Luís Almeida

Patrizia Bagnerini

Multi-scale tissular-cellular model for wound healing

Keywords: Actin cable, Wound healing, Dorsal closure, Movement of epithelia, Forces in embryogenesis

In previous works we developed continuous mathematical models for wound healing and dorsal closure in Drosophila embryos. In this paper we extend this study to the case of non convex wounds in Drosophila pupal epithelium where the sign of the local curvature of the boundary plays an important role in determining the type of acto-myosin contractile structure that is formed. Moreover, we propose a multi-scale model where we combine the previous continuous approach with a cellular-level model that also takes into account interfacial tension between cells. We therefore minimize an extended energy functional so that the junctions of the cells are moved through successive configurations in order to obtain a new mechanical equilibrium. We apply this model to study some simple situations of cell sorting and the movement of genetic clones.

Introduction

Extension of an epithelial membrane to close a hole is a very widespread process both in morphogenesis and in tissue repair. Contraction of actin structures (in one, two or three dimensions) plays an important role in many cellular and tissue movements, both at a multicellular tissue level and at a cellular (and even intracellular) one: from muscle contraction to cell crawling and the contractile ring in cytokinesis. In the [START_REF] Almeida | Tissue repair modeling, Singularities in nonlinear evolution phenomena and applications[END_REF], [START_REF] Almeida | Modeling actin cable contraction[END_REF], [START_REF] Almeida | A mathematical model for dorsal closure[END_REF] we proposed various mathematical models for simulating the contraction of an actin cable structure attached to an external epithelial tissue in different applications such as wound healing or dorsal closure in Drosophila Melanogaster (fruit fly) embryos.

The present work is a natural sequel of these works, but here we are more concerned with wound healing in the pupal stage of Drosophila. An interesting feature of this stage is that the epidermis is under considerably less tension than in embryos. This enables us to make holes which do not become convex while they open, differently to embryo experiments where the the strong pull of the external epidermis renders the hole convex during the opening phase.

In particular, in collaboration with A. Jacinto's lab (CEDOC, Universidade Nova de Lisboa), we generated C-shaped wounds, where part of the boundary is concave and the rest convex (see figure 1). Observing the healing of this type of wound, we realized that the parts of the boundary where the curvature is positive and those where it is negative don't behave in the same way (see also the discussion in [START_REF] Almeida | Predictive power of "a minima" models in biology[END_REF]). We propose here (in section 2) to extend the previous model to describe and simulate this phenomenon. Then, in section 3, we couple the continuum model with a cellular one, where the epidermal tissue is described by a two-dimensional network of cells interacting through their common boundaries.

There are many different cell types in multicellular organisms that result from differentiation from embryo stem cells (through specific gene expression) as part of tissue specialization mechanisms: during morphogenesis cells are separated into compartments that are essential for proper assembly of the body's organs. Inside these compartments, once differentiated, cells are usually committed to a particular lineage and cells belonging to each compartment stay together and do not mix with those from other compartments. Cell sorting is therefore an interesting and open problem. Two main hypothesis have been formulated: first, cell segregation at compartment boundaries could be based on differential cell adhesion or affinity (i.e. cell populations might develop distinct adhesive properties which prevent intermingling), but molecules involved in these processes have still not been completely identified. The second hypothesis is the differential interfacial tension i.e. cells in contact with neighboring cells of a different type, increase the tension at the interfaces and contract the corresponding surfaces.

Motivated by these problems and by some experiences in collaboration with A. Jacinto (work in progress), we consider the tissue formed by a group of cells of two different types, for instance wild type ones and others that are mutant for a certain gene. In section 4, in order to take into account the difference in the interfacial tension between cells of different types, we introduce in the continuum-cellular model an extended energy functional. The junctions of the cells are then moved through successive configurations minimizing this functional in order to obtain a new mechanical equilibrium at each step.

C-shaped wounds in Drosophila pupae

In the few minutes that follow a laser (or a mechanical) circular wounding of the epidermis filamentous actin and myosin II concentrate inside the adjacent cells and give rise to a local acto-myosin cable anchored to the adherens junctions that bind the cell to its neighbors. The result is a continuous, supracellular acto-myosin cable that encircles the wound and reduces the wound's perimeter by a purse-string mechanism. In the (not yet published) experiences in collaboration with A. Jacinto's lab, in the C-shaped wounds on pupae, the acto-myosin cable seems to form mainly on the part of the boundary where the curvature is positive. For this reason, in our model the contractile actin cable term is given by max(κ, 0), where κ represents the curvature.

In [START_REF] Almeida | Implicit scheme for mean curvature flow with obstacles[END_REF], the authors considered the question of arbitrarily shaped wounds from a theoretical point of view: denoting by E(t) (a subset of R 2 ) the position of the wound at time t, they study C 1,1 solutions of the formal geometric equation

V (x) = κ(x) if x ∈ Ω max(κ(x), 0) if x ∈ ∂Ω (1)
where κ denotes the curvature of a closed curves ∂E(t) ⊂ R 2 (boundaries of the sets E(t)) and V the normal inward velocity (i.e. pointing inside E(t)). They prove that the solution E(t) of the mean curvature flow with obstacle is contained at each time t in the corresponding solution of the unconstrained mean curvature flow starting from the same set E(0). In the context of our original biological problem and the proposed models, this result indicates that the strategy of assembling an acto-myosin cable only in the positive curvature part of the boundary of the wound (or hole) allows us close it in a more efficient way than if we had assembled the cable all around the boundary. Flowing only by the positive part of the curvature corresponds (see [START_REF] Almeida | Implicit scheme for mean curvature flow with obstacles[END_REF]) to the special case of problem [START_REF] Almeida | A mathematical model for dorsal closure[END_REF] where the obstacle is taken to be the initial position of the boundary. Another nice feature of this type of flow having a natural biological interpretation is the fact that, during the wound closure phase, the epidermis advances without ever retreating, even locally or temporarily, from a region it has occupied (which would not be always true for a full curvature flow corresponding to having assembled a cable all around the wound closure).

In adult wounds, the main closure mechanism is lamellipodial crawling, i.e. the cells in the first rows extend lamellipodia (which are essentially two dimensional actin structures) that attach to extracellular matrix and pull the epithelium forward into the wounded area. Beneath, at the dermal level, activated fibroblasts proliferate and give rise to the granulation tissue which actively contracts. Both these contributions will be taken into account in the model by introducing in the equation ( 2) the term f pull n, i.e. an active pull (of the lamellipodia or the connective tissue) on the leading edge that moves it inwards to close the hole.

For epidermal wounds and the morphogenetic movements that we consider, the time scale of the closure is very long (hours) while the space scale is very small (cell characteristic length -of the order of a few µm). Therefore, it is reasonable to, in a first approximation, neglect inertial forces, and assume that the dynamic process of wound closure is a succession of static equilibria, i.e. to do a quasi-static approximation.

Taking into account the previous discussion, we propose the following quasistatic model: we consider as simulation domain a rectangle M , which contains the wound at time step i, denoted by W i . Let D i be the part of the domain occupied by the epidermis i.e. D i = M \ W i and ω i = ∂W i the boundary of the wound ("the leading edge"). The acto-myosin cable tension gives rise to a force that is proportional to the curvature. This term will be described by a normal force which is proportional to the local curvature of the leading edge at each point. It points towards the interior of the wound W i at the points of positive curvature and towards the exterior at points of negative curvature. We assume that at each time step i, the corresponding displacement field u i satisfies

     -∆u i = 0 in D i , u i = 0 on ∂M, ∂u i ∂n = f cable max(κ, 0) n + f pull n on ω i . ( 2
)
where n is the external unit normal to ∂D i at each point, κ the curvature, f cable the function associated with the intensity of the cable tension at each point along the leading edge ω i and f pull is function describing the intensity of the inwards pull (of the filopodia/lamellipodia or the connective tissue). We use bold face letters for u i , n and 0 to make it clear that all these quantities are vectors. We choose Dirichlet homogeneous boundary condition on the boundary of the rectangle M , since, differently from embryos, in the pupae stage epidermis is not under a significant tension and the tissue can be considered at equilibrium.

The algorithm is the following. First we compute the solution of problem (2) in the domain D i = M \W i by using finite element methods on a triangular mesh (using Comsol Multiphysics software). We obtain in this way a displacement field u i . Then, we extended it to the domain inside the inner boundary ω i by solving

-∆u int i = 0 in W i , u int i = u i on ω i . (3)
We obtain in this way an extension (the harmonic extension) of the original vector field u i (which for simplicity we will still denote by u i ) defined on rectangular domain M . In order to perform the evolution of contour ω i , we use level set methods. They consist in implicitly representing the front ω i as the zero level set of a function Φ : R 2 × R + → R, solution of the Hamilton-Jacobi equation (HJ)

∂ t Φ(x, t) + u i (x) • ∇Φ(x, t) = 0 in M × [0, T ], Φ(x, 0) = Φ i (x) in M, (4)
where u i (solution of our original problem extended using (3)) gives the direction of front propagation and ∇ denotes the spatial gradient. For the first step, the function Φ 1 (x) is obtained by computing the signed distance to the initial contour ω 1 (positive at the interior of ω 1 ), whereas for the following contours, Φ i (x) is the solution of (4) computed at the previous time step i -1.

We use an Eulerian method instead of a particle (Lagrangian) method since changes of topology are naturally handled and surfaces automatically merge and separate. We solve the HJ problem (4) on a regular cartesian grid by using an upwind second order Essentially Non-Oscillatory (ENO) scheme in space and a second order total variation diminishing Runge-Kutta scheme in time. The value of u i in the regular grid is computed by interpolating u i on a triangular mesh. The level set methods are implemented by using the Matlab toolbox of I. M. Mitchell ([10]).

Since we are doing a quasistatic analysis, the time scale is free for us to fix and thus the numeric values of coefficients f cable and f pull are not physically significant -just the relative values of the different coefficients make a difference in the simulations (up to rescaling time). To have an idea of size and dynamics of the real situation, small wounds with a diameter of a few tens of µm close in a couple of hours. We choose a time interval T in problem (2) (to pass from the time step i to the time step i + 1) equal to 0.1 and a spatial discretization step equal to 0.1 both in x and y directions.

The numerical experiment shown in figure 1 corresponds to the successive positions of the boundary ω i with constant cable tension f cable (q, i) = 0.2, ∀i = 1 . . . N, ∀q ∈ ω i and null lamellipodial force f pull = 0. We notice that, for simplicity, we do not consider a time dependence (where time is the index i as described before) of the values of the different terms, but the model allows such dependence.

Multiscale continuum-cellular model

The model presented above is a macroscopic model that does not take into account the positions of the individual cells constituting each of the tissues considered. It provides a description at a tissue-level space scale and, being a continuous model, has no ambition of describing the changes in geometry of individual cells -for such a detailed description the geometry of each cell and its neighbors should also play an important role.

Therefore, we decided to couple the continuum model with a cellular one. When the number of cells in the tissue is big, people often concentrate on a small group of cells, not being able to follow in detail all the cells in the system. In this situation, the macroscopic models can be useful for providing reasonable boundary or asymptotic conditions for the cellular-level studies. Moreover, the displacement field u i solution of problem (2) can be used to move the junctions of the cells of the tissue. Like in [START_REF] Dahmann | Opposing transcriptional outputs of hedgehog signaling and engrailed control compartmental cell sorting at the drosophila a/p boundary[END_REF][START_REF] Lecuit | Cell surface mechanics and the control of cell shape, tissue patterns and morphogenesis[END_REF][START_REF] Farhadifar | The infuence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing[END_REF], we represent the epidermal tissue as a two-dimensional network of cells discretized as polygons and interacting through their common boundaries (modeled as straight lines connecting vertices).

To perform simulations and test the model, we need to generate a certain number of initial cell configurations. To achieve this, we first generate a quadrilateral mesh in the domain and we compute the barycenters of each elements of the mesh. Next, we jiggle the barycenters of a small random quantity and then we construct the Voronoi triangulation of these points. We choose this procedure to take advantage of the capabilities of mesh generators to obtain elements of the desired size, stretched in one direction, etc.

Let P (i) be the vertices of the cells of the epidermal tissue included in D i at time step i. We displace the set of points P (i) (belonging to ω i ) by using the vector field u i in order to obtain a new set of points P (i + 1) belonging to D i+1 , i.e. the vertices of the cells at the following time. Let p j (i) = (x j (i), y j (i)), j = 1, . . . N be the coordinates of the points of the set P (i) at time step i. We compute the coordinates p i+1 j of the points in P (i + 1) at time step i + 1 by solving (with a fourth order Runge-Kutta scheme) for each of them, the following boundary value problem (we will be solving this problem N times, with a different initial condition for each j) p (t) = u i in [0, T ] p(0) = p j (i) = (x i j , y i j ).

(5)

Cell sorting and genetic clones

We already studied cell segregation in [START_REF] Gettings | JNK signalling controls remodelling of the segment boundary through cell reprogramming during Drosophila morphogenesis[END_REF]. By studying the cellular tension in Drosophila Dorsal Closure (a stage of embryo development), we realized that there were some cells at the segment boundaries (which we called mixer or chameleon cells) with a very peculiar and previously not described behavior: they change their genetic identity making a transdifferentiation. Consequently, they do not respect compartment boundaries and give rise to unexpected cell rearrangements at the leading edge. Motivated by this work and by some experiences on wound healing on Drosophila pupae in collaboration with A. Jacinto (currently in progress), we consider in the tissue a group of cells of different type, i.e. for instance, mutant for a certain gene. Another possible application is the clones in the wing imaginal disc. Like in the segmentation of Drosophila embryos epithelia, wing disc contains a compartment boundary that separates anterior (A) from posterior (P) cells. This compartment boundary is under the control of the secreted protein Hedgehog (Hh), even thought the precise mechanism remains poorly understood. In [START_REF] Dahmann | Opposing transcriptional outputs of hedgehog signaling and engrailed control compartmental cell sorting at the drosophila a/p boundary[END_REF] they generate cells that lost the ability to transduce the Hh signal (becoming mutant for that gene). Clones in the posterior (P) compartment have wiggly borders with their neighbors since neither cells of the clone nor cells in its neighborhood transduce the Hh signal (cells marked "off" in figure 4). In contrast, clones (arrowhead) have smooth borders when situated in the anterior (A) compartment, since cells surrounding the clone respond to the Hh signal (cells marked "on" in figure 4) and therefore clones try to minimize their surface contact with the neighboring cells. The large clone in the middle is of anterior origin and has taken up a position in the posterior segment. Aiming at studying cell sorting at compartment boundaries, we introduce in the continuum-cellular model the possibility to modify the interfacial tension between cells of different types.

The algorithm is the following. At each time step i, we compute the solution of the continuum model, i.e. the displacement field u i . Second, we move the junctions of the cells by solving the system of ordinary differential equation [START_REF] Almeida | Predictive power of "a minima" models in biology[END_REF]. Then, we take into account the difference in the interfacial tension between cells of different types, leading the cells to go through successive configurations in order to obtain a new mechanical equilibrium at each step. Stationary and stable network configurations satisfy a mechanical force balance, i.e. at each junction, the sum of forces vanish. We describe these force balances as local minima of an energy function. So, we compute a new stable network configuration by minimizing the following functional (P represents the cell configuration which is defined by the set of the vertices of all the cells, and α is the cell index):

F (P ) = N C α=1 <mn> Λ mn l <mn> + N C α=1 K 2 (A α -Āα ) 2 + N C α=1 Γ 2 L 2 α , (6)
The first term describes the contributions due to the line tension along each of the edges < mn > (between vertices m and n) of each cell α, the second one to cell area elasticity, and the third one the elasticity of the cell perimeter. The parameter N C is the number of cells where we perform minimization (it can be a subset of the total cells of the tissue), l <mn> is the edge length, L α the perimeter of cell α, Āα the preferred area (the actual area of the cell at time step i) and Λ mn the line tension on edge l <mn> depending on the gradient in the tension. We obtain in this way a new cell configuration at time step i + 1. The minimization of the functional F of ( 6) is performed using the Matlab function fminsearch. In figure 4 we show the result of the simulation. As expected, the clone becomes round, since it minimizes its contact surface with its neighboring cells (which are genetically different).

Figure 4: First row: the image is taken from [START_REF] Dahmann | Opposing transcriptional outputs of hedgehog signaling and engrailed control compartmental cell sorting at the drosophila a/p boundary[END_REF]. Clones in the posterior (P) compartment have wiggly borders with their neighbors since neither cells of the clone nor cells in its neighborhood transduce the Hh signal (cells marked "off"). In contrast, clones (arrowhead) have smooth borders when situated in the anterior (A) compartment, since cells surrounding the clone respond to the Hh signal (cells marked "on") and therefore try to minimize their surface contact with the neighboring cells. Second row, left to right: the simulation of the movement of the clone at successive time steps.

Figure 1 :

 1 Figure 1: The initial wound is a C-shaped curve in the domain [0, 1.7]x[0, 1.7]. We show in the left part of the figure the vector field u i solution of problem (2) at the initial step i = 1 with f cable = 0.05 and f pull = 0. In the right part we show the successive curve positions.

Figure 2 :

 2 Figure 2: Generation of the initial configuration: from the left to the right we show resp. the quadrilateral mesh, the barycenter of the element of the mesh after jiggle and the Voronoi diagram of these points.

Figure 3 :

 3 Figure 3: Coupling of continuum-cellular model: first row, left to right: the initial configuration, the mesh used to compute the solution of (2) and the corresponding displacement field u i at time step i.; second row, left to right: the cells at two successive equilibrium time step and the difference of cells between two successive times.

The authors thank the ANR project REGENR for support


Rama Cont

Yi Lu

Weak approximation of martingale representations

published or not. The documents may come

Introduction

Let W be a standard d-dimensional Brownian motion defined on a probability space (Ω, F, P) and (F W t ) its (P-completed) natural filtration. We consider a Brownian martingale X which takes values in R d :

X(t) = X(0) + t 0 σ(u)dW (u)
where σ is a process adapted to (F W t ) satisfying

E T 0 σ(t) 2 dt < ∞ and det(σ(t)) = 0 dt × dP -a.e
Then X is a square-integrable martingale with the predictable representation property [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF][START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]: for any square-integrable F W T -measurable random variable H, or equivalently, any squareintegrable (F W t )-martingale Y (t) = E[H|F W t ], there exists a unique (F W t )-predictable process φ with E T 0 tr(φ(u) t φ(u)d[X](u)) < ∞ such that:

Y (T ) = Y (0) + T 0 φ • dX, i.e. H = E[H] + T 0 φ • dX (1)
The classical proof of this representation result (see e.g. [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]) is non-constructive. However in many applications, such as stochastic control or mathematical finance, one is interested in an explicit expression for φ, which represents an optimal control or a hedging strategy.

Expressions for the integrand φ have been derived using a variety of methods and assumptions, using Markovian techniques [START_REF] Davis | Functionals of diffusion processes as stochastic integrals[END_REF][START_REF] Elliott | A short proof of a martingale representation result[END_REF][START_REF] Fitzsimmons | A new approach to the martingale representation theorem[END_REF][START_REF] Jacod | Explicit form and robustness of martingale representations[END_REF][START_REF] Pardoux | Backward stochastic differential equations and quasilinear parabolic partial differential equations[END_REF], integration by parts [START_REF] Chen | Malliavin Greeks without Malliavin calculus[END_REF] or, in the general case, using Malliavin calculus [START_REF] Bismut | A generalized formula of Itô and some other properties of stochastic flows[END_REF][START_REF] Clark | Correction to: "The representation of functionals of Brownian motion by stochastic integrals[END_REF][START_REF] Haussmann | On the integral representation of functionals of Itô processes[END_REF][START_REF] Karatzas | An extension of Clark's formula[END_REF][START_REF] Nualart | Malliavin calculus and its applications[END_REF][START_REF] Ocone | Malliavin's calculus and stochastic integral representations of functionals of diffusion processes[END_REF][START_REF] Fournié | Applications of Malliavin calculus to Monte Carlo methods in finance[END_REF]. Some of these methods are limited to the case where X is a Markov process; others require differentiability and/or ellipticity assumptions on σ [START_REF] Fournié | Applications of Malliavin calculus to Monte Carlo methods in finance[END_REF], differentiability assumptions, in the Fréchet or Malliavin sense, on H, or an explicit form for the density of X [START_REF] Chen | Malliavin Greeks without Malliavin calculus[END_REF]. Almost all of these methods invariably involve an approximation step, either through the solution of an auxiliary partial differential equation (PDE) or the simulation of an auxiliary stochastic differential equation.

A systematic approach to obtaining martingale representation formulae, based on the Functional Ito calculus [START_REF] Dupire | Functional Itô calculus[END_REF][START_REF] Cont | A functional extension of the Ito formula[END_REF][START_REF] Cont | Change of variable formulas for non-anticipative functionals on path space[END_REF], has been proposed in [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF], where it is shown [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF]Theorem 5.9] that for any square-integrable (F X t )-martingale Y ,

∀t ∈ [0, T ], Y (t) = Y (0) + t 0 ∇ X Y • dX P-a.s.
where ∇ X Y is the weak vertical derivative of Y with respect to X, constructed as an L 2 limit of pathwise directional derivatives. This approach does not rely on any Markov property nor on the Gaussian structure of the Wiener space and is thus applicable to functionals of a large class of processes.

In the present work we build on this approach to propose a general framework for computing explicit approximations to the integrand φ in a general setting in which X is allowed to be the solution of a stochastic differental equation (SDE) with path-dependent coefficients:

dX(t) = σ(t, X t )dW (t) X(0) = x 0 ∈ R d (2)
where X t = X(t∧.) designates the trajectory stopped at t and σ : [0, T ]×D([0, T ], R d ) → GL d (R) is a Lipschitz map. For any square-integrable variable of the form H = g(X(t), 0 ≤ t ≤ T ) where g : (D([0, T ], R d ), . ∞ ) → R is a continuous functional, we construct an explicit sequence of approximations φ n for the integrand φ in [START_REF] Bismut | A generalized formula of Itô and some other properties of stochastic flows[END_REF]. These approximations are constructed as vertical derivatives, in the sense of the functional Ito calculus, of the weak Euler approximation of the martingale Y . We show that these provide explicit expressions for these approximations and analyze their convergence to the integrand φ. Under a Lipschitz assumption on g, we provide error estimates in L 2p . These approximations are easy to compute and readily integrated in commonly used numerical schemes for approximations of SDEs.

Our approach requires neither the Markov property of the underlying processes nor the differentiability of coefficients, making our approach applicable to functionals of a large class of semimartingales. By contrast to methods based on Malliavin calculus [START_REF] Bismut | A generalized formula of Itô and some other properties of stochastic flows[END_REF][START_REF] Clark | Correction to: "The representation of functionals of Brownian motion by stochastic integrals[END_REF][START_REF] Haussmann | On the integral representation of functionals of Itô processes[END_REF][START_REF] Karatzas | An extension of Clark's formula[END_REF][START_REF] Ocone | Malliavin's calculus and stochastic integral representations of functionals of diffusion processes[END_REF][START_REF] Fournié | Applications of Malliavin calculus to Monte Carlo methods in finance[END_REF], it does not require Malliavin differentiability of the terminal variable H nor does it involve the choice of 'Malliavin weights', a delicate step in these methods.

Ideas based on Functional Ito calculus have also been recently used by Leão and Ohashi [START_REF] Leão | Weak approximations for Wiener functionals[END_REF] for weak approximation of Wiener functionals, using a space-filtration discretization scheme. However, unlike the approach proposed in [START_REF] Leão | Weak approximations for Wiener functionals[END_REF], our approach is based on a Euler approximation on a fixed time grid, rather than the random time grid used in [START_REF] Leão | Weak approximations for Wiener functionals[END_REF], which involves a sequence of first passage times. Our approach is thus much easier to implement and analyze and is readily integrated in commonly used numerical schemes for approximations of SDEs, which are typically based on fixed time grids.

Outline We first recall some key concepts and results from the Functional Itô calculus in section 2. Section 3 provides some estimates for the path-dependent SDE [START_REF] Chen | Malliavin Greeks without Malliavin calculus[END_REF] and studies some properties of the Euler approximation for this SDE. In Section 4 we show that the weak Euler approximation (Definition 9) may be used to approximate any square-integrable martingale adapted to the filtration of X by a sequence of smooth functionals of X, in the sense of the functional Ito calculus. Moreover, we provide explicit expressions for the functional derivatives of these approximations. Section 5 analyzes the convergence of this approximation and provides error estimates in Theorem 5.1. Finally, in Section 6 we compare our approximation method with those based on Malliavin calculus.

Notations:

In the sequel, we shall denote by M d,n (R) the set of all d × n matrices with real coefficients. We simply denote • ω(t) the value of ω at time t,

R d = M d,1 (R) and M d (R) = M d,d (R). For A ∈ M d (R)
• ω(t-) = lim s→t,s<t ω(s) its left limit at t,

• ω t = ω(t ∧ •) the path of ω stopped at t • ω t-= ω1 [0,t) + ω(t-)1 [t,T ] • ω ∞ = sup{|ω(t)|, t ∈ [0, T ]} the supremum norm.
We note that ω t and ω t-are elements of D([0, T ], R d ). For a càdlàg stochastic process X, we shall similarly denote X t (.) = X(t ∧ .) and

X t-= X1 [0,t) + X(t-)1 [t,T ] .

Functional Itô calculus

The Functional Itô calculus [START_REF] Cont | Functional Ito calculus and path-dependent Kolmogorov equations[END_REF] is a functional calculus which extends the Ito calculus to pathdependent functionals of stochastic processes. It was first introduced in a pathwise setting [START_REF] Cont | A functional extension of the Ito formula[END_REF][START_REF] Cont | Change of variable formulas for non-anticipative functionals on path space[END_REF][START_REF] Dupire | Functional Itô calculus[END_REF] using a notion of pathwise derivative for functionals on the space of right-continuous functions with left limits, and extended in [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF] to a weak calculus applicable to all square-integrable martingales, which has a natural connection to the martingale representation theorem. We recall here some key concepts and results of this approach, following [START_REF] Cont | Functional Ito calculus and path-dependent Kolmogorov equations[END_REF].

Let X be the canonical process on Ω = D([0, T ], R d ), and (F 0 t ) t∈[0,T ] be the filtration generated by X. We consider now F a functional defined on

[0, T ] × D([0, T ], R d ) with values in R.
In this paper, we are interested in one particular class of such functionals characterized by the following property: the process t → F (t, ω) defined on Ω is (F 0 t )-adapted. Under this condition, F (t, •) only depends on the path stopped at t:

∀ω ∈ Ω, F (t, ω) = F (t, ω t ). (3)
This motivates us to consider functionals on the space of stopped paths [START_REF] Cont | Functional Ito calculus and path-dependent Kolmogorov equations[END_REF]: a stopped path is an equivalence class in [0, T ] × D([0, T ], R d ) for the following equivalence relation:

(t, ω) ∼ (t , ω ) ⇐⇒ (t = t and ω t = ω t ). (4)
The space of stopped paths is defined as the quotient of [0, T ] × D([0, T ], R d ) by the equivalence relation (4):

Λ T = {(t, ω(t ∧ •)), (t, ω) ∈ [0, T ] × D([0, T ], R d )} = [0, T ] × D([0, T ], R d ) / ∼
We denote W T the subset of Λ T consisting of continuous stopped paths. We endow this set with a metric space structure by defining the following distance:

d ∞ ((t, ω), (t , ω )) = sup u∈[0,T ] |ω(u ∧ t) -ω (u ∧ t )| + |t -t | = ω t -ω t ∞ + |t -t | (Λ T , d ∞ )
is then a complete metric space. Any functional verifying the non-anticipativity condition (3) can be equivalently viewed as a functional on F : Λ T → R:

Definition 1. A non-anticipative functional on D([0, T ], R d ) is a measurable map F : (Λ T , d ∞ ) -→ R on the space (Λ T , d ∞ ) of stopped paths.
Using the metric structure of (Λ T , d ∞ ), one can define various notions of continuity for nonanticipative functionals [START_REF] Cont | A functional extension of the Ito formula[END_REF]: Definition 2. A non-anticipative functional F is said to be:

• continuous at fixed times if for any t ∈ [0, T ], F (t, •) is continuous with respect to the uniform norm • ∞ in [0, T ], i.e. ∀ω ∈ D([0, T ], R d ), ∀ > 0, ∃η > 0, ∀ω ∈ D([0, T ], R d ), sup |ω -ω | < η =⇒ |F (t, ω) -F (t, ω )| < • jointly continuous if F is continuous with respect to d ∞ , i.e. ∀(t, ω) ∈ Λ T , ∀ > 0, ∃η > 0 such that ∀(t , ω ) ∈ Λ T , d ∞ ((t, ω), (t , ω )) < η =⇒ |F (t, ω) -F (t , ω )| <
We denote by C 0,0 (Λ T ) the set of jointly continuous non-anticipative functionals.

• left-continuous if ∀(t, ω) ∈ Λ T , ∀ > 0, ∃η > 0 such that ∀(t , ω ) ∈ Λ T , (t < t and d ∞ ((t, ω), (t , ω )) < η) =⇒ |F (t, ω) -F (t , ω )| <
We denote by C 0,0 l (Λ T ) the set of left-continuous functionals. Similarly, we can define the set C 0,0 r (Λ T ) of right-continuous functionals.

We also introduce a notion of local boundedness for functionals.

Definition 3. A non-anticipative functional F is said to be boundedness-preserving if for every compact subset K of R d , ∀t 0 ∈ [0, T ], ∃C(K, t 0 ) > 0 such that:

∀t ∈ [0, t 0 ], ∀(t, ω) ∈ Λ T , ω([0, t]) ⊂ K =⇒ F (t, ω) < C(K, t 0 ).
We denote by B(Λ T ) the set of boundedness-preserving functionals.

We now recall some notions of differentiability for functionals following [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF][START_REF] Cont | Functional Ito calculus and path-dependent Kolmogorov equations[END_REF]. For e ∈ R d and ω t ∈ D([0, T ], R d ), we define the vertical perturbation ω e t of (t, ω) as the càdlàg path obtained by shifting the path by e after t ω e t = ω t + e1 [t,T ] .

Definition 4. A non-anticipative functional F is said to be:

• horizontally differentiable at (t, ω) ∈ Λ T if DF (t, ω) = lim h→0 + F (t + h, ω) -F (t, ω) h
exists. If DF (t, ω) exists for all (t, ω) ∈ Λ T , then the non-anticipative functional DF is called the horizontal derivative of F .

• vertically differentiable at (t, ω) ∈ Λ T if the map:

R d -→ R e → F (t, ω t + e1 [t,T ] )
is differentiable at 0. Its gradient at 0 is called the vertical derivative of F at (t, ω):

∇ ω F (t, ω) = (∂ i F (t, ω), i = 1, • • • , d) ∈ R d with ∂ i F (t, ω) = lim h→0 F (t, ω t + he i 1 [t,T ] ) -F (t, ω t ) h
where

(e i , i = 1, • • • , d) is the canonical basis of R d . If F is vertically differentiable at all (t, ω) ∈ Λ T , ∇ ω F : (t, ω) → R d defines a non-anticipative map called the vertical derivative of F .
We may repeat the same operation on ∇ ω F and define similarly ∇ 2 ω F , ∇ 3 ω F , • • • . This leads us to define the the following classes of smooth functionals: Definition 5 (Smooth functionals). We define C 1,k b (Λ T ) as the set of non-anticipative functionals F : (Λ T , d ∞ ) → R which are

• horizontally differentiable with DF continuous at fixed times;

• k times vertically differentiable with ∇ j ω F ∈ C 0,0 l (Λ T ) for j = 0, • • • , k; • DF, ∇ ω F, • • • , ∇ k ω F ∈ B(Λ T ). We denote C 1,∞ (Λ T ) = ∩ k≥1 C 1,k (Λ T ), C 1,∞ b (Λ T ) = ∩ k≥1 C 1,k b (Λ T ).
Many examples of functionals may fail to be globaly smooth, but their derivatives may still be well behaved except at certain stopping times, which motivates the following definition [START_REF] Cont | Functional Ito calculus and path-dependent Kolmogorov equations[END_REF]: Definition 6. A non-anticipative functional F is said to be locally regular of class C 1,2 loc (Λ T ) if there exists an increasing sequence (τ n ) n≥0 of stopping times with τ 0 = 0 and τ n -→ n→∞ ∞, and a sequence of functionals

F n ∈ C 1,2
b (Λ T ) such that:

F (t, ω) = n≥0 F n (t, ω)1 [τn(ω),τn+1(ω)) (t), ∀(t, ω) ∈ Λ T
We recall now the functional Itô formula for non-anticipative functionals of a continuous semimartingale [7, Theorem 4.1]: [START_REF] Cont | A functional extension of the Ito formula[END_REF][START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF]). Let S be a continuous semimartingale defined on a probability space (Ω, F, P). For any non-anticipative functional F ∈ C 1,2 loc (Λ T ) and any t ∈ [0, T ], we have:

Proposition 2.1 ([
F (t, S t ) -F (0, S 0 ) = t 0 DF (u, S u )du + t 0 ∇ ω F (u, S u ) • dS(u) + 1 2 t 0 tr ∇ 2 ω F (u, S u )d[S](u)
Actually the same functional Itô formula may also be obtained for functionals whose vertical derivatives are right-continuous rather than left-continuous. We denote by C 1,2 b,r (Λ T ) the set of non-anticipative functionals F satisfying:

• F is horizontally differentiable with DF continuous at fixed times;

• F is twice vertically differentiable with F ∈ C 0,0 l (Λ T ) and ∇ ω F, ∇ 2 ω F ∈ C 0,0 r (Λ T ); • DF, ∇ ω F, ∇ 2 ω F ∈ B(Λ T );
The localization is more delicate in this case, and we are not able to state a local version of the functional Itô formula by simply replacing [START_REF] Fournie | Functional Ito calculus and applications[END_REF]). However if the stopping times τ n are deterministic, then the functional Itô formula is still valid (Proposition 2.4 and Remark 4.2 in [START_REF] Fournie | Functional Ito calculus and applications[END_REF]). Definition 7. A non-anticipative functional is said to be locally regular of class C 1,2 loc,r (Λ T ) if there exists an increasing sequence (t n ) n≥0 of deterministic times with t 0 = 0 and t n -→ n→∞ ∞, and a sequence of functionals

F n ∈ C 1,2 b (Λ T ) by F n ∈ C 1,2 b,r (Λ T ) in Definition 6 (see Remark 4.2 in
F n ∈ C 1,2
b,r (Λ T ) such that:

F (t, ω) = n≥0 F n (t, ω)1 [tn,tn+1) (t), ∀(t, ω) ∈ Λ T Proposition 2.2 ([7]
). Let S be a continuous semimartingale defined on a probability space (Ω, F, P). For any non-anticipative functional F ∈ C 1,2 loc,r (Λ T ) and any t ∈ [0, T ], we have:

F (t, S t ) -F (0, S 0 ) = t 0 DF (u, S u )du + t 0 ∇ ω F (u, S u ) • dS(u) + 1 2 t 0 tr ∇ 2 ω F (u, S u )d[S](u) P-a.s.
Finally we present briefly the martingale representation formula established in [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF]. Let (X t ) t∈[0,T ] be a continuous R d -valued martingale defined on a probability space (Ω, F, P) with absolutely continuous quadratic variation:

[X](t) = t 0 A(u)du where A is a M d (R)-valued process. Denote by (F X t ) the natural filtration of X and C 1,2 b (X) the set of (F X t )-adapted processes Y which admit a functional representation in C 1,2 b (Λ T ): C 1,2 b (X) = {Y, ∃F ∈ C 1,2 b (Λ T ), Y (t) = F (t, X t ) dt × dP-a.e.} (5)
If A(t) is non-singular almost everywhere, i.e. det(A(t)) = 0, dt × dP-a.e., then for any

Y ∈ C 1,2 b (X), the predictable process ∇ X Y (t) = ∇ ω F (t, X t )
is uniquely defined up to an evanescent set, independently of the choice of F ∈ C 1,2 b (Λ T ) in the representation [START_REF] Cont | Change of variable formulas for non-anticipative functionals on path space[END_REF]. This process ∇ X Y is called the vertical derivative of Y with respect to X. For martingales which are smooth functionals of X, the operator ∇ X : C 1,2 b (X) → C 0,0 l (X) yields the integrand in the martingale representation theorem:

Corollary 2.1. If Y ∈ C 1,2 b (X) is a square-integrable martingale, then ∀t ∈ [0, T ], Y (t) = Y (0) + t 0 ∇ X Y • dX P-a.s.
Consider now the case where X is a square-integrable martingale. Let M 2 (X) be the space of square-integrable (F X t )-martingales with initial value zero, equipped with the norm [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF]Theorem 5.8] show that the operator ∇ X : C 1,2 b (X) → C 0,0 l (X) admits a unique continuous extension to a weak derivative ∇ X : M 2 (X) → L 2 (X) which satisfies the following martingale representation formula:

Y 2 = E|Y (T )| 2 . Cont & Fournié
Proposition 2.3 ([7]
). For any square-integrable (F X t )-martingale Y , we have:

∀t ∈ [0, T ], Y (t) = Y (0) + t 0 ∇ X Y • dX P-a.s.
This weak vertical derivative ∇ X Y coincides with the pathwise vertical derivative ∇ ω F (t, X t ) when Y admits a locally regular functional representation, i.e.

Y (t) = F (t, X t ) with F ∈ C 1,2 loc (Λ T ) ∪ C 1,2 loc,r (Λ T ).
For a general square-integrable martingale Y , the weak derivative ∇ X Y is not directly computable through a pathwise perturbation. An approximation procedure is thus necessary for computing ∇ X Y . The result of [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF] guarantees the existence of such approximations; in the sequel we propose explicit constructions of computable versions of such approximations.

Euler approximations for path-dependent SDEs

Let W be a standard d-dimensional Brownian motion defined on a probability space (Ω, F, P) and (F W t ) its (P-completed) natural filtration and denote by GL d (R the set of d × d non-singular real matrices. We consider the following stochastic differential equation with path-dependent coefficient [START_REF] Chen | Malliavin Greeks without Malliavin calculus[END_REF]:

dX(t) = σ(t, X t )dW (t), X(0) = x 0 ∈ R d
where σ : Λ T → GL d (R is a non-anticipative map, assumed to be Lipschitz-continuous:

Assumption 1. σ : (Λ T , d ∞ ) → GL d (R is Lipschitz continuous: ∃σ lip > 0, ∀t, t ∈ [0, T ], ∀ω, ω ∈ D([0, T ], R d ), σ(t, ω) -σ(t , ω ) ≤ σ lip d ∞ ((t, ω), (t , ω )) .
Denote by (F X t ) the natural filtration of X. Under Assumption 1, (2) has a unique strong solution X and F X t = F W t .

Proposition 3.1. Under Assumption 1, there exists a unique (F W t )-adapted process X satisfying (2). Moreover for p ≥ 1, we have:

E X T 2p ∞ ≤ C(1 + |x 0 | 2p )e CT (6)
for some constant C = C(p, T, σ lip ) depending on p, T and σ lip .

Proof. Existence and uniqueness of a strong solution follows from [START_REF] Protter | Stochastic integration and differential equations[END_REF] (Theorem 7 , Chapter 5): see [4, Section 5]. Let us prove [START_REF] Cont | A functional extension of the Ito formula[END_REF]. Using the Burkholder-Davis-Gundy inequality and Hölder's inequality, we have:

E X T 2p ∞ ≤ C(p) |x 0 | 2p + E T 0 σ(t, X t ) 2 dt p ≤ C(p, T ) |x 0 | 2p + E T 0 σ(t, X t ) 2p dt ≤ C(p, T ) |x 0 | 2p + E T 0 ( σ(0, 0) + σ lip (t + X t ∞ )) 2p dt ≤ C(p, T, σ lip ) |x 0 | 2p + 1 + T 0 E X t 2p ∞ dt
where 0 is the path which takes constant value 0. We conclude by Gronwall's inequality.

In the following, we always assume that Assumption 1 holds. The strong solution X of equation ( 2) is then a Brownian martingale and defines a non-anticipative functional X : W T → R d given by the Ito map associated to (2).

Euler approximations as non-anticipative functionals

We now consider an Euler approximation for the SDE (2) and study its properties as a nonanticipative functional. Let n ∈ N, δ = T n . The Euler approximation n X of X on the grid (t j = jδ, j = 0..n) is defined as follows:

Definition 8. [Euler scheme] For ω ∈ D([0, T ], R d ), we denote by n X(ω) ∈ D([0, T ], R d ) the piecewise constant Euler approximation for (2) computed along the path ω, defined as follows:

n X(ω) is constant in each interval [t j , t j+1 ), ∀ 0 ≤ j ≤ n -1 with n X(0, ω) = x 0 and n X(t j+1 , ω) = n X(t j , ω) + σ(t j , n X tj (ω))(ω(t j+1 -) -ω(t j -)), 0 ≤ j ≤ n -1 (7)
where n X t (ω) is the path of n X(ω) stopped at time t, and by convention ω(0-) = ω(0).

When computed along the path of the Brownian motion W, n X(W ) is simply the piecewise constant Euler-Maruyama scheme [START_REF] Pardoux | Discretization and simulation of stochastic differential equations[END_REF] for the stochastic differential equation [START_REF] Chen | Malliavin Greeks without Malliavin calculus[END_REF].

By definition, the path n X(ω) depends only on a finite number of increments of ω:

ω(t 1 -) - ω(0), • • • , ω(t n -) -ω(t n-1 -). We can thus define an application p : M d,n (R) → D([0, T ], R d ) as follows: for y = (y 1 , • • • , y n ) ∈ M d,n (R) with each y l ∈ R d for 1 ≤ l ≤ n, p(y) = p(y 1 , • • • , y n ) = n X(ω) (8)
with ω any path in

D([0, T ], R d ) satisfying ω(t 1 -) -ω(0) = y 1 , • • • , ω(t n -) -ω(t n-1 -) = y n .
And we note p t (y) the path of p(y) stopped at time t.

The application p :

M d,n (R) → D([0, T ], R d ),
• ∞ is locally Lipschitz continuous as shown by the following lemma.

Lemma 3.1. Let y = (y 1 , • • • , y n ) and y = (y 1 , • • • , y n ) ∈ M d,n (R) with y l , y l ∈ R d for 1 ≤ l ≤ n. If max 1≤k≤n |y k -y k | ≤ η, then we have: p(y) -p(y ) ∞ ≤ C(y, η, σ lip , T ) max 1≤k≤n |y k -y k |
for some constant C depending only on y, η, σ lip and T .

Proof. As the two paths p(y) and p(y ) are stepwise constant by definition, it suffices to prove the inequality at times (t j ) 0≤j≤n . We prove by induction that:

p tj (y) -p tj (y ) ∞ ≤ C(y, η, σ lip , T ) max 1≤k≤j |y k -y k | (9)
with some constant C which depends only on y, η, σ lip and T .

For j = 0, this is clearly the case as p(y)(0) = p(y )(0) = x 0 . Assume that ( 9) is verified for some 0 ≤ j ≤ n -1, consider now p tj+1 (y) -p tj+1 (y ) ∞ , we have: where 0 is the path which takes constant value 0. And consequently we have:

p(y)(t j+1 ) = p(y)(t j ) + σ(t j , p tj (y))
p tj+1 (y) -p tj+1 (y ) ∞ ≤ C(y, η, σ lip , T ) max 1≤k≤j+1 |y k -y k |
for some different constant C depending only on y, η, σ lip and T . And we conclude by induction.

Strong convergence

To simplify the notations, n X T (W T ) will be noted simply n X T in the following. The following result, which gives a uniform estimate of the discretization error, X Tn X T extends similar results known in the Markovian case [START_REF] Faure | Simulation du mouvement brownien et des diffusions[END_REF][START_REF] Pardoux | Discretization and simulation of stochastic differential equations[END_REF][START_REF] Higham | Strong convergence of Eulertype methods for nonlinear stochastic differential equations[END_REF] to the path-dependent SDE (2):

Proposition 3.2. Under Assumption 1 we have the following estimate in L 2p for the strong error of the piecewise constant Euler-Maruyama scheme:

E sup s∈[0,T ] X(s) -n X(s) 2p ≤ C(x 0 , p, T, σ lip ) 1 + log n n p , ∀p ≥ 1
with C a constant depending only on x 0 , p, T and σ lip .

Proof. The idea is to construct a 'Brownian interpolation' n XT of the Euler scheme n X T :

n X(s) = x 0 + s 0 σ u, n X u dW (u)
where u = u δ • δ is the largest subdivision point which is smaller or equal to u.

Clearly the process n XT is a continuous martingale and sup

s∈[0,T ] |X(s) -n X(s)| 2p can be
controlled by the sum of the two following terms:

sup s∈[0,T ] |X(s) -n X(s)| 2p ≤ sup s∈[0,T ] |X(s) -n X(s)| 2p + sup s∈[0,T ] | n X(s) -n X(s)| 2p (10)
We start with the term sup

s∈[0,T ] |X(s)-n X(s)| 2p .
Using the Burkholder-Davis-Gundy inequality and Hölder's inequality, we have

E X T -n XT 2p ∞ ≤ C(p) E T 0 σ(s, X s ) -σ(s, n X s ) 2 ds p ≤ C(p, T ) E T 0 σ(s, X s ) -σ(s, n X s ) 2p ds ≤ C(p, T, σ lip ) E T 0 (s -s) 2p + X s -n X s 2p ∞ ds ≤ C(p, T, σ lip ) 1 n 2p + T 0 E X s -n X s 2p ∞ ds
The constants may differ from one line to another, and we have used n X s = n X s as n X is piecewise constant.

Consider now the second term sup

s∈[0,T ] | n X(s) -n X(s)| 2p . Noting that: n X(s) -n X(s) = n X(s) -n X(s) = σ s, n X s (W (s) -W (s)), we have n XT -n X T ∞ ≤ C(σ lip , T )(1 + n X T ∞ ) sup s∈[0,T ] |W (s) -W (s)| and E n XT -n X T 2p ∞ ≤ C(σ lip , T ) 2p E (1 + n X T ∞ ) sup s∈[0,T ] |W (s) -W (s)| 2p .
By the Cauchy-Schwarz inequality, we have:

E n XT -n X T 2p ∞ ≤ C(p, σ lip , T ) 1 + E n X T 4p ∞ E sup s∈[0,T ] |W (s) -W (s)| 4p
We will make use of the following result:

∀p > 0, sup s∈[0,T ] |W (s) -W (s)| p ≤ C(W, p) T n (1 + log n)
which results from the following lemma:

Lemma 3.2. Let Y 1 , • • • ,
Y n be non-negative random variables with the same distribution satisfying E e λY1 < ∞ for some λ > 0. Then we have:

∀p > 0, max(Y 1 , • • • , Y n ) p ≤ 1 λ (log n + C(p, Y 1 , λ))
We have thus:

E sup s∈[0,T ] |W (s) -W (s)| 4p ≤ C(p, T ) 1 + log n n p (11)
Furthermore, using again the Burkholder-Davis-Gundy inequality, we have:

E n X T 4p ∞ ≤ E n XT 4p ∞ ≤ C(p)   x 4p 0 + E T 0 σ(s, n X s ) 2 ds 2p   ≤ C(p, x 0 , T ) 1 + T 0 E σ(s, n X s ) 4p ds ≤ C(p, x 0 , T, σ lip ) 1 + T 0 E n X s 4p ∞ ds
We deduce from Gronwall's inequality that E n X T 4p ∞ is bounded by a constant which depends only on p, x 0 , T and σ lip .

Combining this result with [START_REF] Dupire | Functional Itô calculus[END_REF], we get:

E n XT -n X T 2p ∞ ≤ C(x 0 , p, T, σ lip ) 1 + log n n p Finally (10) becomes: E X T -n X T 2p ∞ ≤ C(p) E X T -n XT 2p ∞ + E n XT -n X T 2p ∞ ≤ C(x 0 , p, T, σ lip ) 1 + log n n p + T 0 E X s -n X s 2p ∞ ds
And we conclude by Gronwall's inequality.

Corollary 3.1. Under Assumption 1, ∀α ∈ [0, 1 2 ), n α X T -n X T ∞ -→ n→∞ 0, P-a.s. Proof. Let α ∈ [0, 1 2
). For a p large enough, by Proposition 3.2, we have:

E   n≥1 n 2pα X T -n X T 2p ∞   < ∞ Thus n≥1 n 2pα X T -n X T 2p ∞ < ∞, P-a.s. and n α X T -n X T ∞ -→ n→∞ 0, P-a.s.

Smooth functional approximations for martingales

Let g : D([0, T ], R d ) -→ R be a functional which satisfies the following condition:

Assumption 2. g : (D([0, T ], R d ), • ∞ ) -→ R is continuous with polynomial growth: ∃q ∈ N, ∃C > 0, ∀ω ∈ D([0, T ], R d ), |g(ω)| ≤ C (1 + ω q ∞ )
and consider the (square-integrable) martingale

Y (t) = E g(X T )|F X t = E g(X T )|F W t .
Y may be represented as a non-anticipative functional of X (or W ):

Y (t) = G(t, X t ) = F (t, W t )
where the functionals F, G are square-integrable but may not have any smoothness property a priori. By Proposition 2.3 we have:

g(X T ) = Y (T ) = Y (t) + T t ∇ X Y (s) • dX(s) = Y (t) + T t ∇ W Y (s) • dW (s) P-a.s.
where ∇ X Y (resp. ∇ W Y ) is the weak vertical derivative of Y with respect to X (resp. W ). The two representations are related [START_REF] Cont | Functional Ito calculus and path-dependent Kolmogorov equations[END_REF]Theorem 4.19] by the equality

t (∇ X Y (s))σ(s, X s ) = t (∇ W Y (s))
outside an evanescent set. So if one of them is computable, the other one is computable as well. However in general neither G nor F is a smooth functional (for example ∈ C 1,2 loc,r (Λ T )) so neither of the two weak derivatives may be computed directly as a pathwise directional derivative.

The main idea is to approximate the martingale Y by a sequence of smooth martingales n Y which admit a functional representation n Y (s) = F n (s, W s ) with F n ∈ C 1,2 loc,r (Λ T ), regular enough to apply the functional Itô formula. Then by the functional Itô formula, we have:

T t ∇ ω F n (s, W s ) • dW (s) = n Y (T ) -n Y (t) -→ n→∞ Y (T ) -Y (t) = T t ∇ X Y (s) • dX(s)
One can then use the following estimator for ∇ X Y :

Z n (s) = t σ -1 (s, X s ) ∇ ω F n (s, W s ),
where the vertical derivative

∇ ω F n (s, W s ) = (∂ i F n (s, W s ), 1 ≤ i ≤ d) may be computed as a pathwise derivative ∂ i F n (s, W s ) = lim h→0 F n (s, W s + he i 1 [s,T ] ) -F n (s, W s ) h ,
yielding a concrete procedure for computing the estimator.

We will show in this section that the familiar weak Euler approximation provides a systematic way of constructing such smooth functional approximations in the sense of Definition 7.

Define the concatenation of two càdlàg paths ω, ω ∈ D([0, T ], R d ) at time s ∈ [0, T ], which we note ω ⊕ t ω , as the following càdlàg path on [0, T ]:

ω ⊕ s ω = ω s ⊕ s ω = ω(u) u ∈ [0, s) ω(s) + ω (u) -ω (s) u ∈ [s, T ] Observe that: ∀z ∈ R d , ω z s ⊕ s ω = (ω s ⊕ s ω ) + z1 [s,T ] .
Definition 9 (Weak Euler approximation). We define the (level-n) weak Euler approximation of F as the functional F n defined by

F n (s, ω s ) = E g( n X(ω s ⊕ s W T )) (12)
Applying this functional to the path of the Wiener process W , we obtain a (F W t ) t≥0 -adapted process:

n Y (s) = F n (s, W s ). Using independence of increments of W , we have n Y (s) = E g( n X(W T ))|F W s = E g( n X(W s ⊕ s W T ))|F W s = E g( n X(W s ⊕ s B T ))|F W s
where B is any Wiener process independent from W. In particular n Y is a square-integrable martingale, so is weakly differentiable in the sense of [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF]Theorem 5.8]. We will now show that F n is in fact a smooth functional in the sense of Definition 7.

Theorem 4.1. Under Assumptions 1 and 2, the functional F n defined in [START_REF] Elliott | A short proof of a martingale representation result[END_REF] is horizontally differentiable and infinitely vertically differentiable.

Proof. Let (s, ω) ∈ Λ T with t k ≤ s < t k+1 for some 0 ≤ k ≤ n -1. We start with the vertical differentiability of F n at (s, ω), which is equivalent to the differentiability at 0 of the following map:

v(z) = F n (s, ω z s ) = E g( n X(ω z s ⊕ s B T )) , z ∈ R d
The main idea of the proof is to absorb z in the density function of Gaussian variables when taking the expectation, which smoothens the dependence of v on z.

As we have already shown, n X(ω z s ⊕

B T )(t k -) = B(t k+1 ) -B(s) + ω(s) + z -ω(t k -) = B(t k+1 ) -B(s) + z + ω(s) -ω(t k -)
And for j > k, we have:

(ω z s ⊕ s B T )(t j+1 -) -(ω z s ⊕ s B T )(t j -) = B(t j+1 ) -B(s) + ω(s) + z -(B(t j ) -B(s) + ω(s) + z) = B(t j+1 ) -B(t j )
Thus we have:

n X(ω z s ⊕ s B T ) = p ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), B(t k+1 ) -B(s) + z + ω(s) -ω(t k -), B(t k+2 ) -B(t k+1 ), • • • , B(t n ) -B(t n-1 )
where p :

M d,n (R) → D([0, T ], R d
) is the map defined by [START_REF] Cvitanić | Efficient computation of hedging portfolios for options with discontinuous payoffs[END_REF].

Observe from the previous equation that, for a fixed z, the value of n X(t k+1 , ω z s ⊕ s B T ) as a random variable depends only on a finite number of Gaussian variables:

B(t k+1 ) -B(s), B(t k+2 ) -B(t k+1 ), • • • , B(t j ) -B(t j-1 ). Since the joint distribution of these Gaussian variables is explicit, v(z) = E g( n X T (ω z s ⊕ s B T )) can be computed explicitly as an integral in finite dimension. Let y = (y 1 , • • • , y n-k ) ∈ M d,n-k (R) with each y l ∈ R d for 1 ≤ l ≤ n -k. We have: v(z) = E g( n X T (ω z s ⊕ s B T )) = E g p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), B(t k+1 ) -B(s) + z + ω(s) -ω(t k -), B(t k+2 ) -B(t k+1 ), • • • , B(t n ) -B(t n-1 ) = R d×(n-k) g p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), y 1 + z + ω(s) -ω(t k -), y 2 , • • • , y n-k ) Φ(y 1 , t k+1 -s) n-k l=2 Φ(y l , δ)dy 1 dy 2 • • • dy n-k = R d×(n-k) g p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), y 1 + ω(s) -ω(t k -), y 2 , • • • , y n-k ) Φ(y 1 -z, t k+1 -s) n-k l=2 Φ(y l , δ)dy 1 dy 2 • • • dy n-k (13) with Φ(x, t) = (2πt) -d 2 exp - |x| 2 2t , x ∈ R d
the density function of a d-dimensional Gaussian variable with covariance matrix tI d .

Since the only term which depends on z in the integrand of ( 13) is Φ(y 1 -z, t k+1 -s), which is a smooth function of z, thus v is differentiable at all z ∈ R d , in particular at 0. Hence F n is vertically differentiable at (s, ω) ∈ Λ T with: for 1

≤ i ≤ d, ∂ i F n (s, ω) = R d×(n-k) g p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), y 1 + ω(s) -ω(t k -), y 2 , • • • , y n-k ) y 1 • e i t k+1 -s Φ(y 1 , t k+1 -s) n-k l=2 Φ(y l , δ)dy 1 dy 2 • • • dy n-k = E g( n X(ω s ⊕ s B T )) (B(t k+1 ) -B(s)) • e i t k+1 -s (14)
Remark that when s tends towards t k+1 , ∇ ω F n (s, ω) may tend to infinity because of the term t k+1 -s in the denominator. However in the interval [t k , t k+1 ), ∇ ω F n (s, ω) behaves well and is locally bounded.

Iterating this procedure, one can show that F n is vertically differentiable to any order. For example, we have: for z ∈ R d ,

∂ i F n (s, ω z s ) = R d×(n-k) g p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), y 1 + z + ω(s) -ω(t k -), y 2 , • • • , y n-k ) y 1 • e i t k+1 -s Φ(y 1 , t k+1 -s) n-k l=2 Φ(y l , δ)dy 1 dy 2 • • • dy n-k with Φ(x, t) = (2πt) -d 2 exp - |x| 2 2t , x ∈ R d
The following result shows that the functional derivatives of F n satisfy the necessary regularity conditions for applying the functional Itô formula to F n : Theorem 4.2. Under Assumptions 1 and 2, F n ∈ C 1,2 loc,r (Λ T ).

Proof. We have already shown in Theorem 4.1 that F n is horizontally differentiable and twice vertically differentiable. Using the expressions of DF n , ∇ ω F n and ∇ 2 ω F n obtained in the proof of 4.1 and the assumption that g has at most polynomial growth with respect to • ∞ , we observe that in each interval [t k , t k+1 ) with 0 ≤ k ≤ n -1, DF n , ∇ ω F n and ∇ 2 ω F n satisfy the boundedness-preserving property. We now prove that F n is left-continuous, ∇ ω F n and ∇ 2 ω F n are right-continuous, and DF n is continuous at fixed times.

Let s ∈ [t k , t k+1 ) for some 0 ≤ k ≤ n -1 and ω ∈ D([0, T ], R d ).
We first prove that F n is right-continuous at (s, ω), and is jointly continuous at (s, ω) for s ∈ (t k , t k+1 ). By definition of joint-continuity (or right-continuous), we want to show that: ∀ > 0, ∃η > 0, ∀(s , ω ) ∈ Λ T (for the right-continuity, we assume in addition that s > s),

d ∞ ((s, ω), (s , ω )) < η) ⇒ |F n (s, ω) -F n (s , ω )| < Let (s , ω ) ∈ Λ T (
with s > s for the right-continuity). We assume that d ∞ ((s, ω), (s , ω )) ≤ η with an η small enough such that s ∈ [t k , t k+1 ) (this is always possible as if s = t k , we are only interested in the right-continuity, thus s > s). It suffices to prove that |F n (s, ω) -F n (s , ω )| ≤ C(s, ω s , η) with C(s, ω s , η) a quantity depending only on s, ω s and η, and C(s, ω s , η) -→ η→0 0.

We use the expression of F n obtained in the proof of Theorem 4.

1. Let y = (y 1 , • • • , y n-k ) ∈ M d,n-k (R) with each y l ∈ R d for 1 ≤ l ≤ n -k, we have: F n (s, ω) = R d×(n-k) g p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), y 1 + ω(s) -ω(t k -), y 2 , • • • , y n ) Φ(y 1 , t k+1 -s) n-k l=2 Φ(y l , δ)dy 1 dy 2 • • • dy n-k and F n (s , ω ) = R d×(n-k) g p(ω (t 1 -) -ω (0), • • • , ω (t k -) -ω (t k-1 -), y 1 + ω (s ) -ω (t k -), y 2 , • • • , y n ) Φ(y 1 , t k+1 -s ) n-k l=2 Φ(y l , δ)dy 1 dy 2 • • • dy n-k with Φ(x, t) = (2πt) -d 2 exp - |x| 2 2t , x ∈ R d
To simplify the notations, we set:

p(ω, s, y) = p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), y 1 + ω(s) -ω(t k -), y 2 , • • • , y n ) and p(ω , s , y) = p(ω (t 1 -) -ω (0), • • • , ω (t k -) -ω (t k-1 -), y 1 + ω (s ) -ω (t k -), y 2 , • • • , y n )
Similarly pt (•) will be the path of p(•) stopped at time t.

As ω s -ω s ∞ ≤ η < δ, by Lemma 3.1, we have:

p(ω, s, y) -p(ω , s , y) ∞ ≤ C(ω s , y, σ lip , T )η
Actually we have the following better estimate of p(ω, s, y) -p(ω , s , y) ∞ :

Lemma 4.1. We have:

p(ω, s, y) -p(ω , s , y) ∞ ≤ C(ω s , σ lip , T ) n-k l=1 (1 + |y l |σ lip )η
for some constant C which depends only on ω s , σ lip and T .

Proof. By Lemma 3.1, we know already that:

pt k (ω, s, y) -pt k (ω , s , y) ∞ ≤ C(ω s , σ lip , T )η
Now we prove by induction that, for any k

+ 1 ≤ j ≤ n, ptj (ω, s, y) -ptj (ω , s , y) ∞ ≤ C(ω s , σ lip , T ) j-k l=1 (1 + |y l |σ lip )η (16)
for some constant C which depends only on ω s , σ lip and T .

Consider first the case where j = k + 1. We have:

p(ω, s, y)(t k+1 ) = p(ω, s, y)(t k ) + σ (t k , pt k (ω, s, y)) (ω(s) -ω(t k -) + y 1 ) and p(ω , s , y)(t k+1 ) = p(ω , s , y)(t k ) + σ (t k , pt k (ω , s , y)) (ω (s ) -ω (t k -) + y 1 )
As σ is Lipschitz continuous with respect to d ∞ , we have:

σ (t k , pt k (ω, s, y)) -σ (t k , pt k (ω , s , y)) ≤ σ lip C(ω s , σ lip , T )η In addition, we have |ω(s) -ω (s )| ≤ η and |ω(t k -) -ω (t k -)| ≤ η as ω s -ω s ∞ ≤ η. Thus we have: |p(ω, s, y)(t k+1 ) -p(ω , s , y)(t k+1 )| ≤ |p(ω, s, y)(t k ) -p(ω , s , y)(t k )| + σ (t k , pt k (ω, s, y)) 2η + σ lip C(ω s , σ lip , T )η • (2 ω s ∞ + |y 1 |) ≤ C(ω s , σ lip , T )η + C(σ lip , T )(1 + pt k (ω, s, y) ∞ )2η + σ lip C(ω s , σ lip , T )η • (2 ω s ∞ + 2η + |y 1 |) ≤ C (ω s , σ lip , T )(1 + |y 1 |σ lip )η
with C a constant which depends only on ω s , σ lip and T .

Assume now that ( 16) holds for some j ≥ k + 1. We have: p(ω, s, y)(t j+1 ) = p(ω, s, y)(t j ) + σ t j , ptj (ω, s, y) y j-k+1 and p(ω , s , y)(t j+1 ) = p(ω , s , y)(t j ) + σ t j , ptj (ω , s , y) y j-k+1

Observe that |Φ(y

1 , t k+1 -s) -Φ(y 1 , t k+1 -s )| ≤ |s -s | • ρ(y 1 , η) ≤ ρ(y 1 , η) • η with ρ(y 1 , η) = sup t∈[t k+1 -s-η,δ] |∂ t Φ(y 1 , t)|
and we have:

ρ(y 1 , η) -→ η→0 sup t∈[t k+1 -s,δ] |∂ t Φ(y 1 , t)| = sup t∈[t k+1 -s,δ] Φ(y 1 , t) |y 1 | 2 2t 2 - d 2t < ∞
So the second part of ( 17) can be controlled by:

R d×(n-k) |g(p(ω, s, y))|•|Φ(y 1 , t k+1 -s)-Φ(y 1 , t k+1 -s )| n-k l=2 Φ(y l , δ)dy 1 dy 2 • • • dy n-k ≤ C(s, ω s , η) with C(t k , ω t k , η) -→ η→0 0.
We conclude that

|F n (t k , ω) -F n (s , ω )| ≤ C(t k , ω t k , η)
with C(t k , ω t k , η) -→ η→0 0, which proves the left-continuity of F n at (t k , ω).

Corollary 4.1. Under Assumptions 1 and 2, for any t ∈ [0, T ) we have:

F n (T, W T ) -F n (t, W t ) = T t ∇ ω F n (s, W s ) • dW (s), P -a.s. (18)
Proof. As F n ∈ C 1,2 loc,r (Λ T ), we can apply the functional Itô formula Proposition 2.2 and we remark that the finite variation term is zero as n Y (s) = F n (s, W s ) is a martingale. Remark 4.1. We can also verify using directly the expressions we have obtained in Theorem 4.1 for DF n and ∇ 2 ω F n that the finite variation terms in (18) cancel each other. By the functional Itô formula, the finite variation term in [START_REF] Gobet | Computation of Greeks for barrier and lookback options using Malliavin calculus[END_REF]

equals to DF n (s, W s ) + 1 2 tr(∇ 2 ω F n (s, W s ))
. And for (s, ω) ∈ Λ T with s ∈ [t k , t k+1 ), we have:

tr ∇ 2 ω F n (s, ω) = d i=1 ∂ 2 i F n (s, ω) = R d×(n-k) g(p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), y 1 + ω(s) -ω(t k -), y 2 , • • • , y n )) d i=1 (y 1 • e i ) 2 (t k+1 -s) 2 - 1 t k+1 -s Φ(y 1 , t k+1 -s) n-k l=2 Φ(y l , δ)dy 1 • • • dy n-k = R d×(n-k) g(p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), y 1 + ω(s) -ω(t k -), y 2 , • • • , y n )) |y 1 | 2 (t k+1 -s) 2 - d t k+1 -s Φ(y 1 , t k+1 -s) n-k l=2 Φ(y l , δ)dy 1 • • • dy n-k = -2DF n (s, ω s )
which confirms that F n is a solution of the path-dependent Kolmogorov equation [START_REF] Cont | Functional Ito calculus and path-dependent Kolmogorov equations[END_REF]Sec. 5]:

DF n (s, W s ) + 1 2 tr(∇ 2 ω F n (s, W s )) = 0.

Convergence and error analysis

In this section, we analyze the convergence rate of our approximation method. After having constructed a sequence of smooth functionals F n ( Theorem 4.1 and Theorem 4.2), we can now approximate ∇ X Y by:

Z n (s) = t (σ -1 (s, X s ))∇ ω F n (s, W s )
which, in contrast to the weak derivative ∇ X Y , is computable as a pathwise directional derivative.

In practice, ∇ ω F n (s, W s ) can be computed numerically via a finite difference method or a Monte-Carlo method using the expression ( 14) of ∇ ω F n .

For t ∈ [0, T ], the quantity we are interested in is the integral of ∇ X Y -Z n along the path of X between t and T , i.e.

T t (∇ X Y -Z n ) • dX = T t ∇ X Y (s) • dX(s) - T t ∇ ω F n (s, W s ) • dW (s)
By the martingale representation formula Proposition 2.3 and Corollary 4.1, we have P-a.s.

T t (∇ X Y -Z n ) • dX = Y (T ) -Y (t) -( n Y (T ) -n Y (t)) = g(X T ) -g( n X T (W T )) -E g(X T ) -g( n X T (W t ⊕ t B T ))|F X t
where n X is the path of the piecewise constant Euler-Maruyama scheme defined in [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF]. Remark that by definition of the concatenation operation and using the fact that B and W are two independent Brownian motions, we have:

E g( n X T (W t ⊕ t B T ))|F X t = E g( n X T (W t ⊕ t W T ))|F X t = E g( n X T (W T )|F X t Corollary 5.1. Under Assumptions 1 and 2, ∀t ∈ [0, T ], T t (∇ X Y -Z n ) • dX -→ n→∞ 0, P-a.s.
Proof. We have already shown that:

T t (∇ X Y -Z n ) • dX = g(X T ) -g( n X T ) -E g(X T ) -g( n X T )|F X t
As g is continuous with respect to • ∞ , by Corollary 3.1, we have:

g(X T ) -g( n X T ) -→ n→∞ 0, P-a.s.
Moreover, g has at most polynomial growth with respect to • ∞ , which, together with Proposition 3.2, ensures the uniform integrability of g( n X T ). And thus

E g(X T ) -g( n X T )|F X t -→ n→∞ 0, P-a.s. Corollary 5.2. Under Assumptions 1 and 2, ∀t ∈ [0, T ], T t (∇ X Y -Z n ) • dX 2p -→ n→∞ 0, ∀p ≥ 1
Under a slightly stronger assumption on g we can obtain a rate of convergence for our approximation: Theorem 5.1 (Rate of convergence). Let p ≥ 1 and assume g :

(D([0, T ], R d ), . ∞ ) → R is Lipschitz-continuous: ∃g lip > 0, ∀ω, ω ∈ D([0, T ], R d ), |g(ω) -g(ω )| ≤ g lip sup |ω -ω | .
Under Assumptions 1 the L 2p -error of the approximation Z n of ∇ X Y along the path of X between t and T is bounded by:

E   T t (∇ X Y -Z n ) • dX 2p   ≤ C(x 0 , p, T, σ lip , g lip ) 1 + log n n p , ∀p ≥ 1
where the constant C depends only on x 0 , p, T, σ lip and g lip . In particular:

∀α ∈ [0, 1 2 ), n α T
t (∇ X Y -Z n ) • dX -→ n→∞ 0, P-a.s.
Proof. This result is a consequence of Proposition 3.2 since

T t (∇ X Y (s) -Z n (s)) • dX(s) 2p ≤ g(X T ) -g( n X T ) 2p + E[g(X T ) -g( n X T )|F X t ] 2p ≤ 2 g(X T ) -g( n X T ) 2p ≤ 2g lip sup s∈[0,T ] |X(s) -n X(s)| 2p .
The following example how our result may be used to construct explicit approximations with cotnroled convergence rates for conditional expectation of non-smooth functionals:

Example 5.1. Let g(ω) = ψ(ω(T ), sup t∈[0,T ] ω(t) ) where ψ ∈ C 0 (R d × R + ,

Comparison with approaches based on the Malliavin calculus

The vertical derivative ∇ X Y (t) which appears in the martingale representation formula may be viewed as a 'sensitivity' of the martingale Y to the initial condition X(t). Thus, our method is related to methods previously proposed for 'sensitivity analysis' of Wiener functionals.

One can roughly classify such methods into two categories [START_REF] Chen | Malliavin Greeks without Malliavin calculus[END_REF]: methods that differentiate paths and methods that differentiate densities. When the density of the functional is known, the sensitivity of an expectation with respect to some parameter is to differentiate directly the density function with respect to the parameter. However, as this is almost never the case in a general diffusion model, let alone a non-Markovian model, alternative methods, are used: these consist of differentiating either the functional g or the process with respect to the parameter under the expectation sign, then estimating the expectation with the Monte-Carlo method. To differentiate process, one required the existence of the so-called first variation process, which requires the regularity of the coefficients of the SDE satisfied by X.

Sensitivity estimators for non-smooth functionals may be computed using Malliavin calculus: this approach, proposed by Fournié et al. [START_REF] Fournié | Applications of Malliavin calculus to Monte Carlo methods in finance[END_REF] and developed by Cvitanic, Ma and Zhang [START_REF] Cvitanić | Efficient computation of hedging portfolios for options with discontinuous payoffs[END_REF], Fournié et al. [START_REF] Fournié | Applications of Malliavin calculus to Monte-Carlo methods in finance[END_REF], Gobet and Kohatsu-Higa [START_REF] Gobet | Computation of Greeks for barrier and lookback options using Malliavin calculus[END_REF], Kohatsu-Higa and Montero [START_REF] Kohatsu | Malliavin calculus in finance[END_REF], Davis and Johansson [START_REF] Mark | Malliavin Monte Carlo Greeks for jump diffusions[END_REF] and others, uses the Malliavin integration-by-parts formula on Wiener space in the case where g is not smooth. These methods require quite demanding regularity assumptions (differentiability and ellipticity condition on σ for example) on the coefficients of the initial SDE satisfied by X.

By contrast, the approximation method presented here allows for any continuous functional g with polynomial growth and requires only mild assumptions on σ: Lipschitz continuity and nonsingularity. It is thus applicable to a wider range of examples than the Malliavin approach, while being arguably simpler from a computational viewpoint. Our method involves discretizing then differentiating, as opposed to the Malliavin approach which involves differentiating in the Malliavin sense, then discretizing the tangent process which, as argued in [START_REF] Chen | Malliavin Greeks without Malliavin calculus[END_REF], has its computational advantages.

In our setting, we have F n ∈ C 1,2 loc,r (Λ T ) which is sufficient for obtaining an approximation of martingale representations via the functional Itô formula. One can ask if the Euler approximation n X can also be used to obtain a Clark-Haussmann-Ocone type formula, and in this case, whether the pathwise vertical derivative ∇ ω F n (t, W t ) leads to the same representation as the Clark-Haussmann-Ocone formula.

For n ∈ N, define H n = g( n X T (W T )) with n X the weak piecewise constant Euler-Maruyama scheme defined by [START_REF] Cont | Functional Itô calculus and stochastic integral representation of martingales[END_REF]. By the definition of n X, the random variable H n actually depends only on a finite number of Gaussian variables: W (t 1 ), W (t 2 ) -W (t 1 ), • • • , W (t n ) -W (t n-1 ), thus it can be written as:

H n = h n (W (t 1 ), W (t 2 ) -W (t 1 ), • • • , W (t n ) -W (t n-1 ))
with h n : M d,n (R) → R (h n is actually g • p with p defined by ( 8)).

Clearly if h n is a smooth function with polynomial growth, then H n ∈ D 1,2 with Malliavin derivative [START_REF] Nualart | Malliavin calculus and its applications[END_REF]: So in the case where h n are smooth, our method provides the same result as given by the Clark-Haussmann-Ocone formula applied to h n . However, in our framework, as the functional g is only assumed to be continuous with polynomial growth, the function h n may fail to be differentiable. So, even in the cylindrical case, it is not clear whether the random variable H n is differentiable in the Malliavin sense, and even if it is the case, it is difficult to obtain an explicit form for E D t H n |F W t using the Malliavin calculus.

D t H n = (D k t H n , 1 ≤ k ≤ d) ∈ R
The reason our approximation method works even in the cases where H n is not differentiable in the Malliavin sense is that our regularity assumptions are not on the terminal variable H n , but on the martingale n Y (t) = E H n |F W t ; as shown in Section 4, n Y is differentiable in the pathwise sense even when H n is not differentiable in the Malliavin sense.

  , we shall denote by t A the transpose of A, and A = tr ( t AA) the Frobenius norm of A. For x, y ∈ R d , x • y is the scalar product on R d . Let T > 0. We denote by D([0, T ], R d ) the space of functions defined on [0, T ] with values in R d which are right continuous with left limits (càdlàg). For a path ω ∈ D([0, T ], R d ) and t ∈ [0, T ], we denote by:

  y j+1 and p(y )(t j+1 ) = p(y )(t j ) + σ(t j , p tj (y ))y j+1 . Thus |p(y)(t j+1 ) -p(y )(t j+1 )| ≤ |p(y)(t j ) -p(y )(t j )| + σ(t j , p tj (y)) • |y j+1 -y j+1 | + σ(t j , p tj (y)) -σ(t j , p tj (y )) • |y j+1 | ≤ C(y, η, σ lip , T ) max 1≤k≤j |y k -y k | + σ(0, 0) + σ lip (t j + p tj (y)) ∞ ) |y j+1 -y j+1 | +σ lip C(y, η, σ lip , T ) max 1≤k≤j |y k -y k |(|y j+1 | + |y j+1 -y j+1 |) ≤ C (y, η, σ lip , T ) max 1≤k≤j+1 |y k -y k |

  ) -p(ω , s , y)(t j )| + σ t j , ptj (ω, s, y) -σ t j , ptj (ω , s , y)• |y j-k+1 | ≤ C(ω s , σ lip , T ) |y l |σ lip )η • (1 + |y j-k+1 |σ lip ) ≤ C(ω s , σ lip , T ) |y l |σ lip )ηfor some constant C which depends only on ω s , σ lip and T . Now we can control the difference between F n (s, ω) and F n (s , ω ).

	Thus
	|p(ω, s, y)(t j+1 ) -p(ω , s , y)(t j+1 )|
	≤ |p(ω, s, y)(t j j-k
		l=1 (1 + j-k+1
		(1 + |y l |σ lip )η
		l=1
	And we conclude by induction that
		n-k
	p(ω, s, y) -p(ω , s , y) ∞ ≤ C(ω s , σ lip , T ) (1 + |F n (s, ω) -F n (s , ω )| l=1
			n-k
	≤	|g(p(ω, s, y))Φ(y 1 , t k+1 -s) -g(p(ω , s , y))Φ(y 1 , t k+1 -s )|	Φ(y l , δ)dy 1 dy 2 • • • dy n-k
	R d×(n-k)		l=2
	≤	|g(p(ω, s, y)) -g(p(ω , s , y))|Φ(y 1 , t k+1 -s )
	R d×(n-k)
		n-k
	+|g(p(ω, s, y))| • |Φ(y 1 , t k+1 -s) -Φ(y 1 , t k+1 -s )| ×	Φ(y l , δ)dy 1 dy 2 • • • dy n-k
		l=2

  R) is a continuous function with polynomial growth, and setY (t) = E[g(X T )|F t ].Then g satisfies Assumption 2, and our approximation method applies.

	Moreover,
	if ψ is Lipschitz-continuous, then Theorem 5.1 yields an explicit control of the approximation error with a 1/ √ n bound.

  ∂ kj h n (W (t 1 ), W (t 2 ) -W (t 1 ), • • • , W (t n ) -W (t n-1 ))1 [tj ,tj+1) (t), t ∈ [0, T )In this case, assume that t ∈ [t j , t j+1 ) for some 0 ≤ j ≤ n -1, we have:for 1 ≤ k ≤ d, E D k t H n |F W t = E ∂ kj h n (W (t 1 ), W (t 2 ) -W (t 1 ), • • • , W (t n ) -W (t n-1 ))|F W t = E ∂ kj h n (ω(t 1 ), ω(t 2 ) -ω(t 1 ), • • • , ω(t j ) -ω(t j-1 ), W (t j+1 ) -W (t) + ω(t) -ω(t j ), W (t j+2 ) -W (t j+1 ), • • • , W (t n ) -W (t n-1 )) | ωt=Wt = E ∂ ∂h h n (ω(t 1 ), ω(t 2 ) -ω(t 1 ), • • • , ω(t j ) -ω(t j-1 ), W (t j+1 ) -W (t) + ω(t) -ω(t j ) + he k , W (t j+2 ) -W (t j+1 ), • • • , W (t n ) -W (t n-1 ))| h=0 | ωt=Wt

			d
	with
			n-1
	D k t H n =
			j=0
	=	∂ ∂h	E h

n (ω(t 1 ), ω(t 2 ) -ω(t 1 ), • • • , ω(t j ) -ω(t j-1 ), W (t j+1 ) -W (t) + ω(t) -ω(t j ) + he k , W (t j+2 ) -W (t j+1 ), • • • , W (t n ) -W (t n-1 )) ωt=Wt | h=0 which is none other than ∂ k F n (t, W t ) = lim h→0 F n (t, W t + he k 1 [t,T ] ) -F n (t, W t ) h .

Thus we have:

And for i = j:

g p(ω(t 1 -) -ω(0), • • • , ω(t k -) -ω(t k-1 -), y 1 + ω(s) -ω(t k -),

The horizontal differentiability of F n can be proved similarly. Consider the following map:

The objective is to show that w is differentiable at 0+.

We assume again that t k ≤ s < t k+1 for some 0 ≤ k ≤ n -1, and we take an h > 0 small enough such that s + h < t k+1 . Using the same argument and the fact that ω s (s + h) = ω(s), we have:

Again the only term which depends on h in the integrand of ( 15) is Φ(y 1 , t k+1 -s -h), which is a smooth function of h. Therefore F n is horizontally differentiable with:

For the first part of (17), we use the continuity of g and Lemma 4.1. As g is continuous at p(ω s , y), we have:

Thus the first part of ( 17) can also be bounded by C(s, ω s , η) with

We conclude that |F n (s, ω s ) -F n (s , ω s )| ≤ C(s, ω s , η) with C(s, ω, η) depending only on s, ω s and η, and C(s, ω s , η) -→ η→0 0, which proves the right-continuity of F n and the joint-continuity of

The right-continuity of ∇ ω F n , ∇ 2 ω F n and the continuity at fixed times of DF n can be deduced as above using the expressions of ∇ ω F n , ∇ 2 ω F n and DF n obtained in Theorem 4.1. Now it remains to show that for 1

We choose an η small enough in order that s ∈ [t k-1 , t k ), and we want to show that

We first decompose |F n (t k , ω) -F n (s , ω )| into two terms:

For the second part, as F n is continuous at fixed time s by the first part of the proof, and

For the first part |F n (t k , ω) -F n (s , ω s )|, the difficulty is that s and t k no longer lie in the same interval, thus we need to perform one more integration for F n (s , ω s ) compared to

Using again the expression of F n we have obtained in the proof of Theorem 4.1, we have:

We now define ζ : R d → R by: for y ∈ R d ,

By Lemma 3.1 and the continuity of g with respect to • ∞ , the map

is continuous. As g has at most polynomial growth with respect to • ∞ , by the dominated convergence theorem, ζ is also continuous. Moreover, ζ has at most polynomial growth. And as t k -s ≤ η, we have

It remains to control the difference between F n (t k , ω) and ζ(0). We remark that:

As ω s -ω t k ∞ ≤ ω s -ω s ∞ + ω t k -ω s ∞ ≤ 2η, again by the continuity of F n at fixed time t k established in the first part of the proof, we have: