˘G u,t pxq " ∇F u,t pxq 1 b u pxq `1 2 ∇ 2 F u,t pxq 1 a u pxq and H u,t pxq " ∇F u,t pxq 1 σ u pxq and letting pu, vq " ps, tq in (1.5). The regularity conditions on the drift and the diffusion function ensure that conditions pH i q i with i " 1, 2, 3 stated in section 1.1.1 are satisfied. We emphasize that the backward diffusion flow discussed in (1.7) and (4.1) is essential to apply theorem 1.1. Section 4.2 also provides a multivariate version of (1.10).

The interpolation formula (1.10) with a fluctuation term given by the Skorohod stochastic integral (1.12) can be seen as a Alekseev-Gröbner formula of Skorohod type.

In this context, the integrability of the fluctuation term and any quantitative type estimates require a refined analysis of the Malliavin derivatives of the integrand. Under our regularity conditions the stochastic flows X s,t pxq and X s,t pxq are Holder-continuous w.r.t. the time parameters as well as twice differentiable w.r.t. the space variables, with almost sure uniformly bounded first and second order derivatives. In addition, for any n ě 1 all the n-absolute moments of the stochastic flows are finite with at most linear growth w.r.t. the initial values. These properties ensure that the Skorohod stochastic integral (1.12) is well defined and they allow to derive several quantitative estimates. Section 5 provides a refined of the fluctuation term; see for instance theorem 5.2.

When σ t " 0 the flow X s,t pxq is deterministic so that the Skorohod fluctuation term (1.12) reduces to the traditional Itô stochastic integral. In this context, quantitative estimates of the fluctuation term are obtained combining Burkholder-Davis-Gundy inequalities with the generalized Minkowski inequality. The resulting interpolation formula (1.10) can be seen as a Alekseev-Gröbner formula of Itô-type.

To distinguish these two classes of models, the interpolation formulae (1.10) associated with the case σ t " 0 will be called an Itô-Alekseev-Gröbner formula; the one associated with the case ∆σ t " 0 will be called a Skorohod-Alekseev-Gröbner formula.

Uniform estimates w.r.t. the time horizon

The final objective of this article is to derive uniform estimates w.r.t. the time parameter. Our methodology is mainly based on two different types of regularity conditions to be defined and discussed in detail in section 2.2:

' The first is a technical condition that ensures that the n-absolute moments of the flows X s,t and X s,t are uniformly bounded w.r.t. the time horizon; we call this condition pM q n . ' The second is a spectral condition on the gradient of the drift and diffusion matrices of the stochastic flows, which we call condition pT q n . Without going into details, we state one usual case of interest: for constant diffusion functions (1.2) the spectral condition pT q n is met for any n ě 2 as soon as the following log-norm conditions are met ∇b t `p∇b t q 1 ď ´2λ I and ∇b t `p∇b t q 1 ď ´2λ I for some λ ^λ ą 0,

To motivate the above condition consider a linear drift function of the form b t pxq " B t x and σ " 0. In this case the tangent process ∇X s,t pxq satisfies a time-varying deterministic linear dynamical system B t ∇X s,t pxq " ∇X s,t pxq B 1 t

The asymptotic behavior of this process cannot be characterized by the statistical properties of the spectral abscissa of the matrices B t . Indeed, unstable semigroups associated with time-varying (deterministic) matrices B t with negative eigenvalues are exemplified in [START_REF] Coppel | Dichotomies in Stability Theory[END_REF][START_REF] Wu | A note on stability of linear time-varying systems[END_REF]. Conversely, stable semigroups with B t having positive eigenvalues are given by Wu in [START_REF] Wu | A note on stability of linear time-varying systems[END_REF]. In contrast, the uniform log-norm condition (1.14) provides a readily verifiable condition.

To describe with some precision the second main result of the article, we need to introduce some additional terminology. When there is no ambiguity, we denote by }.} any (equivalent) norm on some finite dimensional vector space. For some multivariate function f t pxq, for pt, xq P r0, 8q ˆRd , let }f pxq} :" sup t }f t pxq} and the uniform norm be }f } :" sup t,x }f t pxq}. For any n ě 1 we also set |||f pxq||| n :" sup sě0 sup těs E `}f t pX s,t pxqq} n ˘1{n

(1.15)

We denote by κ n and κ δ,n some constants that depend on some parameters n and pδ, nq but do not depend on the time horizon, nor on the space variable. In this notation, the second main result of the article takes basically the following form.

Theorem 1.3. Assume conditions pM q 2n{δ and pT q 2n{p1´δq are satisfied for some parameters n ě 2 and δ Ps0, 1r. In this situation, we have the time-uniform estimates The estimates (1.16) come from (7.5) and (5.9). A more detailed proof is provided in the appendix, on page 51. The estimates (1.17) are direct consequences of (2.17) and (5.11).

When σ t " σ t the Skorohod term is indeed absent and (1.10) reduces to X s,t pxq ´Xs,t pxq " ż t s p∇X u,t q pX s,u pxqq 1 ∆b u pX s,u pxqq du (1.18)

We recover the interpolation formula for nonlinear stochastic flows presented in section 3.1 in the article [START_REF] Arnaudon | A variational approach to nonlinear and interacting diffusions[END_REF]. In this context the analysis of L n -errors will proceed via two-step procedure. In section 3.1 we will derive the exponential bound sup x Ep} p∇X u,t q pxq} n 2 q 1{n ď κ n expp´λpnq pt ´uqq for some λpnq ą 0

Using the Minkowski integral inequality in (1.18) where p m n pxq ď κ n p1 `}x}q.

Proposition 1.4 is proved in section 7.3. To apply proposition 1.4 to a Langevin diffusion with a convex potential U pxq, the drift would be b t pxq " ´∇U pxq and the corresponding assumptions on U pxq are typical.

Comments and comparisons with existing literature

The interpolation formula (1.10) can be interpreted as an extension of Alekseev-Gröbner lemma [START_REF] Alekseev | An estimate for the perturbations of the solution of ordinary differential equations[END_REF][START_REF] Gröbner | Die Lie-Reihen und ihre Anwendungen[END_REF][START_REF] Jentzen | On the Alekseev-Gröbner formula in Banach spaces[END_REF] as well as an extended version of the variation-of-constant and related Gronwall type lemma [START_REF] Bellman | Stability Theory of Differential Equations[END_REF][START_REF] Gronwall | Note on the derivatives with respect to a parameter of the solutions of a system of differential equations[END_REF] to diffusion processes. In this connection we underline that the forward-backward formula (1.10) differs from the stochastic Gronwall lemma presented in [START_REF] Scheutzow | A stochastic Gronwall lemma[END_REF] based on particular classes of stochastic linear inequalities that doesn't involve Skorohod type integrals.

The forward-backward interpolation formula (1.10) can also be seen as an extension of theorem 6.1 in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] on two-sided stochastic integrals to diffusion flows. This interpolation formula can also be interpreted as a backward version of the generalized Itô-Ventzell formula presented in [START_REF] Ocone | A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations[END_REF] (see also theorem 3.2.11 in [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]).

Stochastic interpolation formulae of the form (1.10) and their discrete time version discussed in (4.2) are not really new. To describe their origins, it is worth to mention that the stochastic perturbations may come from auxiliary random sources, uncertainty propagations, as well as time discretization schemes and mean field type particle fluctuations.

The pivotal interpolating telescoping sum formula (4.2) and the second order forward-backward perturbation semigroup methodology discussed in the present article can also be found in chapter 7 in [START_REF] Moral | Feynman-Kac formulae. Genealogical and interacting particle systems with applications[END_REF] for discrete time models as well as in the series of articles [START_REF] Del Moral | On the stability of measure valued processes with applications to filtering[END_REF][START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF][START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF] published at the beginning of the 2000s, see also chapter 10 in [START_REF] Moral | Mean field simulation for Monte Carlo integration[END_REF]. In this context, the random perturbations come from the fluctuations of a genetic type particle interpretation of nonlinear Feynman-Kac semigroups.

The more recent articles [START_REF] Bishop | On the Stability of Matrix-Valued Riccati Diffusions[END_REF][START_REF] Bishop | Perturbations and projections of Kalman-Bucy semigroups[END_REF][START_REF] Bishop | A perturbation analysis of stochastic matrix Riccati diffusions[END_REF] also provide a series of backward-forward interpolation formulae of the same form as (1.10) for stochastic matrix Riccati diffusion flows arising in data assimilation theory (cf. for instance theorem 1.3 in [START_REF] Bishop | A perturbation analysis of stochastic matrix Riccati diffusions[END_REF] as well as section 2.2 in [START_REF] Bishop | Perturbations and projections of Kalman-Bucy semigroups[END_REF] and the proof of theorem 2.3 in [START_REF] Bishop | On the Stability of Matrix-Valued Riccati Diffusions[END_REF]). In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional.

We underline that the Itô-Alekseev-Gröbner formula (4.6) discussed in [START_REF] Bishop | A perturbation analysis of stochastic matrix Riccati diffusions[END_REF] is an extension of the interpolation formula (1.10) to stochastic diffusion flows in matrix spaces. In this context the unperturbed model is given by the flow of a deterministic matrix Riccati differential equation and the random perturbations are described by matrix-valued diffusion martingales. The corresponding Itô-Alekseev-Gröbner formulae can be seen as a matrix version of theorem 1.2 in the present article when σ " 0. These stochastic interpolation formulae were used in [START_REF] Bishop | A perturbation analysis of stochastic matrix Riccati diffusions[END_REF] to quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. We will briefly discuss the analog of these Taylor type expansions in section 7.1 in the context of Euclidian diffusions.

The forward-backward perturbation methodology discussed in the present article has also been used in [START_REF] Arnaudon | A variational approach to nonlinear and interacting diffusions[END_REF][START_REF] Arnaudon | A second order analysis of McKean-Vlasov semigroups[END_REF] in the context of nonlinear diffusions and their mean field type interacting particle interpretations, see for instance section 2.3 in [START_REF] Arnaudon | A second order analysis of McKean-Vlasov semigroups[END_REF]. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions. The extended version of the Itô-Alekseev-Gröbner formula (1.18) to nonlinear diffusions is also discussed in section 3.1 in the article [START_REF] Arnaudon | A variational approach to nonlinear and interacting diffusions[END_REF]. In this situation, the time varying drift and diffusion functions of the stochastic flows depend on some possibly different nonlinear measure valued semigroups which may start from two possibly different initial distributions. For a more thorough discussion on this class of nonlinear diffusions, we refer to the Itô-Alekseev-Gröbner formula (3.2) and corollary 3.2 in the article [START_REF] Arnaudon | A variational approach to nonlinear and interacting diffusions[END_REF]. These Itô-Alekseev-Gröbner formulae correspond to theorem 1.2 in the present article when σ " 0.

The interpolating stochastic semigroup techniques discussed in the present article are also applied to mean field particle systems and deterministic nonlinear measure valued semigroups. In this context, the process X s,t is given a deterministic measure-valued process and X s,t represents the evolution of the particle density profiles associated with an approximating mean field particle interpretation of X s,t . For instance, the article [START_REF] Arnaudon | A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces[END_REF] is concerned with interacting jumps models on path spaces, the second article [START_REF] Arnaudon | A second order analysis of McKean-Vlasov semigroups[END_REF] discusses the propagation of chaos properties of mean field type interacting diffusions. The stochastic interpolation formulae discussed in [START_REF] Arnaudon | A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces[END_REF][START_REF] Arnaudon | A second order analysis of McKean-Vlasov semigroups[END_REF] correspond to the case (1.10) with σ " 0 and or σ " σ (see for instance the interpolation formula (3.5), theorem 2.6, theorem 2.7 and the interpolating telescoping sum in section 1.2 in [START_REF] Arnaudon | A second order analysis of McKean-Vlasov semigroups[END_REF])

In the series of articles discussed above, as in (1.9) the central common idea is to analyse the evolution of the interpolating process (1.13) between a given process X s,t and some stochastic flow X s,t with an extra level of randomness. In discrete time settings, the differential interpolation formula (1.9) can also recasted in terms of a telescoping sum of the same form as (4.2) combined with a second order Taylor expansion reflecting the differences between a stochastic semigroup and its perturbations, see for instance chapter 7 in [START_REF] Moral | Feynman-Kac formulae. Genealogical and interacting particle systems with applications[END_REF].

In most of the application domains discussed above, this second order stochastic perturbation methodology has been developed to quantify uniformly w.r.t. the time horizon the propagations of some stochastic perturbations entering in some deterministic and stable reference or unperturbed process. In the context of Euclidian diffusions, this corresponds to the situation where the diffusion function σ " 0 (the case σ " 0 can be treated by symmetry arguments). The Itô-Alekseev-Gröbner type formulae discussed in section 3.1 in the article [START_REF] Arnaudon | A variational approach to nonlinear and interacting diffusions[END_REF] correspond to theorem 1.2 in the present article when σ " σ.

The present article can be seen as a natural extension of the second order perturbation methodology developed in the above referenced articles to diffusion type perturbed processes when σ " σ.

To the best of our knowledge, the first article considering the case σ " σ with σ " 0 and σ " 0 is the independent work of Hudde-Hutzenthaler-Jentzen-Mazzonetto [START_REF] Hudde | On the Itô-Alekseev-Gröbner formula for stochastic differential equations[END_REF]. In this article, the authors discuss an Itô-Alekseev-Gröbner formula for abstract diffusion perturbation models of the form (4.11). Here again, as in the list of referenced articles discussed above, the common central idea is to use discrete time approximations and combine the pivotal interpolating telescoping sum formulae (4.2) with a second order Taylor expansion. Besides this fact and in contrast with our analysis, the fluctuation term (1.12) discussed in [START_REF] Hudde | On the Itô-Alekseev-Gröbner formula for stochastic differential equations[END_REF] cannot be interpreted in terms of the extended two-sided stochastic integral defined in (4.3) (see also proposition 6.2) but only in terms of a Skorohod stochastic integral. The study [START_REF] Hudde | On the Itô-Alekseev-Gröbner formula for stochastic differential equations[END_REF] is also based on a series of particularly chosen and custom regularity conditions. For instance, the authors assume that the abstract diffusion perturbation models are chosen so that the Skorohod fluctuation term exists without providing any quantitative type estimate. This work is also not connected to the two-sided stochastic integration calculus developed by Pardoux and Protter in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] nor to any type of backward Itô-Ventzell formula.

We feel that our approach is more direct and intuitive as it relies on an extended version of Itô's change rule formula (1.9) to interpolating stochastic flows. It also allows to interpret the fluctuation term (1.12) as an extended two-sided stochastic integral.

In section 5 in the present article, we will also see that any quantitative analysis requires to estimate the absolute moments of the Malliavin derivatives of the stochastic integrands of the Brownian motion arising in the Skorohod fluctuation term. In our framework, these Malliavin derivatives depend on the gradient of both of the diffusion functions pσ, σq as well as on the tangent process of the perturbed diffusion flow. The quantitative analysis developed in 5 can be extended without difficulties to abstract diffusion perturbation models satisfying appropriate differentiability and integrability conditions.

The article [START_REF] Hudde | On the Itô-Alekseev-Gröbner formula for stochastic differential equations[END_REF] also presents an application to tamed Euler type discrete time approximations of a stochastic van-der-Pol process introduced in [START_REF] Timmer | Pathological tremors: Deterministic chaos or nonlinear stochastic oscillators[END_REF], simplifying the analysis provided in an earlier work [START_REF] Hutzenthaler | On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients[END_REF]. In this situation, we underline that the Skorohod fluctuation term is null so that the resulting Alekseev-Gröbner type formula resumes to the simple and elementary case discussed in (1.18) and in the article [START_REF] Arnaudon | A variational approach to nonlinear and interacting diffusions[END_REF]. As expected for this class of "unstable processes", the authors recast a series of L 2 -estimates discussed in [START_REF] Hutzenthaler | On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients[END_REF] into a series of estimates that grow exponentially fast with respect to the time horizon.

In contrast with the present work, the above article doesn't discuss any quantitative uniform estimates w.r.t. the time horizon. The analysis presented in [START_REF] Hudde | On the Itô-Alekseev-Gröbner formula for stochastic differential equations[END_REF] is mainly concerned with the proof of a Skorohod-Alekseev-Gröbner type formula for abstract diffusion perturbation models and it doesn't apply to derive any type of estimates to general diffusion perturbation models without adding regularity conditions.

Besides its elegance the forward-backward interpolation formula (1.10) is clearly of rather poor mathematical and numerical interest without a better understanding of the variational processes and the Skorohod fluctuation term (1.12). A crucial problem is to avoid exceedingly pessimistic exponential estimates that grow exponentially fast w.r.t. the time horizon.

One advantage of the second order perturbation methodology developed in the present article is that it takes advantage of the stability properties of the tangent and the Hessian flow in the estimation of Skorohod fluctuation term and this sharpen analysis of the difference between stochastic flows. Our main contribution is to develop a refined analysis of these variational processes and the Skorohod fluctuation terms. We also deduce several uniform perturbation propagation estimates with respect to the time horizon, yielding what seems to be the first results of this type for this class of models.

The forward-backward stochastic interpolation formula (1.10) can also be extended to more general classes of stochastic flows on abstract state spaces. For instance the recent article [START_REF] Jentzen | On the Alekseev-Gröbner formula in Banach spaces[END_REF] provides a deterministic first order version of (1.10) on abstract Banach spaces. The stochastic perturbation analysis developed in the series of articles [START_REF] Arnaudon | A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces[END_REF][START_REF] Arnaudon | A second order analysis of McKean-Vlasov semigroups[END_REF][START_REF] Bishop | On the Stability of Matrix-Valued Riccati Diffusions[END_REF][START_REF] Bishop | Perturbations and projections of Kalman-Bucy semigroups[END_REF][START_REF] Bishop | A perturbation analysis of stochastic matrix Riccati diffusions[END_REF][START_REF] Del Moral | On the stability of measure valued processes with applications to filtering[END_REF][START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF][START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF] and the books [START_REF] Moral | Feynman-Kac formulae. Genealogical and interacting particle systems with applications[END_REF][START_REF] Moral | Mean field simulation for Monte Carlo integration[END_REF] is applied to matrix-valued diffusions and measure valued processes, including mean field type interacting diffusions and Feynman-Kac type interacting jumps models.

The stability properties of these abstract models discussed above depend on the problem at hand. To focus on the main ideas without clouding the article with unnecessary technical details and sophisticated mathematical tools based on abstract ad hoc regularity conditions we have chosen to concentrate the article on diffusion flows on Euclidian spaces with simple and easily checked regularity conditions.

Preliminary results

Some basic notation

With a slight abuse of notation, we denote by I the identity pd ˆdq-matrix, for any d ě 1. We also denote by }.} any (equivalent) norm on a finite dimensional vector space over R. All vectors are column vectors by default.

We introduce some matrix notation needed from the onset. We denote by TrpAq, }A} 2 :" λ max pAA 1 q 1{2 " λ max pA 1 Aq 1{2 , resp. }A} F " TrpAA 1 q 1{2 and ρpAq " λ max ppA `A1 q{2q the trace, the spectral norm, the Frobenius norm, and the logarithmic norm of some matrix A. A 1 is the transpose of A and λ max p.q the largest eigenvalue. The spectral norm is sub-multiplicative or }AB} 2 ď }A} 2 }B} 2 and compatible with the Euclidean norm for vectors, by that we mean for a vector x we have }Ax} ď }A} 2 }x}.

Let rns be the set of n multiple indexes i " pi 1 , . . . , i n q P I n over some finite set I. We denote by pA i,j q pi,jqPrpsˆrqs the entries of a pp, qq-tensor A with index set I for rps and J for rqs. For the sake of brevity, the index sets will be implicitly defined through the context.

For a given pp 1 , qq-tensor A and a given pq, p 2 q tensor B, AB and B 1 is a pp 1 , p 2 q-tensor resp. a pp 2 , qq-tensor with entries given by @pi, jq P rp 1 s ˆrp 2 s pABq i,j "

ÿ kPrqs A i,k B k,j and B 1 j,k :" B k,j . (2.1)
The symmetric part A sym of a pp, pq-tensor is the pp, pq-tensor A sym with entries @pi, jq P rps ˆrps pA sym q i,j " pA i,j `Aj,i q{2

We consider the Frobenius inner product given for any pp, qq-tensors A and B by xA, By F " TrpAB 1 q " ÿ i pAB 1 q i,i and the norm }A} F "

a TrpAA 1 q
For any pp, qq-tensors A and B we also check the Cauchy-Schwartz inequality

xA, By 2 F ď }A} F }B} F and }A} 2 ď }A} F ď CardpIq p }A} 2 with }A} 2 :" λ max pAA 1 q 1{2
For any tensors A, B with appropriate dimensions we have the inequality

}AB} F ď }A} F }B} F
Given some tensor valued function T : pt, xq Þ Ñ T t pxq we also set }T } F :" sup

}T t pxq}

Given some smooth function hpxq from R p into R q we denote by ∇h " " ∇h 1 , . . . , ∇h q ‰ with ∇h i " » --

B x 1 h i . . . B xp h i fi ffi fl (2.2)
the gradient pp, qq-matrix associated with the column vector-valued function h " ph i q 1ďiďq . Building on this notation: let b : R n Ñ R p and let the mapping x Ñ Gpxq " hpbpxqq. Then ∇Gpxq " ∇bpxq ˆ∇hpbpxqq. Let

∇ 2 h " " ∇ 2 h 1 , . . . , ∇ 2 h q ‰ with ∇ 2 h i " » - - B x 1 ,x 1 h i . . . B x 1 ,xp h i . . . . . . . . . B xp,x 1 h i . . . B xp,xp h i fi ffi fl (2.3)
The Hessian H " ∇ 2 h associated with the function h " ph i q 1ďiďq is a p2, 1q-tensor where H pi,jq,k " p∇ 2 h k q i,j " B x i ,x j h k . In this notation we can compactly represent the second order term of the Taylor expansion of the the vector valued function h. For a vector y " py 1 , . . . , y p q 1 » --

y 1 ∇ 2 h 1 pxq y . . . y 1 ∇ 2 h q pxq y fi ffi fl " ∇ 2 hpxq 1 yy 1
where we have regarded the matrix yy 1 as the p2, 1q-tensor Y with Y pi,jq,1 " y i y j .

In the same vein, in terms of the tensor product (2.1), for any pair of column vector-valued function h " ph k q 1ďkďq and b " pb i q 1ďiďp and any matrix function a " pa i,j q 1ďi,jďp from R p into R q , for any parameter 1 ď k ď q we also have

`∇hpxq 1 bpxq ˘k " ÿ 1ďiďp p∇hpxqq 1 k,i b i pxq " ÿ 1ďjďp B x i h k pxq b i pxq " x∇h k pxq, bpxqy `∇2 hpxq 1 apxq ˘k " ÿ 1ďi,jďp p∇ 2 hpxqq 1 k,pi,jq a i,j pxq " ÿ 1ďi,jďp B x i ,x j h k pxq a i,j pxq " x∇ 2 h k pxq, apxqy F
In a more compact form, the above formula takes the form

∇hpxq 1 bpxq " » - - x∇h 1 pxq, bpxqy . . . x∇h q pxq, bpxqy fi ffi fl and ∇ 2 hpxq 1 apxq " » - - x∇ 2 h 1 pxq, apxqy F . . . x∇ 2 h q pxq, apxqy F fi ffi fl (2.4)
For any n ě 1 we let P n pR d q be the convex set of probability measures µ 1 , µ 2 on R d with absolute n-th moment and equipped with the Wasserstein distance of order n denoted by

W n pµ 1 , µ 2 q :" inf Ep}X 1 ´X2 } n q 1{n
In the above display the infimum is taken over all pair or random variables pX 1 , X 2 q with marginal distributions pµ 1 , µ 2 q. The stochastic transition semigroups associated with the flows X s,t pxq and X s,t pxq are defined for any measurable function f on R d by the formulae P s,t pf qpxq :" f pX s,t pxqq and P s,t pf qpxq :" f pX s,t pxqq Given some column vector-valued function f " pf i q 1ďiďp , let P s,t pf q and P s,t pf q denote the column vector-valued functions with entries P s,t pf i q and P s,t pf i q. Building on the tensor notation, let P s,t p∇f q and P s,t p∇ 2 f q respectively denote the p1, 1q and p2, 1q-tensor valued functions with entries P s,t p∇f qpxq i,k :" P s,t pB x i f k qpxq and P s,t p∇ 2 f qpxq pi,jq,k :" P s,t pB x i ,x j f k qpxq We also consider the random p2, 1q and p2, 2q-tensors given by Throughout the rest of the article, unless otherwise stated κ, κ , κ n , κ n, denote constants whose values may vary from line to line but only depend on the parameters in their subscripts, i.e. n ě 0 and ą 0, as well as on the parameters of the model; that is, on the drift and diffusion functions. We also use the letters c, c , c n , c n, to denote universal constants. Importantly these contants do not depend on the time horizon. We also consider the uniform log-norm parameters ρp∇σq 2 :" ÿ 1ďkďr ρp∇σ k q 2 and ρ ‹ p∇σq :" sup 1ďkďr ρp∇σ k q with ρp∇σ k q :" sup t,x ρp∇σ t,k pxqq (2.5) and the parameters χ pb, σq defined by χ pb, σq :" c `}∇ 2 b} `}∇ 2 σ} 2 `ρ‹ p∇σq 2 (2.6)

∇ 2 X s,t pxq pi,jq,k " B x i ,x j X k s,t pxq " " ∇ 2 X s,t pxq ‰ 1 k,pi

Regularity conditions and some preliminary results

We consider two different types of regularity conditions (M) n and pT q n , indexed by some parameter n P r2, 8r, for the diffusion pb t , σ t q.

pMq n There exists some parameter κ n ě 0 such that for any x P R d we have m n pxq :" sup sďt E p}X s,t pxq} n q 1{n ď κ n p1 _ }x}q pT q n There exists some parameter λ A ą 0 such that

A t :" ∇b t `p∇b t q 1 `ÿ 1ďkďr ∇σ k,t p∇σ k,t q 1 ď ´2λ A I (2.7)
where σ k,t denotes the k-th column of σ t . In addition, the following condition is satisfied

λ A pnq :" λ A ´dpn ´2q 2 ρ ‹ p∇σq 2 ą 0 (2.8)
We now define the corresponding assumptions for the diffusion pb t , σ t q.

pMq n The regularity condition defined as in pMq n for the diffusion pb t , σ t q.

pT q n Let A t be the symmetric matrix defined as A t in (2.7) when pb t , σ t q " pb t , σ t q. Assume there exists some λ A ą 0 such that A t ď ´2λ A I. Furthermore, assume λ A pnq ą 0 where λ A pnq is defined as λ A pnq when pλ A , σ t q " pλ A , σ t q. pM q n We write pM q n when both conditions pMq n and pMq n are satisfied.

pT q n Both conditions pT q n and pT q n are met, and let λ A,A pnq :" λ A pnq ^λA pnq

In practice, the uniform moment condition pMq n is often checked using Lyapunov techniques. For example we can use the following polynomial growth condition.

pPq n There exists some parameters α i , β i ě 0 with i " 0, 1, 2 such that for any t ě 0 and any

x P R d we have

}σ t pxq} 2 F ď α 0 `α1 }x} `α2 }x} 2 and xx, b t pxqy ď β 0 `β1 }x} ´β2 }x} 2 (2.9)
for some norm }σ t pxq} of the matrix-valued diffusion function. In addition, we have

β 2 pnq :" β 2 ´pn ´1q 2 α 2 ą 0 Lemma 2.1.
For any n ě 2 we have

pPq n ùñ pMq n with κ n " 1 `pγ 1 `pn ´2qα 1 q `pγ 0 `pn ´2qα 0 q 1{2 2β 2 pnq 1{2 (2.10)
The proof of the above assertion follows standard stochastic calculations, thus it is housed in the appendix, on page 47.

For one-dimensional geometric Brownian motions the condition pPq n is a sufficient and necessary condition for the existence of uniformly bounded absolute n-moments. In this case pT q n coincides with pPq n by setting λ A " β 2 ´α2 {2 and α 2 " ρ ‹ p∇σq 2

Whenever condition pM q n is met for some n ě 2, we also check the uniform estimates

E `}rX u,t ˝Xs,u spxq} n ˘1{n ď κ n p1 `}x}q (2.11)
with the same parameter κ n as the one associated with the condition pM q n . Recalling that the functions pb t , b t q and pσ t , σ t q have at most linear growth, with the L n -norms |||.||| n introduced in (1.15) we also have that

|||∆bpxq||| n ď κ 1,n p1 _ }x}q and |||∆apxq||| n{2 ď κ 2,n p1 _ }x}q 2 (2.12)
To give more insight where these assumptions will be used, we now briefly state the stability results that stem from them. Condition pT q n ensures that the exponential decays of the absolute and uniform n-moments of the tangent and the Hessian processes; that is, when pT q n is met for some n ě 2 we have that E p}∇X s,t pxq} n q 1{n _ E `}∇ 2 X s,t pxq} n ˘1{n ď κ n e ´λpnqpt´sq for some λpnq ą 0 (2.13)

A more precise statement is provided in proposition 3.2 and proposition 3.10. These uniform estimates clearly imply, via a conditioning argument, that for any n ě 2 and s ď u ď t we have

E `}p∇X u,t qpX s,u pxqq} n ˘1{n _ E `}p∇ 2 X u,t qpX s,u pxqq} n ˘1{n ď κ n e ´λpnqpt´uq (2.14)
with the same parameters pκ n , λpnqq as in (2.13).

The case ∇σ " 0 will also serve a useful purpose, for example in analysing the error of a numerical implementation as in proposition 1.4. For instance whenever pT q 2 is met we have the almost sure and uniform gradient estimates

}∇X s,t } 2 :" sup x }∇X s,t pxq} 2 ď e ´λA pt´sq
(2.15)

In addition, we have the almost sure and uniform Hessian estimates

}∇ 2 X s,t } F :" sup x }∇ 2 X s,t pxq} F ď d λ A }∇ 2 b} F e ´λA pt´sq (2.16)
A proof of the above estimates is provided in the beginning of section 3.1 and section 3.2. In this situation, whenever pT q 2 is met we have E r}T s,t p∆a, ∆bqpxq} n s 1{n ď κ p|||∆bpxq||| n `|||∆apxq||| n q .

(2.17)

In the above display, T s,t p∆a, ∆bqpxq stands for the stochastic process discussed in (1.11), and κ stands for some finite constant that doesn't depend on the parameter n. For instance, for a Langevin diffusion associated with some convex potential function U we have b " ´∇U and ∇σ " 0. Then assuming

∇ 2 U ě λ I ùñ pT q 2 is met ùñ }∇X s,t } 2 ď e ´λpt´sq and }∇ 2 X s,t } F ď d λ }∇ 3 U } F e ´λpt´sq (2.18)
where the almost sure tangent and Hessian bounds follow from (2.15) and (2.16) respectively. In practice, it is often easier to work with a t pxq " σ t pxqσ t pxq 1 than σ t pxq and we now discuss some ways of estimating ∆σ t pxq " σ t pxq ´σt pxq in terms of ∆a t pxq " a t pxq ´at pxq and in the reverse direction. The latter is straightforward: }∆a t pxq} ď }∆σ t pxq} r}σ t pxq} `}σ t pxq}s .

To estimate ∆σ t in terms of ∆a t , assume the following ellipticity condition is satisfied a t pxq ě υ I and a t pxq ě υ I for some parameter υ ą 0.

(2.19)

We recall the Ando-Hemmen inequality [START_REF] Ando | An inequality for trace ideals[END_REF] for any symmetric positive definite matrices

Q 1 , Q 2 }Q 1{2 1 ´Q1{2 2 } ď " λ 1{2 min pQ 1 q `λ1{2 min pQ 2 q ı ´1 }Q 1 ´Q2 } (2.20)
for any unitary invariant matrix norm }.}. In the above display, λ min p.q stands for the minimal eigenvalue. We also have the square root inequality

Q 1 ě Q 2 ùñ Q 1{2 1 ě Q 1{2 2
(2.21)

See for instance theorem 6.2 on page 135 in [START_REF] Higham | Functions of Matrices : Theory and Computation[END_REF], as well as proposition 3.2 in [START_REF] Ando | An inequality for trace ideals[END_REF]. A proof of (2.21) can be found in [START_REF] Bellman | Some inequalities for the square Root of a Positive Definite Matrix[END_REF]. In this situation, using (2.20)

Some results on anticipating stochastic calculus

In this section we review some results on Malliavin derivatives and Skorohod integration calculus which will be needed below. We restrict the presentation to unit time intervals. Let pΩ, Wq be the canonical space equipped with the Wiener measure P associated with the r-dimensional Brownian motion W t discussed in the introduction. The Malliavin derivative D t is a linear operator from some dense domain D 2,1 Ă L 2 pΩq into the space L 2 pΩ ˆr0, 1s; R r q of r-dimensional processes with square integrable states on the unit time interval. For multivariate d-column vector random variables F with entries F i , we use the same rules as for the gradient and we set

D t F " " D t F 1 , . . . , D t F d ı with D t F i " » - - D 1 t F i . . . D r F i fi ffi fl
For pp ˆqq-matrices F with entries F j k we let D t F be the tensor with entries

pD t F q i,j,k " D i t F j k
It is clearly out of the scope of this article to review the analytical construction of Malliavin differential calculus. For a more thorough discussion we refer the reader to the seminal book by Nualart [START_REF] Nualart | The Malliavin calculus and related topics[END_REF], see also the more synthetic presentation in the articles [START_REF] Nualart | Stochastic calculus with anticipating integrands[END_REF][START_REF] Ocone | A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations[END_REF]. Formally, one can think the Malliavin derivatives D i t F of some F P D 2,1 as way to extract from the random variable F the integrand of Brownian increment dW i t . For instance, when s ď t we have

D i t X s,t pxq " σ t,i pX s,t pxqq pD t ∇X s,t pxqq i,j,k " D i t p∇X s,t pxqq j,k :" p∇X s,t pxq ∇σ t,i pX s,t pxqqq j,k (2.23)
As conventional differentials, for any smooth function G from R d into R pˆq , Malliavin derivatives satisfy the chain rule properties

D i t pG j k ˝F q " ÿ 1ďlďd ´Bx l G j k ¯pF q ˆDi t F l ðñ D t pG ˝F q " D t F pp∇Gq ˝F q
For instance, for any s ď u ď v we have D u pX u,t ˝Xs,u q " pD u X s,u q rp∇X u,t q ˝Xs,u s and D u pς t ˝Xs,t q " pD u X s,t q rp∇ς t q ˝Xs,t s (2.24)

In the same vein, we have D u p∇X s,u rp∇X u,t q ˝Xs,u sq " pD u ∇X s,u q rp∇X u,t q ˝Xs,u s `pD u X s,u b ∇X s,u q "`∇ 2 X u,t ˘˝X s,u ‰

(2.25)

Let L 2,1 pR r q Ă L 2 pΩˆr0, 1s; R r q be the Hilbert space of r-dimensional process U t with Malliavin differentiable entries U i t P D 2,1 equipped with the norm

|||U ||| :" E ˜żr0,1s }U t } 2 dt ¸1{2 `E ˜żr0,1s 2 }D s U t } 2 ds dt

¸1{2

The Skorohod integral w.r.t. the Brownian motion W i t on the unit interval is defined a linear and continuous mapping from

V P L 2,1 pRq Þ Ñ ż 1 0 V t dW i t P L 2 pΩq
characterized by the two following properties

E ˆż 1 0 V t dW i t ˙" 0 E ˜ˆż 1 0 V t dW i t ˙2¸" E ˜żr0,1s V 2 t dt ¸`E ˜żr0,1s 2 D i s V t D i t V s ds dt ¸(2.26)
The above formula can be seen as an extended version of the Itô isometry to Skorohod integrals, for instance [START_REF] Nualart | Generalized multiple stochastic integrals and the representation of wiener functionals[END_REF], as well as chapters 1.3 to 1.5 in the book by Nualart [START_REF] Nualart | The Malliavin calculus and related topics[END_REF].

As for the Itô integral, the Skorohod integral w.r.t. the r-dimensional Brownian motion W t of a matrix valued process with entries V i k P L 2,1 pRq is defined by the column vector with entries ˆż 1 0 V t dW t ˙i :" These expansions, when combined with condition pT q 2 , yield the following estimates of the difference between X s,t pxq and X s,t pyq. combined with (3.4) and (3.5) completes the proof.

ż 1 0 V i t dW t :" ÿ 1ďkďr ż 1 0 V i t,
These contraction inequalities quantify the stability of the stochastic flow X s,t pxq w.r.t. the initial state x. For instance, the estimate (3.2) ensures that the Markov transition semigroup is exponentially stable; that is, we have that W 2 pµ 0 P s,t , µ 1 P s,t q ď c exp r´λ A pt ´sqs W 2 pµ 0 , µ 1 q (3.6)

For the Langevin diffusions discussed in (2.18) the stochastic flow is time homogeneous; that is we have that X s,t " X t´s :" X 0,pt´sq and P s,t " P t´s :" P 0,pt´sq . In addition when σpxq " σ I, the diffusion flow X t pxq has a single invariant measure on R d given by the Boltzmann-Gibbs measure

πpdxq " 1 Z exp ˆ´2 σ 2 U pxq ˙dx with Z :" ż e ´2 σ 2 U pxq dx (3.7)
From (2.18), it follows that henever pT q 2 is met, we have the estimate

∇ 2 U ě λ I ùñ W n pµP s,
B t E `}∇X s,t pxq} 4 F ˘ď ´4 " λ A ´ρp∇σq 2 ‰ E `}∇X s,t pxq} 4
F with the uniform log-norm parameter ρp∇σq defined in (2.5). This yields the estimate

B t E `}∇X s,t pxq} 4 F ˘1{4 ď ´"λ A ´ρp∇σq 2 ‰ E `}∇X s,t pxq} 4 F ˘1{4
More generally, we readily check the following result.

Proposition 3.2. When condition pT q n is met we have the following time-uniform bounds, E p}∇X s,t pxq} n F q 1{n ď ? d e ´rλ A ´pn´2qρp∇σq 2 {2spt´sq (3.8)

The Hessian process

In terms of the tensor product (2.1), we have the matrix diffusion equation p∇ 2 σ t,k q p∇ 2 σ t,k q 1 and τ t :" ∇ 2 b t `ÿ 1ďkďd p∇ 2 σ t,k q p∇σ t,k q 1 (3.9)

d ∇ 2 X s,
After some computations, we check that with the parameters χ pb, σq and λ A pnq defined in (2.6) and (2.8).

d " ∇ 2 X s,t pxq∇ 2 X s,t pxq 1 ‰ " ! " ∇ 2 X s,
In the above display, ρ ‹ p∇σq is defined in (2.5). The proof of the above estimate is technical and thus housed in the appendix on page 48

Bismut-Elworthy-Li formulae

We further assume that ellipticity condition (2.19) t ´s e ´λA pt´sqp1´ q p}f } `}∇f }q (3.16)

In the same vein, using (3.15) for any u Pss, tr and any bounded measurable function f s.t. }f } ď 1 we also check the rather crude uniform estimate

}∇ 2 P s,t pf q} ď κ 1 1 ?
u ´s e ´λA pu´sqp1´ q `κ2 2 1 a pt ´uqpu ´sq e ´λA pu´sq e ´λA pt´sqp1´ q Choosing u " s `p1 ´ qpt ´sq in the above display we check that for any Ps0, 1r we obtain the uniform estimate

}∇ 2 P s,t pf q} ď c 1 ? 1 ´ 1 ?
t ´s e ´λA p1´ q 2 pt´sq `c2

2 1 a p1 ´ q 1 t ´s e ´2λ A p1´ qpt´sq (3.17)
The extended versions of the above formulae in the context of diffusions on differentiable manifolds can be found in the series of articles [START_REF] Arnaudon | A Bismut type formula for the Hessian of heat semigroups[END_REF][START_REF] Bismut | Large deviations and the Malliavin calculus[END_REF][START_REF] Elworthy | Formulae for the Derivative of Heat Eemigroups[END_REF][START_REF] Li | Doubly Damped Stochastic Parallel Translations and Hessian Formulas[END_REF][START_REF] Thompson | Derivatives of Feynman-Kac semigroups[END_REF].

4 Backward semigroup analysis

The two-sided stochastic integration

For any given time horizon s ď t we have the rather well known backward stochastic flow equation

X s,t pxq " x `ż t s " ∇X u,t pxq 1 b u pxq `1 2 ∇ 2 X u,t pxq 1 a u pxq  du `ż t s ∇X u,t pxq 1 σ u pxq dW u (4.1)
The right hand side integral is understood as a conventional backward Itô-integral. In a more synthetic form, the above backward formula reduces to (1.7). An elementary proof of the above formula based on Taylor expansions is presented in [START_REF] Prato | Some remarks about backward Itô formula and applications[END_REF], different approaches can also be found in [START_REF] Kunita | First order stochastic partial differential equations[END_REF] and [START_REF] Krylov | On the first integrals and Liouville equations for diffusion processes[END_REF]. Extensions of the backward Itô formula (4.1) to jump type diffusion models as well as nonlinear diffusion flows can also be found in [START_REF] Da Prato | Some results of backward Itô formula[END_REF] and in the appendix of [START_REF] Arnaudon | A variational approach to nonlinear and interacting diffusions[END_REF].

Consider the discrete time interval rs, ts h :" tu 0 , . . . , u n´1 u associated with some refining time mesh u i`1 " u i `h from u 0 " s to u n " t, for some time step h ą 0. In this notation, combining (1.6) with (1.8) for any u P rs, ts h we have the Taylor type approximation X u`h,t ˝Xs,u`h ´Xu,t ˝Xs,u » ´ˆp∇X u`h,t q pX s,u pxqq 1 ∆b u pX s,u pxqq `1 2 `∇2 X u`h,t ˘pX s,u pxqq 1 ∆a u pX s,u pxqq ˙h ´p∇X u`h,t q pX s,u pxqq 1 ∆σ u pX s,u pxqq pW u`h ´Wu q

This yields the interpolating forward-backward telescoping sum formula X s,t pxq ´Xs,t pxq " ´ÿ uPrs,ts h " X u`h,t pX s,u`h pxqq ´Xu,t `Xs,u pxq ˘‰ » ÿ uPrs,ts h ˆp∇X u`h,t q pX s,u pxqq 1 ∆b u pX s,u pxqq `1 2 `∇2 X u`h,t ˘pX s,u pxqq 1 ∆a u pX s,u pxqq ˙h `ÿ uPrs,ts h p∇X u`h,t q pX s,u pxqq 1 ∆σ u pX s,u pxqq pW u`h ´Wu q (4.2) We obtain formally (1.10) by summing the above terms and passing to the limit h Ó 0. To be more precise, we follow the two-sided stochastic integration calculus introduced by Pardoux and Protter in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF]. As mentioned by the authors this methodology can be seen as a variation of Itô original construction of the stochastic integral. In this framework, the Skorohod stochastic integral (1.12) arising in (1.9) is defined by the L 2 -convergence S s,t pςqpxq :" lim hÑ0 ÿ uPrs,ts h p∇X u`h,t q pX s,u pxqq 1 ς u pX s,u pxqq pW u`h ´Wu q with ς u " ∆σ u (4.

3)

The proof of the above assertion is based on a slight extension of proposition 3.3 in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] to Skorohod integrals of the form (1.12). For the convenience of the reader, a detailed proof of the above assertion for one dimensional models is provided in section 6.1. Using (4.3), the complete proof of (1.9) now follows the same line of arguments as the ones used in the proof of Itô-type change rule formula stated in theorem 6.1 in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF], thus it is skipped.

A multivariate stochastic interpolation formulae

In terms of the tensor product (2.1), for any p ě 1 and any twice differentiable function f from R d into R p with at most polynomial growth the function F s,t :" P s,t pf q satisfies the backward formula (1.4) with the random fields G u,t pxq :" ∇F u,t pxq 1 b u pxq `1 2 ∇ 2 F u,t pxq 1 a u pxq and H u,t pxq :" ∇F u,t pxq 1 σ u pxq

Using the quantitative estimates presented in section 5.2, we checked that the regularity conditions pH 1 q, pH 2 q and pH 3 q stated in section 1. This shows that T s,t pf, ∆a, ∆bq and S s,t pf, ∆σq have the same form as the integrals T s,t p∆a, ∆bq and S s,t p∆a, ∆bq defined in (1.10) and (1.11) up to some terms involving the gradient and the Hessian of the function f . For instance, we have the two-sided stochastic integral formula S s,t pf, ∆σqpxq " ż t s P u,t p∇f qpX s,u pxqq 1 ∇X u,t pX s,u pxqq 1 ∆σ u pX s,u pxqq dW u Also observe that (4.4) coincides with (1.10) for the identity function; that is, we have that f pxq " x ùñ T s,t pf, ∆a, ∆bq " T s,t p∆a, ∆bq and S s,t pf, ∆σq " S s,t p∆σq

The above discussion shows that the analysis of the differences of the stochastic semigroups pP s,t ´Ps,t q in terms of the tangent and the Hessian processes is essentially the same as the one of the difference of the stochastic flows pX s,t ´Xs,t q. For instance using the discussion provided section 5.3, when the gradient and the Hessian of the function f are uniformly bounded the estimates stated in theorem 1.3 can be easily extended at the level of the stochastic semigroups.

The L 2 -norm of the two-sided stochastic integrals in (1.10) and (4.4) are uniformly estimated as soon as the pair of drift and diffusion functions pb t , σ t q and pb, σ t q satisfy condition pT q 2 . For a more thorough discussion we refer to section 5.1, see for instance the L n -norm estimates presented in theorem 5.2 applied to the difference function ς t " ∆σ t .

Semigroup perturbation formulae

Besides the fact that the Skorohod integral in the r.h.s. of (4.4) is not a martingale (w.r.t. the Brownian motion filtration) it is centered (see for instance (2.26) and the argument provided in the beginning of section 5.1). Thus, taking the expectation in the univariate version of (4.4) we obtain the following interpolation semigroup decomposition. Rewritten in terms of the infinitesimal generators pL t , L t q of the stochastic flows pX s,t , X s,t q we recover the rather well known semigroup perturbation formula P s,t " P s,t `ż t s P s,u pL u ´Lu qP u,t du ðñ p4.8q

The above formula can be readily checked using the interpolating formula given for any s ď u ă t by the evolution equation B u pP s,u P u,t q " pB u P s,u qP u,t `P s,u pB u P u,t q " P s,u L u P u,t ´P s,u L u P u,t

Now we come to the proof of (4.9). Whenever pT q 2 is met, combining (3. This ends the proof of (4.9).

After some elementary manipulations the forward-backward interpolation formula (4.8) yields the following corollary.

Corollary 4.2. Let X t and X t be some ergodic diffusions associated with some time homogeneous drift and diffusion functions pb, σq and pb, σq. The invariant probability measures π and π of X t and X t are connected for any twice differentiable function f from R d into R with bounded derivatives by the following interpolation formula pπ ´πqpf q " ż 8 0 E ˆ@∇P t pf qpY q, ∆bpY q D `1 2

Tr " ∇ 2 P t pf qpY q ∆apY q ‰ ˙dt (4.10)

In the above display Y stands for a random variable with distribution π and P t stands for the Markov transition semigroup of the process X t .

The formula (4.10) can be used to estimate the invariant measure of a stochastic flow associated with some perturbations of the drift and the diffusion function.

For instance, for homogeneous Langevin diffusions X t associated with some convex potential function U we have b " ´∇U and σ " I ùñ πpdxq 9 exp p´2 U pxqq dx In the above display, dx stands for the Lebesgue measure on R d . In this situation, using (4.10), for any ergodic diffusion flow X t with some drift b and an unit diffusion matrix we have πpf q " πpf q `ż 8 0 E `@pb `∇U qpY q, ∇P t pf qpY q D˘d t Notice that the above formula is implicit as the r.h.s. term depends on π. By symmetry arguments, we also have the following more explicit perturbation formula πpf q " πpf q `ż 8 0 E `@pb `∇U qpY q, ∇P t pf qpY q

D˘d t

In the above display Y stands for a random variable with distribution π and P t stands for the Markov transition semigroup of the process X t .

Some extensions

Several extensions of the forward-backward stochastic interpolation formula (1.10) to more general stochastic perturbation processes can be developed. For instance, suppose we are given some stochastic processes Y s,t pxq P R d and Z s,t pxq P R dˆr adapted to the filtration of the Brownian motion W t , and let X s,t pxq be the stochastic flow defined by the stochastic differential equation dX s,t pxq " Y s,t pxq dt `Zs,t pxq dW t (4.11) In this situation, the interpolation formula (1.9) remains valid when a u pX s,u pxqq is replaced by the stochastic matrices Z s,t pxqZ s,t pxq 1 . This yields without further work the forward-backward stochastic interpolation formula (1.10) with the local perturbations ∆b u pX s,u pxqq :" b u pX s,u pxqq ´Y s,u pxq ∆σ u pX s,u pxqq :" σ u pX s,u pxqq ´Zs,u pxq and ∆a u pX s,u pxqq :" a u pX s,u pxqq ´Zs,u pxqZ s,u pxq 1

The corresponding interpolation formula should be used with some caution as the L 2 -norm of the two-sided stochastic integral (1.12) depends on the Malliavin differential of the integrand process of the Brownian motion; see for instance the variance formula provided in lemma 5.1. Assume that σ " I and the regularity condition pT q 2 is met. Also suppose X s,t pxq is given by a stochastic differential equation of the form (4.11) with r " d and Z s,t pxq " I. Arguing as above, in terms of the tensor product (2.1) we have X s,t pxq ´Xs,t pxq " ż t s p∇X u,t q pX s,u pxqq 1 pb u pX s,u pxqq ´Y s,u pxqq du (4.12)

Combining (2.15) with the generalized Minkowski inequality, we check the following proposition.

Proposition 4.3. Assume that pT q 2 is met for some λ A ą 0. In this situation, for any Recalling that pW u`h ´Wu q is independent of the flows X s,u and ∇X u`h,t , the discrete time approximation (4.3) shows that Skorohod stochastic integral is centered; that is, we have that EpS s,t pςqpxqq " 0.

Following (4.3), the variance can be computed using the following approximation formula

E " }S s,t pςqpxq} 2 ‰ " lim hÑ0 ÿ u,v Prs,ts h ÿ 1ďiďd ÿ 1ďj,kďr E ! " p∇X u`h,t q pX s,u pxqq 1 ς u pX s,u pxqq ‰ i,j " p∇X v`h,t q pX s,v pxqq 1 ς v pX s,v pxqq ‰ i,k pW j u`h ´W j u qpW k v`h ´W k v q ) (5.
2) The proof of the above assertion is provided in section 6.1, see for instance proposition 6.2.

Consider the matrix valued function

Σ s,u,t pxq :" " p∇X u,t q 1 ˝Xs,u ‰ pxq ς u pX s,u pxqq (5.3)
In this notation, the limiting diagonal term u " v in the r.h.s. of (5.2) is clearly equal to

ż t s E « ÿ i,j
Σ s,u,t pxq i,j Σ s,u,t pxq i,j ff du "

ż t s E " }Σ s,u,t pxq} 2 F ‰ du
In addition, whenever condition pT q 2 is met and ς is bounded, (3.4) readily yields the estimate

"ż t s E " }Σ s,u,t pxq} 2 F ‰ du  1{2 ď }ς} 2 a d{p2λ A q (5.4)
More generally, using (3.8) whenever pMq 2{δ and pT q 2{p1´δq are met for some δ Ps0, 1r we have the estimate

E " }Σ s,u,t pxq} 2 ‰ ď c 1,δ " }ςp0q} 2 `}∇ς} 2 p1 `}x}q 2 ‰ e ´2λ A p2{p1´δqqpt´uq
This implies that

"ż t s E " }Σ s,u,t pxq} 2 F ‰ du  1{2 ď c 2,δ r}ςp0q} `}∇ς} p1 `}x}qs { a λ A (5.5)
The non-diagonal term can be computed in a more direct way using Malliavin derivatives of the functions Σ s,u,t . For any s ď u ď v ď t we have

D v " p∇X u,t q 1 ˝Xs,u ‰ " ς u ˝Xs,u ‰( " "`D v p∇X u,t q 1 ˘˝X s,u ‰ " ς u ˝Xs,u ‰ (5.6)
As expected, observe that ∇σ " 0 ùñ D v Σ s,u,t pxq " 0

In the reverse angle, whenever s ď v ď u ď t we have the chain rule formula D v `"ς u ˝Xs,u ‰ " p∇X u,t q ˝Xs,u ‰:

" " D v `ςu ˝Xs,u ˘‰ " p∇X u,t q ˝Xs,u ‰ `"D v X s,u b pς u ˝Xs,u q ‰ "`∇ 2 X u,t ˘˝X s,u ‰ (5.7)
As above, Malliavin differentials D v `ςu ˝Xs,u ˘and D v X s,u can be computed using the chain rule formulae (2.24).

A more detailed analysis of the chain rules formulae (2.24), (2.25) and (5.7) for one dimensional models is provided in section 6.1 (cf. lemma 6.1).

Observe that

∇ς " 0 ùñ D v " Σ 1 s,u,t ‰ " " D v X s,u b pς u ˝Xs,u q ‰ "`∇ 2 X u,t ˘˝X s,u ‰
We consider the inner product

xD u Σ s,v,t pxq, D v Σ s,u,t pxqy :" ÿ i,j,k pD v Σ s,u,t pxqq k,i,j pD u Σ s,v,t pxqq j,i,k
In this notation, an explicit description of the L 2 -norm of the two-sided stochastic integral in terms of Malliavin derivatives is given below.

Lemma

‰ du

The above lemma can be interpreted as a matrix version of the isometry property (2.26). A proof of the above lemma based on the L 2 -approximation of two-sided stochastic integrals is provided in section 6.1 (see for instance proposition 6.2).

Quantitative estimates

For any p ą 1 and any tensor norms we also quote the rather well known L p -norm estimates

E r}S s,t pςqpxq} p s 2{p ď c 1,p ż rs,ts E " }Σ s,u,t pxq} 2 ‰ du `c2,p E » - ˜żrs,ts 2 }D v Σ s,u,t pxq} 2 du dv ¸p{2 fi fl 2{p
for some finite constants c i,p whose values only depend on p. A proof of these estimates can be found in [START_REF] Nualart | Stochastic calculus with anticipating integrands[END_REF][START_REF] Watanabe | Lectures on stochastic differential equations and Malliavin Calculus[END_REF], see also [START_REF] Nualart | Generalized multiple stochastic integrals and the representation of wiener functionals[END_REF] for multiple Skorohod integrals. By the generalized Minkowski inequality, for any n ě 2 we also have the estimate

E r}S s,t pςqpxq} n s 2{n ď c 1,n ż rs,ts E " }Σ s,u,t pxq} 2 ‰ du `c2,n ż rs,ts 2 E r}D v Σ s,u,t pxq} n s 2{n du dv (5.8)
Observe that for any n ě 2 we have

pMq n ùñ |||ςpxq||| n ď κ n p}ςp0q} `}∇ς}q p1 _ }x}q
The main objective of this section is to prove the following theorem.

Theorem 5.2. Assume that pM q 2n{δ and pT q 2n{p1´δq are satisfied for some parameter n ě 2 and some δ Ps0, 1r. In this situation, we have the uniform estimate E r}S s,t pςqpxq} n s 1{n ď κ δ,n |||ςpxq||| 2n{δ p1 _ }x}q (5.9)

For uniformly bounded diffusion functions pς, σ, σq whenever pT q 2n is met for some n ě 2 we have E r}S s,t pςqpxq} n s 1{n ď κ n p}ς} `}∇ς}q (5.10)

In addition, for constant diffusion functions pς, σ, σq whenever pT q 2 is met, for any n ě 2 we have the uniform estimate E r}S s,t pςqpxq} n s 1{n ď κ n }ς} (

The proof of the above theorem, including a more detailed description of the parameters κ δ,n and κ n is provided below.

Next, we estimate the L n -norm of the Malliavin differential D v Σ s,u,t pxq in the two cases ps ď u ď v ď tq and ps ď v ď u ď tq.

Case ps ď u ď v ď tq: Using (5.6) we have }D v Σ s,u,t pxq} ď c }ς u pX s,u pxqq} }pD v ∇X u,t qpX s,u pxqq} Using (2.24) and (2.25) this yields the estimate }D v Σ s,u,t pxq} ď c 1 I s,u,t pxq `c2 J s,u,t pxq with the functions

I s,u,t pxq :" }∇σ} }ς u pX s,u pxqq} }p∇X u,v qpX s,u pxqq} }p∇X v,t qpZ s,v u pxqq} J s,u,t pxq :" }σ v pZ s,v u pxqq} }ς u pX s,u pxqq} }p∇X u,v qpX s,u pxqq} }p∇ 2 X v,t qpZ s,v u pxqq}
In the above display, Z s,v u pxq stands for the interpolating flow defined in (1.13). • Firstly assume that }ς} _ }σ} ă 8 and pT q 2n is satisfied for some parameter n ě 1. In this situation, applying proposition 3.2 and proposition 3.3, for any Ps0, 1r we have the uniform estimates E p}D v Σ s,u,t pxq} n q 1{n ď }ς} χ n, pb, σq exp p´p1 ´ qλ A p2nqpt ´uqq with the parameter χ n, pb, σq given by χ n, pb, σq :" c r}σ} _ }∇σ}s

" 1 `1 n λ A p2nq
χ pb, σq  with χ pb, σq given in (2.6).

• More generally, when }∇ς} _ }∇σ} ă 8 the functions ς t pxq and σ t pxq may grow at the most linearly with respect to }x}. Assume that conditions pM q 2n{δ and condition pT q 2n{p1´δq are satisfied for some parameters n ě 1 and δ Ps0, 1r. In this situation, applying Hölder inequality we check that E p}I s,u,t pxq} n q 1{n ď c }∇σ} E ´}ς u pX s,u pxqq} n{δ ¯δ{n

ˆE ´}p∇X u,v qpX s,u pxqq} 2n{p1´δq ¯p1´δq{p2nq E ´}p∇X v,t qpZ s,v u pxqq} 2n{p1´δq ¯p1´δq{p2nq
Applying proposition 3.2 we check that E p}I s,u,t pxq} n q 1{n ď c n,δ }∇σ} |||ςpxq||| n{δ e ´λA p2n{p1´δqqpt´uq

In the same vein, combining proposition 3.2 and proposition 3.3 with the uniform moment estimates (2.11) we check that E p}J s,u,t pxq} n q 1{n ď c n,δ r}σp0q} `}∇σ}s

1 χ pb, σq λ A p2n{p1 ´δqq ˆ|||ςpxq||| 2n{δ r1 `}x}s e ´p1´ qλ A p2n{p1´δqqpt´uq
We conclude that

E p}D v Σ s,u,t pxq} n q 1{n ď χ n,δ, pb, σq |||ςpxq||| 2n{δ r1 `}x}s e ´p1´ qλ A p2n{p1´δqqpt´uq
with the parameter χ n,δ, pb, σq :" c n,δ r}σp0q} `}∇σ}s ˆ1 `1 χ pb, σq λ A p2n{p1 ´δqq Ċase ps ď v ď u ď tq:

We use (5.7) to check that

}D v Σ s,u,t pxq} ď }rD v `ςu ˝Xs,u ˘spxq} } p∇X u,t q pX s,u pxqq} `}rD v X s,u spxq b ς u pX s,u qpxq} } `∇2 X u,t ˘pX s,u pxqq}
On the other hand, using the chain rules (2.24) we have

D v X s,u :" `Dv X s,v ˘"`∇ X v,u ˘˝X s,v ‰ D v `ςu ˝Xs,u ˘" pD v X s,u q " p∇ς u q ˝Xs,u ‰
This yields the estimate

}D v Σ s,u,t pxq} ď c 1 }σ v pX s,v pxqq} }∇ς} }p∇X v,u qpX s,v pxqq} } p∇X u,t q pX s,u pxqq} `c2 }σ v pX s,v pxqq} }ς u pX s,u pxqq} }p∇X v,u qpX s,v pxqq} } `∇2 X u,t ˘pX s,u pxqq}
• Firstly assume that }ς} _ }σ} ă 8 and condition pT q 2n is satisfied for some n ě 1. In this situation, arguing as above for any Ps0, 1r we have the uniform estimates E p}D v Σ s,u,t pxq} n q 1{n ď p}ς} `}∇ς}q χ n, pb, σq exp ´´p1 ´ qλ A,A p2nqpt ´vq for some universal constant c and the parameter χ n, pb, σq given by χ n, pb, σq :" c }σ}

« 1 `1 n λ A,A p2nq
χ pb, σq ff with χ pb, σq given in (2.6).

• More generally assume that }∇ς} _ }∇σ} ă 8. Also assume that conditions pM q 2n{δ and pT q 2n{p1´δq are satisfied for some parameters n ě 1 and δ Ps0, 1r. In this situation, we have

E p}D v Σ s,u,t pxq} n q 1{n ď χ n,δ, pb, σ, σq |||ςpxq||| 2n{δ r1 `}x}s e ´p1´ qλ A,A p2n{p1´δqqpt´vq
with the parameter

χ n,δ, pb, σ, σq :" c n,δ r}σp0q} `}∇σ}s ˜1 `1 χ pb, σq λ A,A p2n{p1 ´δqq
Ţhe end of the proof of theorem 5.2 is a direct consequence of the estimates discussed above combined with (5.8) and the diagonal estimates presented in (5.4).

Some extensions

This section is concerned with the two-sided stochastic integral (4.6). Using the gradient formula in (4.7) the Skorohod stochastic integral in (4.6) takes the form S s,t pf, ∆σqpxq " ż t s Σ s,u,t pf qpxq dW u with the integrands Σ s,u,t pf qpxq :" ∇f pZ s,t u pxqq 1 Σ s,u,t pxq and Σ s,u,t pxq :"

" p∇X u,t q 1 ˝Xs,u ‰ " ∆σ u ˝Xs,u ‰
As in (2.25), using the chain rule properties of Malliavin derivatives we check that

D i v Σ s,u,t pf q " `Di v ∇f pZ s,t u q 1 ˘Σs,u,t `∇f pZ s,t u q 1 D i v Σ s,u,t
as well as

D i v ∇f pZ s,t u q 1 " ∇ 2 f pZ s,t u q 1 D i v Z s,t u
This yields the differential formula

D i v Σ s,u,t pf q " ∇f pZ s,t u q 1 D i v Σ s,u,t `∇2 f pZ s,t u q 1 pD i v Z s,t u q Σ s,u,t
The Malliavin derivatives D i v Σ s,u,t are computed using formulae (5.6) and (5.7); thus, it remains to compute the Malliavin derivatives D v Z s,t u of the interpolating path.

' When u ď v we have Z s,t u " pX v,t ˝Xu,v q ˝Xs,u " X v,t ˝Zs,v u In this situation, as in (2.24) using the chain rule properties of Malliavin derivatives we check that

D v Z s,t u " D v Z s,v
u pp∇X v,t q ˝Zs,v u q " ppD v X u,v q ˝Xs,u q pp∇X v,t q ˝Zs,v u q By (2.23) we conclude that

D v Z s,t u " pσ v ˝Zs,v u q pp∇X v,t q ˝Zs,v u q ' When v ď u we have Z s,t u " X u,t ˝pX v,u ˝Xs,v q " Z v,t u ˝Xs,v
In this situation, arguing as above we check that

D v Z s,t u " D v X s,v pp∇Z v,t
u q ˝Xs,v q " D v X s,v pp∇X v,u q ˝Xs,v q pp∇X u,t q ˝Xs,u q By (2.23) we conclude that D v Z s,t u " pσ v ˝Xs,v q pp∇X v,u q ˝Xs,v q pp∇X u,t q ˝Xs,u q 6 Some anticipative calculus

For clarity and to avoid unnecessary sophisticated multi-index notation, we only consider one dimensional model. The proof of the results presented in this section in the general case can be reproduced word-for-word for multidimensional models.

To simplify the presentation, we write B n f the derivative of order n ě 1 of a smooth function f . We also set Y s,t pxq :" X s,t pxq. We also reduce the analysis to the unit interval. In this context, for any t P r0, 1s we set Y t :" Y 0,t and X t :" X t,1 (6.1)

Extended two-sided stochastic integrals

The aim of this section is to extend the two-sided stochastic integration introduced in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] to Skorohod integrals of the form (4.3), for some time homogeneous function ς u " ς satisfying (5.1). For any t P r0, 1s we set ΦpX t , Y t pxqq :" BX t pY t pxqq ςpY t pxqq (6.2)

In this notation the limiting integral in (4.3) takes formally the following form S 0,1 pςqpxq :"

ż 1 0 ΦpX t , Y t pxqq dW t
The existence of this two-sided stochastic integral is discussed below in (6.4).

To simplify the presentation, we fix the state variable x and we write Y t and ΦpX t , Y t q instead of Y t pxq and ΦpX t , Y t pxqq. Next technical lemma provided a more explicit description of the Malliavin derivatives of the processes ΦpX t , Y t q. Lemma 6.1. For any s ă t we have D s ΦpX t , Y t q " " BppBX t q ˝Ys,t qpY s q pς ˝Ys,t qpY s q `ppBX t q ˝Ys,t qpY s q ˆBpς ˝Ys,t qpY s q ‰ σpY s q

In addition, we have D t ΦpX s , Y s q " " BppBX t q ˝Xs,t qpY s q pσ ˝Xs,t qpY s q `BpX t ˝Xs,t qpY s q BσpX s,t pY s qq ‰ ςpY s q

Proof. Using the chain rules properties, for any s ă t we have D s ΦpX t , Y t q " D s `pBX t qpY s,t pY s qq pς ˝Ys,t qpY s q " D s `pBX t q ˝Ys,t qpY s q ˘pς ˝Ys,t qpY s q `pBX t q ˝Ys,t qpY s q D s pς ˝Ys,t qpY s q

The end of the proof of the first assertion comes from the fact that D s `pBX t q ˝Ys,t qpY s q ˘" BppBX t q ˝Ys,t qpY s q D s Y s with D s Y s " σpY s q

In the same vein, we have D s pς ˝Ys,t qpY s q " Bpς ˝Ys,t qpY s q σpY s q

We also have that D t ΦpX s , Y s q " D t ppBX s qpY s q ςpY s qq " D t `BpX t ˝Xs,t qpY s q ˘ςpY s q " D t ``pBX t q ˝Xs,t ˘pY s q pBX s,t qpY s q ˘ςpY s q

The last assertion comes from the fact that D t ``pBX t q ˝Xs,t ˘pY s q pBX s,t qpY s q " D t `pBX t q ˝Xs,t ˘pY s q pBX s,t qpY s q ``pBX t q ˝Xs,t ˘pY s q D t pBX s,t qpY s q

The r.h.s. term in the above display can be rewritten as follows D t pBX s,t qpY s q " BσpX s,t pY s qq pBX s,t qpY s q ùñ ppBX t q ˝Xs,t qpY s q D t pBX s,t qpY s q " BpX t ˝Xs,t qpY s q BσpX s,t pY s qq

In the same vein, we have D t `pBX t q ˝Xs,t ˘pY s q " ppB 2 X t q ˝Xs,t qpY s q D t X s,t pY s q " ppB 2 X t q ˝Xs,t qpY s q σpX s,t pY s qq ùñ D t `pBX t q ˝Xs,t ˘pY s q pBX s,t qpY s q " BppBX t q ˝Xs,t qpY s q σpX s,t pY s qq

This ends the proof of the second assertion. The proof of the lemma is now completed.

From the above lemma, we also check that all the n-absolute moments of the Malliavin derivatives D s ΦpX t , Y t q are finite with at most quadratic growth w.r.t. the initial values.

Next proposition extends proposition 3.3 in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] to stochastic processes of the form (6.2). Proposition 6.2. Let r0, 1s h be any refining sequence of partitions of the unit interval. For any h ą 0 we define S h pΦq :" ÿ tPr0,1s h ΦpX t`h , Y t q pW t`h ´Wt q

Then S h pΦq is a Cauchy sequence in L 2 pΩq. In addition, for any decreasing sequence of time steps h 1 ą h 2 we have the formula

lim h 1 Ñ0 E ´Sh 1 pΦq S h 2 pΦq ¯" E ˜ż 1 0 ΦpX t , Y t q 2 dt `żr0,1s 2 D s ΦpX t , Y t q D t ΦpX s , Y s q ds dt (6.3)
Before entering into the details of the proof of the proposition, we give a couple of comments. The hypothesis that r0, 1s h is a refining sequence indexed by h is not essential but it simplifies the proof of the proposition, see for instance lemma 3.1.1 in [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]. Arguing as in the proof of theorem 3.3 and theorem 7.1 in [START_REF] Pardoux | A two-sided stochastic integral and its calculus[END_REF] the above proposition ensures that the two-sided integral defined by the L 2 pΩq-limit coincides with the two-sided stochastic integral of the process ΦpX t , Y t q over the unit interval; that is, we have that

ż 1 0 ΦpX t , Y t q dW t :" L 2 ´lim hÑ0 ÿ tPr0,1s h ΦpX t`h , Y t q pW t`h ´Wt q (6.4)
In this context, proposition 6.2 can be interpreted as a version of the isometry property (2.26) for the generalized two-sided integral defined above.

Proof of proposition 6.2:

We fix h 1 ą h 2 and we assume that r0, 1s h 2 is a refinement of P r0, 1s h 1 . For any ps, tq P pr0, 1s h 1 ˆr0, 1s h 2 q we also set Π h 1 ,h 2 s,t :" ΦpX s`h 1 , Y s q ΦpX t`h 2 , Y t q pW s`h 1 ´Ws q pW t`h 2 ´Wt q With a slight abuse of notation we set ∆W s :" pW s`h 1 ´Ws q and ∆W t :" pW t`h 2 ´Wt q ' For any overlapping pair s ă t ă t `h2 ă s `h1 using the decomposition ∆W s " pW s`h 1 ´Wt`h 2 q `∆W t `pW t ´Ws q we have

E ´ΦpX s`h 1 , Y s q ΦpX t`h 2 , Y t q ∆W t ∆W s | W t _ W t`h 2 ¯" ΦpX s`h 1 , Y s q ΦpX t`h 2 , Y t q h 2
It follows from the continuity properties of the processes that

E ˜ÿ sătăt`h 2 ăs`h 1 Π h 1 ,h 2 s,t ¸ÝÑ h 1 Ñ0 E ˆż 1 0 ΦpX t , Y t q 2 dt ' When s `h1 ă t we have BX s`h 1 " BpX t ˝Xs`h 1 ,t q " BpX t`h 2 ˝Xs`h 1 ,t q `´´B X t ´BX t`h 2 ¯˝X s`h 1 ,t ¯ˆBX s`h 1 ,t
On the other hand, we have the decomposition BX t ´BX t`h 2 " ´pBX t`h 2 q ˝Xt,t`h 2 ¯ˆBX t,t`h 2 ´BX t`h 2 " ´pBX t`h 2 q ˝pI `∆X t q ´BX t`h 2 ¯`BX t`h 2 ˆ∆X 1 t `´pBX t`h 2 q ˝pI `∆X t q ´BX t`h 2 ¯ˆ∆X 1 t with the increment functions

∆X 1 t :" BX t,t`h 2 ´1 and ∆X t :" X t,t`h 2 ´I
With a slight abuse of notation, we shall denote by Oph p q some possible random variable with any n-absolute moment of order h p , for some p ą 0 with 0 ă h ă 1. In this notation, we have

∆X 1 t pxq " ż t`h 2 t BσpX t,u pxqq BX t,u pxq dW u `Oph 2 q " Oph 1{2 2 q ∆X t pxq " ż t`h 2 t σpX t,u pxqq dW u `Oph 2 q " Oph 1{2 2 q
Given a smooth function θ we set B n θpx, yq :"

ż 1 0 p1 ´ q n´1 pn ´1q! θ 2 px ` yq d
In this notation, we have the first and second order decompositions `pBX t`h 2 q ˝pI `∆X t q ´BX t`h 2 ˘pxq " pB 2 X t`h 2 qpx, ∆X t pxqq ∆X t pxq " pB 2 X t`h 2 qpxq ∆X t pxq `pB

BX s`h 1 " BpX t`h 2 ˝Xs`h 1 ,t q
`"BppBX t`h 2 q ˝Xs`h 1 ,t q ˆpp∆X t q ˝Xs`h 1 ,t q `BpX t`h 2 ˝Xs`h 1 ,t q ˆpp∆X 1 t q ˝Xs`h 1 ,t q ı `Oph 2 q

This yields the first order decomposition ΦpX s`h 1 , Y s q " ψ 0 s,t pY s q `ψ1 s,t pY s q pp∆X t q ˝Xs`h 1 ,t qpY s q `ψ2 s,t pY s q pp∆X 1 t q ˝Xs`h 1 ,t qpY s q `Oph 2 q with the functions ψ 0 s,t pY s q :" BpX t`h 2 ˝Xs`h 1 ,t qpY s q ςpY s q ψ 1 s,t pY s q :" BppBX t`h 2 q ˝Xs`h 1 ,t qpY s q ςpY s q and ψ 2 s,t pY s q :" BpX t`h 2 ˝Xs`h 1 ,t qpY s q ςpY s q Notice that none of the functions but the increment functions p∆X t q and p∆X 1 t q depend on W t,t`h 2 , nor on W s,s`h 1 .

In the reverse angle, we have pBX t`h 2 q ˝Yt " pBX t`h 2 q ˝pY s`h 1 ,t ˝Ys q `"ppBX t`h 2 q ˝Ys`h 1 ,t q ˝pI `∆Y s q ´ppBX t`h 2 q ˝Ys`h 1 ,t q ı ˝Ys with ∆Y s :" pY s,s`h 1 ´Iq ùñ Y s`h 1 " pI `∆Y s q ˝Ys

Arguing as above, we have " ppBX t`h 2 q ˝Ys`h 1 ,t q ˝py `∆Y s pyqq ´ppBX t`h 2 q ˝Ys`h 1 ,t qpyq ı " BppBX t`h 2 q ˝Ys`h 1 ,t qpyq ∆Y s pyq `B2 ppBX t`h 2 q ˝Ys`h 1 ,t qpy, ∆Y s pyqq p∆Y s pyqq 2

We conclude that pBX t`h 2 q ˝Yt " pBX t`h 2 q ˝pY s`h 1 ,t ˝Ys q `BppBX t`h 2 q ˝Ys`h 1 ,t qpY s q pp∆Y s q ˝Ys q `Oph 1 q

In the same vein, we have ς ˝Yt " pς ˝Ys`h 1 ,t ˝Ys q `Bpς ˝Ys`h 1 ,t qpY s q pp∆Y s q ˝Ys q `Oph 1 q

Multiplying these terms, we check that ΦpX t`h 2 , Y t q " Ψ 0 s,t pY s q `Ψ1 s,t pY s q pp∆Y s q ˝Ys q `Oph 1 q with the functions Ψ 0 s,t pY s q :" ppBX t`h 2 q ˝Ys`h 1 ,t qpY s q ˆpς ˝Ys`h 1 ,t qpY s q Ψ 1 s,t pY s q :" " BppBX t`h 2 q ˝Ys`h 1 ,t qpY s q ˆpς ˝Ys`h 1 ,t qpY s q `ppBX t`h 2 q ˝Ys`h 1 ,t qpY s q ˆBpς ˝Ys`h 1 ,t qpY s q ‰ None of the functions but the increment ∆Y s depend on W s,s`h 1 , nor on W t,t`h 2 .

Recall that the functions ΦpX t`h 2 , Y t q and ψ 0 s,t pY s q don't depend on ∆W t . In addition, the functions ΦpX s`h 1 , Y s q and Ψ 0 s,t pY s q don't depend on ∆W s . This yields the formula

E ´ΦpX s`h 1 , Y s q ΦpX t`h 2 , Y t q ∆W s ∆W t " E ´"ΦpX s`h 1 , Y s q ´ψ0 s,t pY s q ı " ΦpX t`h 2 , Y t q ´Ψ0 s,t pY s q ı ∆W s ∆W t " E `Ψ1
s,t pY s q ψ 1 s,t pY s q rpp∆Y s q ˝Ys q ∆W s s rpp∆X t q ˝Xs`h 1 ,t qpY s q ∆W t s Ȇ `Ψ1 s,t pY s q ψ 2 s,t pY s q rpp∆Y s q ˝Ys q ∆W s s " pp∆X 1 t q ˝Xs`h 1 ,t qpY s q ∆W t ‰˘`O ´h2`1{2

This

shows that

h ´1 1 h ´1 2 E ´ΦpX s`h 1 , Y s q ΦpX t`h 2 , Y t q ∆W s ∆W t " E ˆΨ1 s,t pY s q ψ 1 s,t pY s q h ´1 1 "ż s`h 1 s σpY u q du  h ´1 2 "ż t`h 2 t σpX s`h 1 ,u pY s qq du Ė ˆΨ1 s,t pY s q ψ 2 s,t pY s q h ´1 1 "ż s`h 1 s σpY u q du  h ´1 2 "ż t`h 2 t
B pσ ˝Xt,u q pX s`h 1 ,t pY s qq du

˙`O ´h1{2 1 Īt follows that lim h 1 Ñ0 E ˜ÿ s`h 1 ăt Π h 1 ,h 2 s,t " E ˆż 1 0 ż t 0 "
BppBX t q ˝Ys,t qpY s q pς ˝Ys,t qpY s q `ppBX t q ˝Ys,t qpY s q ˆBpς ˝Ys,t qpY s q ‰ σpY s q ˆ"BppBX t q ˝Xs,t qpY s q σpX s,t pY s qq `BpX t ˝Xs,t qpY s q BσpX s,t pY s qq ‰ ςpY s q ds dt We end the proof of (6.3) using lemma 6.1 and symmetry arguments. This ends the proof of the proposition.

Generalized backward Itô-Ventzell formula

This section is mainly concerned with the proof of theorem 1.1. Before entering into the details of the proof we discuss how it applies to the process pX t , Y t q introduced in (6.1).

Consider the random fields F t pxq :" X t pxq ùñ BF t " BX t and B 2 F t " B 2 X t G t pxq :" BX t pxq bpxq `1 2 B 2 X t pxq apxq and H t pxq :" BX t pxq σpxq (6.5)

In this notation, the backward random field formula (4.1) with t P r0, 1s takes the form

F t pxq :" F 1 pxq `ż 1 t G s pxq ds `ż 1 t H s pxq dW s with F 1 pxq " x (6.6)
We fix some given Y 0 " y P R and we write Y t instead of Y t pyq and set pA u , B u , Σ u q :" `apY u q, bpY u q, σpY u q Ȋn this notation, we have

Y t " y `ż t 0 B u du `ż t 0 Σ u dW u (6.7)
Observe that B u , Σ u as well as the Malliavin derivatives D v Σ u " BσpY u q D v Y u have moments of any order. Consider the processes U t :" BF t pY t q B t `1 2 B 2 F t pY t q A t ´Gt pY t q " BX t pY t q pb ´bqpY t q `1 2 B 2 X t pY t q pa ´aqpY t q V t :" BF t pY t q Σ t ´Ht pY t q " BX t pY t q pσ ´σqpY t q with A t :" Σ 2 t

In this notation, up to a change of sign and replacing x by Y 0 in (1.10) the stochastic interpolation formula stated in theorem 1.2 on the unit interval takes the following form

F 1 pY 1 q ´F0 pY 0 q " ż 1 0 U s ds `ż 1 0 V s dW s
More generally, suppose we are given a forward real valued continuous semi-martingale Y t of the form (6.7) for some W 0,t -adapted functions B t and Σ t , and a backward random field models of the form (6.6) for some W t,1 -adapted functions F t pxq, G t pxq, H t pxq.

We consider the following conditions: (6.9) pH 3 q: The random processes B u , Σ u as well as D v Σ u are continuous w.r.t. the time parameter and they have moments of any order.

pH 1 q 1 :
The next theorem is a slight extension of theorem 1.1 applied to the semi-martingale and the random fields models discussed in (6.7) and (6.5). Theorem 6.3. Consider a backward random field models of the form (6.6) for some functions F t pxq, G t pxq, H t pxq satisfying pH 1 q 1 and pH 2 q 1 . Also let Y t be a continuous semi-martingale of the form (6.7) functions B t and Σ t satisfying pH 3 q. In this situation, for any t P r0, 1s we have the generalized backward Itô-Ventzell formula

F t pY t q ´F0 pY 0 q " ż t 0 ˆBF s pY s q B s `1 2 B 2 F s pY s q A s ´Gs pY s q ˙ds `ż t 0 pBF s pY s q Σ s ´Hs pY s qq dW s (6.10)
The r.h.s. term in the above display is understood as a Skorohod integral.

Proof: We use the same approximation technique as in [START_REF] Bismut | A generalized formula of Itô and some other properties of stochastic flows[END_REF][START_REF] Ocone | A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations[END_REF] and [START_REF] Pardoux | Applications of anticipating stochastic calculus to stochastic differential equations[END_REF] (see also the proof of theorem 3.2.11 in [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]). Consider a mollifier type approximation of the identify given for any ą 0 by the function ϕ pxq :" ϕpx{ q{ for some smooth compactly supported function ϕ s.t. ş 8 ´8 ϕpxqdx " 1. For any x, applying the Itô-type change rule formula stated in proposition 8.2 in [START_REF] Nualart | Stochastic calculus with anticipating integrands[END_REF] pV s ´Vs q 2 dsq `E ˜żr0,ts 2 pD r V s ´Dr V s q pD s V r ´Ds V r q dr ds ¸ÝÑ Ñ0 0 (6.12)

Observe that ż t 0 pV s ´Vs q On the other hand, we have

D t V s " D t pBF s pY s qq Σ s `BF s pY s q D t Σ s ´Dt pH s pY s qq " `pD t BF s qpY s q `B2 F s pY s q D t Y s ˘Σs `BF s pY s q D t Σ s ´ppD t H s qpY s q `BH s pY s q D t Y s q
Arguing as above, we have the estimate E ˜żr0,1s 2 pD r V s ´Dr V s q pD s V r ´Ds V r q dr ds ḑ 2 E ˜żr0,1s 2 pD t V s ´Dt V s q 2 ds dt ¸ď 2 4 ÿ 1ďiď5 J i p q

In the above display, J i p q stands for the sequences J 1 p q :" E ˜żr0,1s 2 ˆR pBF s pxq ´BF s pY s qq 2 pD t Σ s q 2 ϕ pY s ´xq ds dt dx J2 p q :" E ˜żr0,1s 2 ˆR pBH s pxq ´BH s pY s qq 2 pD t Y s q 2 ϕ pY s ´xq ds dt dx J3 p q :" E ˜żr0,1s 2 ˆR `B2 F s pxq ´B2 F s pY s q ˘2 pD t Y s q 2 A s ϕ pY s ´xq ds dt dx Ţhe last two terms depend on the Malliavin derivatives of BF s and H s are they are given by J 4 p q :" E ˜żr0,1s

¯2

for some universal constant c ă 8 and the parameter χ pb, σq introduced in (2.6). This ends the proof of (7.1). The proof of the theorem is completed.

for some fluctuation parameter σ ě 0. For any s `kh ď u ă s `pk `1qh we have X h s,u pxq ´Xh s,s`kh pxq " Y h s,u pxq pu ´ps `khqq `σ pW u ´Ws`kh q Using (4.12), in terms of the tensor product (2. The L n -norm of the second integral term is bounded by }∇b}σ ? h. The assumption xx, bpxqy ď ´β }x} 2 , for some β ą 0, implies the stochastic flows X s,t pxq has uniform absolute moments of any order n ě 1 w.r.t. the time horizon, that is, we have that m n pxq ď κ n p1 `}x}q with m n pxq defined in (2.10). for some parameters λ ą 0 and v t ě 0, and some non negative processes pα t , β t , u t q. In this situation, for any ą 0 we have EpZ n t q 1{n ď e ş t 0 λn,sp qds EpZ n 0 q 1{n `ż t 0 e ş t s λn,up qdu z n s p q ds (7.4) This yields the estimate B t EpZ n t q 1{n " EpZ n t q ´p1´1{nq n ´1B t EpZ n t q ď λ n,t p q EpZ n t q 1{n `zn t This ends the proof of the lemma.

The stochastic flows

with
We set Y s,t pxq :" }∇ 2 X s,t pxq} 2 F and T s,t pxq :" }∇X s,t pxq} F and we also consider the collection of parameters }τ } F :" sup t,x }τ t pxq} F ρpυq :" sup t,x λ max pυ t pxqq with the tensor functions pτ t , υ t q introduced in (3.9). Observe that }τ } F ď }∇ The Skorohod integral S s,t p∆σqpxq is estimated using theorem 5.2. Using (7.5) and (5.9) we check that E " }X s,t pxq ´Xs,t pxq} n ‰ 1{n ď κ pδ 1 ,δ 2 q,n ´|||∆apxq||| n{p1´δ 1 q `|||∆bpxq||| n{p1´δ 1 q `|||∆σpxq||| 2n{δ 2 p1 _ }x}q ās soon as the regularity conditions pT q n{δ 1 , pM q 2n{δ 2 and pT q 2n{p1´δ 2 q are satisfied for some parameter n ě 2 and some δ 1 , δ 2 Ps0, 1r. Choosing δ 1 " p1 ´δ2 q{2 and setting δ " δ 2 we check that E " }X s,t pxq ´Xs,t pxq} n ‰ 1{n ď κ δ,n ´|||∆apxq||| 2n{p1`δq `|||∆bpxq||| 2n{p1`δq `|||∆σpxq||| 2n{δ p1 _ }x}q ās soon as pM q 2n{δ and pT q 2n{p1´δq are satisfied for some parameter n ě 2 and some δ Ps0, 1r. For instance, pMq 2n{δ and pT q 2n{p1´δq are satisfied as soon as β 2 ´α2 {2 ą pn{δ ´1q α 2 and λ A ą dpn{p1 ´δq ´1q ρ ‹ p∇σq 2

This ends the proof of (1.16).

  q pX s,u pxqq} n ˆ}∆b u pX s,u pxqq} n ‰ 1{n du. diffusion matrix σ t pxq " Σ. Here X h s,t pxq is the discretisation of X s,t pxq with resolution h. Note that that the drift at time t is not a function of the instantaneous value of X h s,t pxq, at time t, but rather the value it took at the largest discrete time-point before t. In section 4.4 we discuss how the formula in (1.10) also applies in this context and establish that

	for a fixed X h s,t pxq ´Xs,t pxq "	ż t	p∇X u,t q pX h s,u pxqq 1	" Y h s,u pxq ´bpX h s,u pxqq ı	du.
					s
	This comparison result when combined with the regularity assumptions (1.19) yields the moment
	bound below.
	Proposition 1.4. Assume that
	∇b `p∇bq 1 ď ´2λ I	}∇b} :" sup	}∇bpxq} ă 8 and xx, bpxqy ď ´β }x} 2	(1.19)
							x
	for some λ ą 0, β ą 0. In this situation, for any n ě 1 we have the uniform estimates
		E ´}X h s,t pxq ´Xs,t pxq} n ¯1{n	ď }∇b} ´r}bp0q} `p m n pxq }∇b}s h	`σ ?	h ¯{λ
							yields
	E } p∇X u,t A further conditioning argument and the above exponential bound on the tangent process yields " ż t }X s,t pxq ´Xs,t pxq} n ‰ 1{n ď s E "
	E	" }X s,t pxq ´Xs,t pxq} n ‰ 1{n	ď κ n	ż t	expp´λpnq pt ´uqqdu sup	Er}∆b u pX s,u pxqq} n s 1{n .
							s	sďu
	Replacing the term outside the time integral with |||∆bpxq||| n yields the stated result in (1.16)
	excluding the terms representing the difference in the diffusions.
	We illustrate one use of theorem 1.2 in the context of analyzing the error in discretising the
	diffusion X s,t pxq for some initial time point s ě 0. Let h ą 0 denote the discretisation interval size
	and for any t P rs `kh, s `pk `1qhr let
		dX h s,t pxq " Y h s,t pxq dt `Σ dW t with Y h s,t pxq :" b ´Xh s,s`kh pxq

  ,jq r∇X s,t pxq b ∇X s,t pxqs pi,jq,pk,lq " ∇X s,t pxq i,k ∇X s,t pxq j,l " r∇X s,t pxq b ∇X s,t pxqs 1 pk,lq,pi,jq

  This provides a way to estimate the growth of σ t pxq in terms of the one of a t pxq. For instance the estimate(1.16) combined with (2.22) implies that E " }X s,t pxq ´Xs,t pxq} n ‰ 1{n ď κ δ,n ´|||∆bpxq||| 2n{p1`δq `|||∆apxq||| 2n{δ p1 _ }x}q ' Assume that pMq n is satisfied for some n ě 1. Also let f t pxq be some multivariate function In this situation, we have the estimates |||f pxq||| n ď }f p0q} `}∇f } m n pxq and therefore |||f pxq||| n ď κ n p}f p0q} `}∇f }q p1 _ }x}q

	such that
	}f p0q} :" sup	}f t p0q} ă 8 and }∇f } :" sup	}∇f t pxq} ă 8
	t	t,x

and (2.21) we check that }∆σ t pxq} ď 1 ? υ }∆a t pxq} and }σ t pxq} ď }σ t p0q} `1 ? υ r}a t pxq} `}a t p0q}s (2.22)

  In terms of the tensor product(2.4), the gradient ∇X s,t pxq of the diffusion flow X s,t pxq is given by the gradient pd ˆdq-matrixd ∇X s,t pxq " ∇X s,t pxq « ∇b t pX s,t pxqq dt `ÿ 1ďkďr ∇σ t,k pX s,t pxqq dW k t ff where W k t is the k-th component of the Brownian motion. After some calculations we check that d " ∇X s,t pxq ∇X s,t pxq 1 ‰ " ∇X s,t pxq A t pX s,t pxqq ∇X s,t pxq 1 dt `dM s,t pxq (3.1) with the matrix function A t pxq defined in (2.7) and the symmetric matrix valued martingale

	k dW k t
	3 Variational equations
	3.1 The tangent process

dM s,t pxq :" ÿ 1ďkďr ∇X s,t pxq " ∇σ t,k pX s,t pxqq `∇σ t,k pX s,t pxqq 1 ‰ ∇X s,t pxq 1 dW k t

  Proposition 3.1. Assume pT q 2 is satisfied. Then E `}X s,t pxq ´Xs,t pyq} 2 ˘1{2 ďIn addition, we have the almost sure estimate ∇σ " 0 ùñ }X s,t pxq ´Xs,t pyq} ď e ´λA pt´sq }x ´y}(3.3)Proof of Prop. 3.1. Whenever pT q 2 is met, we have the following uniform estimate from (3.1)pT q 2 ùñ E `}∇X s,t pxq} 2 from imposing the initial condition ∇X s,s pxq " I on the resulting differential equation for B t E `}∇X s,t pxq} 2 F ˘1{2 . In addition, when ∇σ " 0 the martingale M s,t pxq " 0 is null and as a consequence of (3.1) we have the following almost sure estimate }∇X s,t } 2 :" sup

				?	d e ´λA pt´sq }x ´y}.	(3.2)
		2	˘1{2	ď E `}∇X s,t pxq} 2 F	˘1{2	ď	? d e ´λA pt´sq	(3.4)
	where the	? d term arises "ż 1				
				}∇X s,t p x `p1 ´ qyq} 2 2 d	}x ´y} 2
				0

x }∇X s,t pxq} 2 ď e ´λA pt´sq (3.5)

The Taylor expansion X s,t pxq ´Xs,t pyq "

ż 1 0 ∇X s,t p x `p1

´ qyq 1 px ´yq d ùñ }X s,t pxq ´Xs,t pyq} 2 ď

  ∇X s,t pxq " ∇σ t,k pX s,t pxqq `∇σ t,k pX s,t pxqq 1 ‰ ∇X s,t pxq 1 ˘dW k Tr `∇X s,t pxq " ∇σ t,k pX s,t pxqq `∇σ t,k pX s,t pxqq 1 ‰ ∇X s,t pxq 1 ˘2 This implies that B t E `}∇X s,t pxq} 4 F ˘" 2 E `}∇X s,t pxq} 2 F Tr " ∇X s,t pxq A t pX s,t pxqq ∇X s,t pxq 1 ‰ÿ 1ďkďr E ´Tr `∇X s,t pxq " ∇σ t,k pX s,t pxqq `∇σ t,k pX s,t pxqq 1 ‰ ∇X s,t pxq 1 ˘2W

	t
	Observe that
	ÿ
	B t xN s,. pxqy t "
	k

t , πq ď exp r´λpt ´sqs W n pµ, πq for all n ě 1. Taking the trace in (3.1) we also find that d }∇X s,t pxq} 2 F " Tr " ∇X s,t pxq A t pX s,t pxqq ∇X s,t pxq 1 ‰ dt `dN s,t pxq with the martingale dN s,t pxq " ÿ 1ďkďr Tr `

  ∇X s,t pxqs ∇ 2 σ t,k pX s,t pxqq `∇2 X s,t pxq∇σ t,k pX s,t pxqq ˘dW k

	t
	Consider the tensor functions
	ÿ
	υ t :"
	1ďkďd

t pxq " " r∇X s,t pxq b ∇X s,t pxqs ∇ 2 b t pX s,t pxqq `∇2 X s,t pxq∇b t pX s,t pxqq ‰ dt `dM s,t pxq with the null matrix initial condition ∇ 2 X s,s pxq " 0 and the matrix-valued martingale dM s,t pxq " ÿ 1ďkďr `r∇X s,t pxq b

  ∇X s,t pxqs ∇ 2 σ t,k pX s,t pxqq ∇ 2 X s,t pxq 1 `∇2 X s,t pxq ∇σ t,k pX s,t pxqq ∇ 2 X s,t pxq 1 ( X s,t pxqA t pX s,t pxqq ∇ 2 X s,t pxq 1 ‰ `2 " r∇X s,t pxq b ∇X s,t pxqs ∇ 2 b t pX s,t pxqq ∇ 2 X s,t pxq 1 ‰ symWhenever pT q 2 is met, taking the trace in the above display we check thatB t }∇ 2 X s,t pxq} 2 F ď ´2λ A }∇ 2 X s,t pxq} 2 F `2}∇ 2 b} F }∇X s,t pxq} 2 F }∇ 2 X s,t pxq} FThis yields the estimateB t }∇ 2 X s,t pxq} F ď ´λA }∇ 2 X s,t pxq} F `}∇ 2 b} F }∇X s,t pxq} 2 ∇X s,t pxqs ∇ 2 σ t,k pX s,t pxqq ∇ 2 X s,t pxq 1 `∇2 X s,t pxq ∇σ t,k pX s,t pxqq ∇ 2 X s,t pxq 1 ( 2 Proposition 3.3.Assume pT q n is met. In this situation, for any ą 0 s.t. λ A pnq ą we have

					sym dW k t
	When ∇σ " 0 the above equation reduces to
	B t	"	∇ 2 X s,t pxq∇ 2 X s,t pxq 1 ‰
	"	" ∇ 2 E `}∇ 2 X s,t pxq} n F	˘1{n	ď n ´1 χ pb, σq exp p´rλ A pnq ´ s pt ´sqq	(3.10)

t pxq A t pX s,t pxqq ∇ 2 X s,t pxq 1 ‰ `2 " r∇X s,t pxq b ∇X s,t pxqs τ t pX s,t pxqq ∇ 2 X s,t pxq 1 ‰ sym `"r∇X s,t pxq b ∇X s,t pxqs υ t pX s,t pxqq r∇X s,t pxq b ∇X s,t pxqs 1 ‰( dt `dN s,t pxq with the matrix function A t pxq defined in (2.7) and the tensor-valued martingale dN s,t px :q " 2 ÿ 1ďkďr r∇X s,t pxq b F Using (2.15) this implies that }∇ 2 X s,t pxq} F ď }∇ 2 b} F e ´λA pt´sq ż t s e λ A pu´sq }∇X s,u pxq} 2 F du ď d λ A }∇ 2 b} F e ´λA pt´sq This ends the proof of the almost sure estimate (2.16). For more general models, we have that d }∇ 2 X s,t pxq} 2 F " Tr " ∇ 2 X s,t pxq A t pX s,t pxqq ∇ 2 X s,t pxq 1 ‰ `2 Tr " r∇X s,t pxq b ∇X s,t pxqs τ t pX s,t pxqq ∇ 2 X s,t pxq 1 ‰ `Tr " r∇X s,t pxq b ∇X s,t pxqs υ t pX s,t pxqq r∇X s,t pxq b ∇X s,t pxqs 1 ‰( dt `dM s,t pxq with a continuous martingale M s,t pxq with angle bracket B t xM s,. pxqy t " 4 ÿ 1ďkďr Tr r∇X s,t pxq b

  is met. In this situation, we can extend gradient semigroup formulae to measurable functions using the Bismut-Elworthy-Li formula ∇P s,t pf qpxq " E `f pX s,t pxqq τ ω s,t pxq ˘(3.11)B u ω s,t puq ∇X s,u pxq a u pX s,u pxqq ´1{2 dW uThe above formula is valid for any function ω s,t : u P rs, ts Þ Ñ ω s,t puq P R of the following form ω s,t puq " ϕ ppu ´sq{pt ´sqq ùñ B u ω s,t puq "Let ϕ with Ps0, 1r be some differentiable function on r0, 1s null on r0, 1 ´ s and such that |Bϕ puq| ď c{ and pϕ p1 ´ q, ϕp1qq " p0, 1q. For instance we can choose

	In the above display ∇a ´1{2 u	stands for the tensor function
	p∇a ´1{2 u	pxqq pi,jq,k :" B x i a ´1{2 u	pxq j,k " ´´a ´1{2 u	pxq	" B x i a 1{2 u pxq ı	a ´1{2 u	pxq	¯j,k
	A detailed proof of the formulae (3.14) and (3.15) in the context of nonlinear diffusion flows can be
	found in the appendix in [5].
	Observe that with the stochastic process τ ω s,t pxq :" p2.19q ùñ sup ż t i s Whenever pT q 2 is met, using the estimate (3.3) for any Ps0, 1r }B x i a ´1{2 pxq} ď c }∇σ}{υ u
		}∇ 2 P s,t pf q} ď	κ	?	1
									t	1 ´s Bϕ ppu ´sq{pt ´sqq	(3.12)
									t	κ 1 ´s ż 1 0	e ´2λ A pt´sqv pBϕpvqq 2 dv
			$ &						0	if u P r0, 1 ´ s
		ϕ puq "	%	1 `cos ˆˆ1	`1 ´u	˙π 2	˙if u P r1 ´ , 1s
	In this situation, we check that
		}∇P s,t pf q} 2 ď		κ 2 2	t	1 ´s ż 1 1´	e ´2λ A pt´sqv dv
	from which we find the rather crude uniform estimate
			}∇P s,t pf q} ď	κ	1 t ? ´s e ´λA p1´ qpt´sq	(3.13)
	In the same vein, for any s ď u ď t we have the formulae
	∇ 2 P s,t pf qpxq " E ´f pX s,t pxqq τ s,t pxq `∇X s,t pxq ∇f pX s,t pxqq τ ω r2s,ω s,t pxq 1 ¯(3.14)
		" E ´f pX s,t pxqq	" τ r2s,ω s,u pxq `∇X s,u pxq τ ω u,t pX s,u pxqq τ ω s,u pxq 1	ı¯( 3.15)
	with the process
	τ s,t pxq r2s,ω
	ż t								ı
	:"								dW u

for some non decreasing differentiable function ϕ on r0, 1s with bounded continuous derivatives and such that pϕp0q, ϕp1qq " p0, 1q ùñ ω s,t ptq ´ωs,t psq " 1

Whenever pT q 2 is met, combining (3.4) with (3.11), for any f s.t. }f } ď 1 we check that }∇P s,t pf q} 2 ď E `}τ ω s,t pxq} 2 ď κ 1 ż t s e ´2λ A pu´sq }B u ω s,t puq} 2 du " s B u ω s,t puq " ∇ 2 X s,u pxq a u pX s,u pxqq ´1{2 `p∇X s,u pxq b ∇X s,u pxqq p∇a ´1{2 u qpX s,u pxqq

  1.1 are satisfied. Rewritten in terms of the stochastic semigroups P s,t and P s,t we obtain the forward-backward multivariate interpolation formula ∇P u,t pf qpX s,u pxqq 1 ∆σ u pX s,u pxqq dW u (4.6) Using elementary differential calculus, for twice differentiable (column vector-valued) function f from R d into R p we readily check the gradient and the Hessian formulae ∇ P s,t pf qpxq " ∇X s,t pxq P s,t p∇f qpxq ∇ 2 P s,t pf qpxq " r∇X s,t pxq b ∇X s,t pxqs P s,t p∇ 2 f qpxq `∇2 X s,t pxq P s,t p∇f qpxq (4.7)

	and the two-sided stochastic integral term given by
	ż t
	S s,t pf, ∆σqpxq :"
	s
	P s,t pf qpxq ´Ps,t pf qpxq " T s,t pf, ∆a, ∆bqpxq `Ss,t pf, ∆σqpxq		(4.4)
	with the stochastic integro-differential operator
	T s,t pf, ∆a, ∆bqpxq
	ż t		(4.5)
	:"	du

s " ∇P u,t pf qpX s,u pxqq 1 ∆b u pX s,u pxqq `1 2 ∇ 2 P u,t pf qpX s,u pxqq 1 ∆a u pX s,u pxqq

  Corollary 4.1. For any twice differentiable function f from R d into R with bounded derivatives we have the forward-backward semigroup interpolation formula P s,t pf qpxq ´P s,t pf qpxq " E `x∇P u,t pf qpX s,u pxqq, ∆b u pX s,u pxqqy ˘du

	ż t
	s
				(4.8)
		ż t
	`1 2	s	E `Tr	" ∇ 2 P u,t pf qpX s,u pxqq ∆a u pX s,u pxqq ‰˘d u

In addition, under some appropriate regularity conditions for any differentiable function f such that }f } ď 1 and }∇f } ď 1 we have the uniform estimate |P s,t pf qpxq ´P s,t pf qpxq| ď κ r|||∆apxq||| 1 `|||∆bpxq||| 1 s (4.9)

  13) with (3.16) for any differentiable function f s.t. }f } ď 1 and }∇f } ď 1 and for any Ps0, 1r we check that |P s,s`t pf qpxq ´P s,s`t pf qpxq| ď κ r|||∆apxq||| 1 `|||∆bpxq||| 1 s

	ż t 0	1 ? u	e ´λA p1´ qu du

  Let ς t pxq be some differentiable pd ˆrq-matrix valued function on R d such that

	5 Skorohod fluctuation processes
	5.1 A variance formula
				}∇ς} ă 8 and }ςp0q} :" sup	}ς t p0q} ă 8	(5.1)
									t
									1 ď n ď 8
	we have the estimates
	E	"	}X s,t pxq ´Xs,t pxq} n ‰ 1{n	ď	ż t	e ´λA pt´uq E	" }b u pX s,u pxqq ´Y s,u pxq} ‰ 1{n du	(4.13)
							s
	In the same vein, we have
					ż t
	P s,t pf qpxq ´P s,t pf qpxq "
	For instance, for the Langevin diffusion discussed in (2.18) and (3.7) the weak expansion (4.14)
	implies that
			ż t	ż
	rπP s,t ´πspf q "	πpdxq E `x∇P t´u pf qpX s,u pxqq, ∇U pX s,u pxqq `Y s,u pxqy ˘du	(4.15)
			s
	This yields the W 1 -Wasserstein estimate
				ż t
			W 1 pπP s,t , πq| ď	e ´λA pt´uq
				s

s E `x∇P u,t pf qpX s,u pxqq, b u pX s,u pxqq ´Y s,u pxqy ˘du (4.14) ż πpdxq E `}∇U pX s,u pxqq `Y s,u pxq} ˘du Combining (3.13) with (4.15), for any Ps0, 1r we also have the total variation norm estimate }πP s,t ´π} tv ď c ż t s 1 ? t ´u e ´λA p1´ qpt´uq "ż πpdxq E `}∇U pX s,u pxqq `Y s,u pxq} ˘ du (4.16)

  3 X t`h 2 qpx, ∆X t pxqq ∆X t pxq 2 X t`h 2 qpxq ∆X t pxq `BX t`h 2 pxq ˆ∆X 1 t pxq `pB 3 X t`h 2 qpx, ∆X t pxqq ∆X t pxq 2 `pB 2 X t`h 2 qpx, ∆X t pxqq ∆X t pxq ˆ∆X 1

	This implies that
	pBX t ´BX t`h 2 qpxq
	" pB 2 t pxq
	from which we conclude that

1

  To take the final step, observe thatE p∆Y s pyq ∆W s q E pp∆X t qpX s`h 1 ,t pyqq ∆W t | W s`h 1 ,t q BσpX s`h 1 ,u pyqq pBX t,u qpX s`h 1 ,t pyqq du ˙`O

	" E	ˆż s`h 1	bpY s,u pyqq pW s`h 1 ´Wu q du ˙`E	ˆż s`h 1	σpY s,u pyqq du "
			s			s
	E	ˆż s`h 1 s	σpY s,u pyqq du ˙`O	1 ´h1`1{2	Īn
	the same vein, we have
			" E	ˆż t`h 2 t	σpX s`h 1 ,u pyqq du | W s`h 1 ,t ˙`O	2 ´h1`1{2	ānd
	E `p∆X 1 t qpX s`h 1 ,t pyqq ∆W t | W s`h 1 ,t	"
			ˆż t`h 2		´h1`1{2	"
		E				2
		E	ˆż t`h 2 t	B pσ ˝Xt,u q pX s`h 1 ,t pyqq du ˙`O	´h1`1{2 2

t

  The functions F t pxq, G t pxq and H t pxq as well as the differentials BH t pxq and B 2 F t pxq are continuous w.r.t. pt, xq for any given ω P Ω. In addition, for any n ě 1 we have sup |y|ďn p|F t pY t `yq| _ |H t pY t `yq| _ |G t pY t `yq|q ď g n ptq sup |y|ďn `|BH t pY t `yq| _ |BF t pY t `yq| _ |B 2 F t pY t `yq| ˘ď g n ptq with E pH 2 q 1 : The Malliavin derivatives D s BF t pxq and D s H t pxq are continuous w.r.t. x and ps, tq for any given ω P Ω. In addition, for any n ě 1 we have sup |y|ďn p|pD s F t qpY t `yq| _ |pD s H t qpY t `yq|q ď h n ps, tq sup |y|ďn p|pD s BF t qpY t `yq|q ď h n ps, tq with E ˜żr0,1s 2 h 4 n ps, tq dsdt ¸ă 8

	ˆż 1	g 4 n ptq dt ˙ă 8
	0
		(6.8)

  to the product function ΓpX t pxq, ϕ pY t ´xqq :" X t pxq ϕ pY t ´xq we check that pF t pxq ϕ pY t ´xqq ´pF 0 pxq ϕ pY 0 ´xqq " F s pxq Bϕ pY s ´xq B s `1 2 F s pxq B 2 ϕ pY s ´xq A s ´ϕ pY s ´xq G s pxq v s pxq :" F s pxq Bϕ pY s ´xq Σ s ´ϕ pY s ´xq H s pxq The stochastic integral in the r.h.s. of (6.11) can be interpreted as a two-sided stochastic integral. Recalling that D t ϕ pY s ´xq " D t Y s Bϕ pY s ´xq Bϕ pY s ´xq H s pxq ´ϕ pY s ´xq D t H s pxq 40Condition pH 3 q ensures that the processes Y t and D t Y s have moments of any order. In addition, under the regularity conditions pH 1 q 1 and pH 2 q 1 we check that U s ds as well as F t pY t q ÝÑ Ñ0 F t pY t q

	ż t 0 ˜"ż t u s pxq ds 0 v s pxq dW s `ż t 0  2 v s pxq dW s ¸dx ă 8 Applying the Fubini theorem for Skorohod and measure theory integrals (see for instance [34, 37, 44] (6.11) with u s pxq :" we check that E ˆż t 0 u s pxq 2 ds ˙dx ă 8 and ż E and the work by Leon [35]) we check that F t pY t q :" ż F t pxq ϕ pY t ´xq dx " ż F 0 pxq ϕ pY 0 ´xq dx `ż t 0 U s ds " ż t 0 V s dW s with U s :" ż u s pxq dx and V s :" ż v s pxq dx Integrating by parts where derivatives of ϕ appear we check that U s :" ż ˆBF s pxq B s `1 2 B 2 F we have ż t 0 U s ds ÝÑ Ñ0 ż t 0 It remains to check that Ep ż t D ż 0

t v s pxq " D t F s pxq Bϕ pY s ´xq Σ s `Fs pxq D t Y s B 2 ϕ pY s ´xq Σ s `Fs pxq Bϕ pY s ´xq D t Σ s ´Dt Y s s pxq A s ´Gs pxq ˙ϕ pY s ´xq dx V s :" ż pBF s pxq Σ s ´Hs pxqq ϕ pY s ´xq dx

From the a.s. continuity of F t pxq in x for each t ě 0, we have

F t pY t q ´Ft pY t q " ż pF t pY t ´ xq ´Ft pY t qq ϕpxq dx ÝÑ Ñ0 0

The functions BF t pxq, B 2 F t pxq and G t pxq are almost surely continuous w.r.t. x and uniformly locally bounded. In addition, the random variables A t and B t are integrable at any order. Moreover, under pH 1 q 1 there exists some parameter n ě 0 depending on the support of ϕ such that for any ą 0 we have the estimate

|U s | ď

sup |y|ďn |BF s pY s `yq| |B s | `1 2 sup |y|ďn |B 2 F s pY s `yq| |A s | `sup |y|ďn |G s pY s `yq| ď g n ptq p1 `|A s | `|B s |q Thus, by the dominated convergence theorem on pΩˆr0, 1sq equipped with the measure pPpdωqbdtq

  D t pBF s pxq Σ s ´Hs pxqq ϕ pY s ´xq dx `ż pBF s pxq Σ s ´Hs pxqq D t ϕ pY s ´xq dx pBF s pxq Σ s ´Hs pxqq ``B 2 F s pxq Σ s ´BH s pxq ˘Dt Y s ‰ ϕ pY s ´xq dx Observe that D t pBF s pxq Σ s ´Hs pxqq ``B 2 F s pxq Σ s ´BH s pxq ˘Dt Y s " ppD t BF s qpxq `B2 F s pxq D t Y s q Σ s `BF s pxq D t Σ s ´ppD t H s qpxq `BH s pxq D t Y s q

	Using the chain rule property we have
	D t V s
	ż
	:"
	Integrating by parts, we check that
	ż
	D t V s "	"	D t

2 ds ď 2 ż t 0 ż pBF s pxq ´BF s pY s qq 2 Σ 2 s ϕ pY s ´xq dx ds `2 ż t 0 pH s pxq ´Hs pY s qq 2 ϕ pY s ´xq dx ds

  2 ˆR ppD t BF s qpxq ´pD t BF s qpY s qq 2 A s ϕ pY s ´xq ds dt dx J5 p q :" E ˜żr0,1s 2 ˆR ppD t H s qpxq ´pD t H s qpY s qq 2 ϕ pY s ´xq ds dt dx p∇X u,t q pX s,u pxqq 1 b p1q u pX s,u pxqq du The remainder second order term B 2 δ X s,t pxq in the above display is such that for any n ě 2 s.t. λ A pnq ą 0 we have the uniform estimate p∇X u,t q pX s,u pxqq 1 ∇b p1q u ´Xs,u pxq ` pX δ s,u pxq ´Xs,u pxqq, y ¯1 DX δ s,u pxq du d Combining proposition 3.3 with the estimate (7.2) for any n ě 2 s.t. λ A pnq ą 0 we check that X s,t pxq} n s 1{n ď c p1 `n χ pb, σq{λ A pnqq ´}b p1q }{λ A pnq

	with the first order stochastic flow
								ż t
								BX s,t pxq :"
								s
								sup s,t,x	Er}B 2 δ X s,t pxq} n s 1{n ď c n
	Proof. Using (4.12) we readily check that
			DX δ s,t pxq :" δ ´1rX δ s,t pxq ´Xs,t pxqs "	ż t s	p∇X u,t q pX δ s,u pxqq 1 b p1q δ,u pX δ s,u pxqq du
	By proposition 3.2 for any n ě 2 we have
			λ Àpnq :" λ A ´pn ´2qρp∇σq 2 {2 ą 0 ùñ E ´}DX δ s,t pxq} n ¯1{n	ď c }b p1q }{λ Àpnq	(7.2)
	This yields the first order Taylor expansion (7.1) with
								B 2 δ X s,t pxq :" B δ p2,1q	X s,t pxq	`Bp2,2q δ	X s,t pxq
	and the second order remainder terms
	B δ p2,2q	X s,t pxq :"	ż t s	p∇X u,t q pX δ s,u pxqq 1 b p2q δ,t pX δ s,u pxqq du
	B δ p2,1q	X s,t pxq :" 2δ	s ´1 ż t	" p∇X u,t q pX δ s,u pxqq ´p∇X u,t q pX s,u pxqq ı 1	b p1q u pX δ s,u pxqq du
								`2δ	´1 ż t	p∇X u,t q pX s,u pxqq 1 rb p1q u pX δ s,u pxqq ´bp1q
								s
								E ´}B δ p2,2q	X s,t pxq} n ¯1{n	ď c }b p2q }{λ Àpnq
	To estimate B δ p2,1q	X s,t pxq we need to consider the second order decompositions
	2 ´1 B δ p2,1q	X s,t pxq
	"	ż 1	ż t	" ∇ 2 X u,t	‰ ´Xs,u pxq ` pX δ s,u pyq ´Xs,u pxqq	¯1 " b p1q u pX δ s,u pxqq b DX δ s,u pxq ı	du d
		0	s
	`ż 1	ż t
		0	s
						p2,1q δ Er}B

u pX s,u pxqqs du Arguing as above, for any n ě 2 s.t. λ Àpnq ą 0 we have the uniform estimate

  [START_REF] Alekseev | An estimate for the perturbations of the solution of ordinary differential equations[END_REF] we readily check that ´}Y h s,u pxq ´bpX h s,u pxqq} n ¯1{n du where the second line follows from the exponential estimate of the tangent process from proposition 3.1. The integrand will be bounded as follows: for any s `kh ď u ă s `pk `1qh and any n ě 1 we have E ´}bpX h s,u pxqq ´Y h s,u pxqq} n ¯1{n ď }∇b} ´r}bp0q} `p m n pxq }∇b}s h `σ ? h which then yields the stated result of the proposition. We now prove the stated bound on the difference of the drift processes. For any s `kh ď u ă s `pk `1qh we have

		X h s,t pxq ´Xs,t pxq "	ż t	p∇X u,t q pX h s,u pxqq 1	" Y h s,u pxq ´bpX h s,u pxqq	ı	du
						s
	Combining (3.5) with the Minkowski integral inequality we check that
	E ´}X h s,t pxq ´Xs,t pxq} n ¯1{n	"	ż t	E ´} p∇X u,t q pX h s,u pxqq 1	" Y h s,u pxq ´bpX h s,u pxqq	ı	} n ¯1{n	du
						s
						ż t
	" e ´λpt´uq E bpX h s s,u pxqq ´Y h s,u pxq
	"	"ż 1	∇b ´Xh s,s`kh pxq ` pX h s,u pxq ´Xh s,s`kh pxqq ¯1 b ´Xh s,s`kh pxq	¯d 	pu ´ps `khqq
		0
			`"ż 1	∇b ´Xh s,s`kh pxq ` pX h s,u pxq ´Xh
			0

s,s`kh pxqq ¯1 d  σ pW u ´Ws`kh q (7.3)

  X h s,t pxq also obey a similar moment bound: observe that for any t P rs `kh, s pk `1qhr we have We can check that the stochastic flows X h s,t pxq also have uniform moments w.r.t. the time horizon; that is, for any n ě 1 we have that p m n pxq :" sup }X s,t pxq} 2n ‰ ď ´rγ 2 ´2pn ´1qα 2 s E " }X s,t pxq} 2n ‰ `rγ 1 `2pn ´1qα 1 s E " }X s,t pxq} 2n´1 ‰ `rγ 0 `2pn ´1qα 0 s E " }X s,t pxq} 2pn´1q ‰ }X s,t pxq} 2n ‰ 1{n ď ´rγ 2 ´2pn ´1qα 2 s E " }X s,t pxq} 2n ‰ 1{n `rγ 1 `2pn ´1qα 1 s E " }X s,t pxq} 2n ‰ 1{p2nq `rγ 0 `2pn ´1qα 0 s from which we check that for any ą 0 we have B t E " }X s,t pxq} 2n ‰ 1{n ď ´rγ 2 ´2pn ´1qα 2 ´2 s E " }X s,t pxq} 2n ‰ 1{n `1 8 rγ 1 `2pn ´1qα 1 s 2 `rγ 0 `2pn ´1qα 0 s }X s,t pxq} 2n ‰ 1{n ď ´2 rβ 2 ´pn ´1{2qα 2 ´ s E " }X s,t pxq} 2n ‰ 1{n `1 8 rγ 1 `2pn ´1qα 1 s 2 `rγ 0 `2pn ´1qα 0 s from which we check that E " }X s,t pxq} 2n ‰ 1{n ď e ´2rβ 2 ´pn´1{2qα 2 ´ spt´sq }x} 2 `1 8 rγ 1 `2pn ´1qα 1 s 2 `rγ 0 `2pn ´1qα 0 s 2 rβ 2 ´pn ´1{2qα 2 ´ s as soon as ă β 2 ´pn ´1{2qα 2 and n ě 1. Replacing by pβ 2 ´pn ´1{2qα 2 q and then p2nq by n we check that E r}X s,t pxq} n s 1{n ď e ´p1´ qβ 2 pnqpt´sq }x} `1 4 a p1 ´ q γ 1 pnq `γ0 pnq 1{2 β 2 pnq 1{2 with γ i pnq :" γ i `pn ´2qα i This ends the proof of (2.10). The proof of the estimate (3.10) is mainly based on the following technical lemma of its own interest. Lemma 7.2. Let Z t be a non negative diffusion process satisfying in integral sense an inequality of the following form dZ t ď p´λZ t `αt a Z t `βt q dt `dM t with B t xM y t ď pu t a Z t `vt Z t q 2

	Proof of proposition 3.3
	This implies that
		B t E	"
	d}X h s,t pxq} 2
		"					ı
	ď	´2λ 0 }X h s,t pxq} 2 `2 xX h s,t pxq, bpX h s,s`kh pxqq ´bpX h s,t pxqqy `σ2 d	dt `2σ X h s,t pxq 1 dW t
	Thus, for any ą 0 we have
		d}X h s,t pxq} 2 ď	" p´2λ 0 ` q}X h s,t pxq} 2 ` ´1}∇b} `σ2 d ı	dt `2σ X h s,t pxq 1 dW t
	This implies that
	B t E	"
				hě0	sup těs	E	" }X h s,t pxq} n	ı 1{n	ď c n p1 `}x}q

F ‰ dt `2 ř k xX s,t pxq, σ k,t pX s,t pxqqy dW k t After some elementary computations, for any n ě 1 we check that n ´1B t E "

  Ōnthe other hand, for any ą 0 we have the almost sure inequality rα t `pn ´1qu t v t s Z

	This implies that
					n ´1B t EpZ n t q
					ď λ n,t p q EpZ n t q	`E "ˆβ t	`n	´1 2	u 2 t `1 2	t rα t `pn ´1qu t v t s 2 ˙Zn´1	
	Applying Hölder inequality we check that
	E	"ˆβ t	`n	´1 2	u 2 t `1 2	t rα t `pn ´1qu t v t s 2 ˙Zn´1	
	ď E	"ˆβ	t	`n	´1 2	u 2 t `1 2	rα t `pn ´1qu t v t s 2	˙n 1{n	EpZ n t q 1´1{n ď z n t EpZ n t q 1´1{n
	the parameters
	λ n,t p q :" ´λ	`n	´1 2	v 2 t `	2
	z n t p q :" E rβ n t s 1{n `n	´1 2	E	"	u 2n t	‰ 1{n	`1	´E " α 2n t	‰ 1{n	`pn ´1q 2 E	"	pu t v t q 2n ‰ 1{n	Proof.
	Applying Itô's formula, for any n ě 2, we have
	n ´1B t EpZ n t q
	ď E	"	Z n´1 t	p´λZ t `αt	a Z t `βt q	`n	´1 2	pu t	a Z t `vt Z t q 2 Z n´2 t	
	" ˆ´λ	`n	´1 2	v 2 t ˙EpZ n t q	`E "ˆβ	t	`n	´1 2	u 2 t	˙Zn´1 t		`E ´rα t `pn ´1qu t v t s Z	n´1{2 t
														pn´1q{2 t	Z	n{2 t	ď	1 2	rα t `pn ´1qu t v t s 2 Z n´1 t	`	2	Z n t

  2 b} F `d }∇ 2 σ} 2 Whenever pT q 2 is met we have Tr " ∇ 2 X s,t pxq A t pX s,t pxqq ∇ 2 X s,t pxq 1 ‰ ď ´2λ A Y s,t pxq Also observe that |Tr " r∇X s,t pxq b ∇X s,t pxqs τ t pX s,t pxqq ∇ 2 X s,t pxq 1 ‰ | ď }τ } F Y s,t pxq 1{2 T s,t pxq 2 and Tr " r∇X s,t pxq b ∇X s,t pxqs υ t pX s,t pxqq r∇X s,t pxq b ∇X s,t pxqs 1 ‰ ď ρpυq T s,t pxq 4In the same vein, we have|Tr r∇X s,t pxq b ∇X s,t pxqs ∇ 2 σ t,k pX s,t pxqq ∇ 2 X s,t pxq 1 `∇2 X s,t pxq ∇σ t,k pX s,t pxqq ∇ 2 X s,t pxq 1 ( | ď }∇ 2 σ k } F T s,t pxq 2 Y s,t pxq 1{2 `ρp∇σ k q Y s,t pxqWe are now in position to prove proposition 3.3.Proof of proposition 3.3:Applying the above lemma to the processesZ t " Y s,t pxq λ " 2λ A α t " 2}τ } F T s,t pxq 2 β t " ρpυq T s,t pxq 4 and the parameters u t " 2 ? d }∇ 2 σ} F T s,t pxq 2 and v t " 2 ? d ρ ‹ p∇σq we obtain the estimate (7.4) with the parameters λ n,t p q :" ´2 " λ A ´dpn ´1qρ ‹ p∇σq 2 ´ 4 ı z n t p q :" ρpυq `2dpn ´1q }∇ 2 σ} 2 F `4 `}τ } 2 F `4 d 2 pn ´1q 2 ρ ‹ p∇σq 2 }∇ 2 σ} 2 cn 2 p1 _ ´1q χ pb, σq 2 E " }∇X s,t pxq} 4n F ‰ 1{n for some universal constant c ă 8 and the parameter χ pb, σq defined in (2.6). Using (3.8) we check that E `}∇ 2 X s,t pxq} 2n ´ 4 spt´uq e ´4rλA´pn´1qρp∇σq 2 spu´sq du " cn 2 p1 _ ´1q χ pb, σq 2 e ´2rλA´dpn´1qρ‹p∇σq 2 ´ 4 spt´sq ż t s e ´2rλA´pn´1qρp∇σq 2 `pn´1qrdρ‹p∇σq 2 ´ρp∇σq 2 s` 4 spu´sq du Assume that λ A ą dpn ´1qρ ‹ p∇σq 2In this case there exists some 0 ă n ď 1 such that for any 0 ă ď n we haveλ A ´dpn ´1qρ ‹ p∇σq 2 ąUsing (2.14), the generalized Minkowski inequality applied to (1.10) whenever pT q n{δ is met for some δ Ps0, 1r and n ě 2 gives E r}T s,t p∆a, ∆bqpxq} n s 1{n ď κ n{δ λpn{δq ´|||∆bpxq||| n{p1´δq `|||∆apxq||| n{p1´δq ¯with pκ n , λpnqq given in (2.13).(7.5)

	Proof of (1.16)
							˘*
							F
			ˆE " }∇X s,t pxq} 4n F	‰ 1{n
	Observe that
	z n t p q ď F ˘1{n
	ď cn 2 p1 _ ´1q χ pb, σq 2 e ´2rλA´dpn´1qρ‹p∇σq 2 and therefore ż t s
	E `}∇ 2 X s,t pxq} 2n F	˘1{p2nq	ď c n ´1 χ pb, σq exp	`´" λ A ´dpn ´1qρ ‹ p∇σq 2 ´	‰	pt ´sq This
	ends the proof of the proposition.

F and ρpυq ď d }∇ 2 σ} 2 2

Acknowledgments

P. Del Moral is supported in part from the Chair Stress Test, RISK Management and Financial Steering, led by the French Ecole polytechnique and its Foundation and sponsored by BNP Paribas, and by the ANR Quamprocs on quantitative analysis of metastable processes.

We also thank the anonymous reviewers for their excellent suggestions for improving the paper. Their detailed comments greatly improved the presentation of the article.

* P. Del Moral was supported in part from the Chair Stress Test, RISK Management and Financial Steering, led by the French Ecole polytechnique and its Foundation and sponsored by BNP Paribas, and by the ANR Quamprocs on quantitative analysis of metastable processes.

Using this bounds, we check that

Ep}bpX h s,s`kh pxqq} n q 1{n " }bp0q} `m n pxq}∇b}

The end of the proof now follows elementary manipulations, thus it is skipped. The proof of proposition 1.4 is now completed.

Arguing as above, by the dominated convergence theorem we conclude that the Skorohod integral ż t 0 V s dW s converges in L 2 pΩq as Ñ 0 to the Skorohod integral

This ends the proof of (6.12), and the proof of the theorem is now easily completed.

We end this section with some comments. Remark 6.4. Recalling that the diffusion flow Y t introduced in (6.1) has finite absolute moments of any order, the integrability conditions stated in (6.8) and (6.9) are satisfied as soon as the functions F t , G t , H t , the differentials BF t , B 2 F t , BH t , and the Malliavin derivatives D s H t , D s BF t have at most polynomial growth w.r.t. the state variable.

It is now readily check that pH 1 q 1 and pH 2 q 1 are met for the random fields introduced in (6.5).

The proof can be also be extended without difficulties to multivariate models. Following the proof of proposition 3.1 in [START_REF] Ocone | A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations[END_REF], an alternative proof of theorem 6.3 based on Itô formula for Hilbert space valued processes can be developed. This elegant functional approach requires to introduce a custom Hilbert-space valued processes framework but this approach avoids to do explicitly the interchange of integration using the Fubini theorem for Skorohod and measure theory integrals. As the statement of proposition 3.1 in [START_REF] Ocone | A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations[END_REF], the assumptions of theorem 6.3 can also be weaken when expressed in terms of this generalized stochastic calculus for Hilbert-space valued processes. for some perturbation parameter δ P r0, 1s and some functions b piq δ,t pxq with i " 1, 2. In this context, the stochastic flow X s,t pxq :" X δ s,t pxq can be seen as a δ-perturbation of X s,t pxq :" X 0 s,t pxq. We further assume that the unperturbed diffusion satisfies condition pT q 2 . To avoid unnecessary technical discussions on the existence of absolute moments of the flows we also assume that b With some additional work to estimate the absolute moments of the flows, the perturbation analysis presented below allows to handle more general models. The methodology described in this section can also be extended to expand the flow X δ s,t pxq at any order as soon as δ Þ Ñ b δ,t pxq is sufficiently smooth.

Illustrations

The first order approximation is given by the following theorem.