Discrete-time approximation of doubly reflected BSDEs

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Discrete-time approximation of doubly reflected BSDEs

Jean-François Chassagneux

To cite this version:

Jean-François Chassagneux. Discrete-time approximation of doubly reflected BSDEs. 2007. �hal- 00152231�

Discrete-time approximation of doubly reflected BSDEs

Jean-François Chassagneux^∗

Université Paris VII, PMA, and CREST Paris, France [email protected]

This version: May, 2007

Abstract

We study the discrete time approximation of doubly reflected BSDEs in a multidimensional setting. As in Ma and Zhang (2005) or Bouchard and Chassagneux (2006), we introduce the discretely reflected counterpart of these equations. We then provide representation formulae which allow us to obtain new regularity results. We also propose an Euler scheme’s type approximation and give new convergence results for both discretely and continuously reflected BSDEs.

Key words: Reflected BSDEs, discrete-time approximation schemes, Game Option, regularity.

MSC Classification (2000): 65C99, 60H35, 60G40.

∗The author would like to thank Bruno Bouchard for fruitful discussions.

1 Introduction

The main motivation of this paper is the discrete time approximation of Backward Stochastic Differential Equations (BSDEs) with two reflecting barriers, also known as doubly reflected BSDEs:







Y_t=g(X_T) +R_T

t f(X_u, Y_u, Z_u)du−R_T

t (Z_u)⁰dW_u+R_T

t dK_u⁺−R_T

t dK_u⁻ l(X_t)≤Y_t≤h(X_t), ∀t∈[0, T], a.s. (C)

R_T

0 (Y_s−l(X_s))dK_s⁺=R_T

0 (Y_s−h(X_s))dK_s⁻= 0.

(1.1)

where f, g are Lipschitz-continuous functions,h, l are smooth functions (say C_b²), and the processX is the solution of a forward SDE

X_t=X₀+ Z _t

0

b(X_s)ds+ Z _t

0

σ(X_s)dW_s, withband σ Lipschitz-continuous.

These equations can be considered as extensions of simply reflected BSDEs, which are related to optimal stopping problem (American option in finance), see e.g. [10], and whose numerical approximation has been widely studied, see e.g. [2, 3, 5, 15].

Existence and uniqueness of solutions to (1.1) have been first studied by Cvitanic- Karatzas in [7]. There has been a lot of contributions on this subject since then, consisting essentially in weakening the assumptions for the existence of (1.1), see e.g. [1] and the references therein. In economics, [7], among others, shows that these equations are related to stochastic stopping games (Dynkin games) and Ma-Cvitanic [6] connects them to the pricing of Game Options (or Israeli Options), introduced in [12].

In this Markovian setting, [6] shows that the solution of (1.1) is associated to vari- ational inequalities (or obstacles problem) of the type

(

(u−l)∧ {(u−h)∨ −[∂_tu+b∂_xu+ ¹₂T r(σσ⁰∂_xxu) +f(t, x, u, σ∂_xu)]}= 0

u(T, x) =g(x) (1.2)

in the sense that (Y_t, Z_t) = (u(t, X_t), ∂_xuσ(t, X_t)) for t ∈ [0, T]. Thus, studying the discrete time approximation of (1.1) offers alternative numerical methods to estimate the solution of (1.2).

While studying the discrete-time approximation of (1.1), it appeared that the tech- niques we used, can be applied to a multidimensional setting. Namely, Y takes values inR^d and each componentY^` verifies:

(

Y_t^{`</sup> =g<sup>`}(X_T) +R_T

t f^`(X_u, Y_u, Z_u)du−R_T

t (Z_u^`)⁰dW_u+R_T

t dK_u^`+−R_T

t dK_u^`− R_T

0 (Y_s^{`</sup>−l<sup>`}(X_s))dK_s^`−=R_T

0 (Y_s^{`</sup>−h<sup>`}(X_s))dK_s<sup>`−</sup>= 0, `∈ {1, . . . , d}, (1.3)

and,almost surely, for all t≤T,Y_t is constrained to take values in the domainO_Xt where

O_x:={y ∈R^d| ∀`∈ {1, . . . , d}, l<sup>`</sup>(x)≤y^{`</sup>≤h<sup>`}(x)}.

Following [3, 15], we first introduce “discretely reflected” versions of (1.1), meaning that condition (C) is imposed only on a deterministic set of times <={0 =: r₀ <

. . . < r_κ:=T}:

Y_T^<=Ye_T^< := g(X_T) ∈ O_XT and, forj≤κ−1and t∈[r_j, r_j+1),

( Ye_t^< = Y_r^<_j+1+R_rj+1

t f(X_s,Ye_s^<, Z_s^<)ds−R_rj+1

t (Z_s^<)⁰dW_s , Y_t^< = Ye_t^<1_{{t /∈<}}+P(X_t,Ye_t^<)1_{t∈<},

whereP(x, y) is the projection ofy∈R^donto O_x.

In the framework of doubly reflected BSDEs, i.e. d= 1, this corresponds to stochas- tic stopping games, where the stopping is allowed only on< \ {T}.

We then focus on the discrete-time approximation of such equations. As in [3, 5, 15], we introduce a partitionπ ={0 =:t₀ <· · ·< t_n:=T} such that< ⊂π and define (Y^π,Z¯^π) by the backward induction:









Z¯_t^π_i = (t_i+1−t_i)⁻¹E h

(W_ti+1−W_ti)(Y_t^π_i+1)⁰ | F_ti i Ye_t^π_i = E

h

Y_t^π_i+1 | F_ti i

+ (t_i+1−t_i)f(X_t^π_i,Ye_t^π_i,Z¯_t^π_i) Y_t^π_i = Ye_t^π_i 1_{ti∈<}/ +P(X_t^π_i,Ye_t^π_i)1_{ti∈<}, i≤n−1,

(1.4)

with terminal condition (recall thatt_n=T)

Ye_T^π = Y_T^π :=g(X_T^π). Here,X^π is the Euler scheme associated toX.

As in [3, 5, 15], we show that the error induced by this scheme:

maxi<n sup

t∈[ti,ti+1)

E h

|Ye_t^<−Ye_t^π_i|² i

+ E

"_n−1 X

i=0

Z _ti+1

ti

|Z_t^<−Z¯_t^π_i|²dt

#

(1.5) is intimately related to the regularity of the process (Y^<, Z^<), or equivalently (Ye^<, Z^<), through the quantities

maxi<n sup

t∈[ti,ti+1)

E h

|Ye_t^<−Ye_t^<_i|² i

and E

"

n−1X

i=0

Z _ti+1

ti

|Z_t^<−Z_t^<_i|²dt

# ,

for which we provide new controls in terms of |π|, the modulus ofπ. This is based on a generalization of the representation ofZ^< derived in [3].

In this paper, we essentially rely on the basic concepts developed in [3], but we face two new difficulties:

(i) Contrary to [3] where O_x is of the form {y ∈ R : y ≥ ψ(x)}, we do not have an exact expression of the projected processP(X_t,Ye_t^<)and the reflection terms are much more intricate to handle.

(ii) In the one dimensional case, a simple Girsanov transformation allows to get rid of the Malliavin derivatives ofY^< andZ^< which enter in the representation formula ofZ^< (see section 3). This is no more possible, in general, in our multidimensional setting.

Yet, in the discretely reflected case, we are able to extend the regularity result of [3]. This allows to show that the scheme (1.4) has a convergence rate of at least

|π|¹⁴. Under stronger regularity conditions on the boundaries and the coefficient of the SDE solved byX, we obtain a convergence rate of at least|π|¹² (see section 5.3).

Using an approximation argument, we then extend these results to continuously reflected BSDEs. The convergence is obtained under minimal Lipschitz-continuity assumptions with a control of order |π|¹²¹ . Under stronger regularity conditions, we extend the one dimensional result of [15], but without their uniform ellipticity assumption. Namely, we provide an upper bound of order|π|¹⁴ for the approximation error. When the system of BSDEs is decoupled, which is the most important case for financial applications, we improve it to|π|¹³.

We would like to conclude this introduction by observing that the scheme (1.4) is obviously not directly implementable since it requires the computation of conditional expectations. The global numerical error is then the sum of the discrete time ap- proximation error (1.5) and the numerical error induced by the approximation of the conditional expectations. However, this approximation problem is well understood and [2, 5, 9] among others propose efficient numerical methods, which can be easily adapted to our framework. This paper being already long, we shall not detail this part here and only focus on the discretization error.

The rest of the paper is organized as follows. In Section 2, we define BSDEs which are discretely reflected in a convex domainO_xof the above form. In Section 3, we provide different representations ofZ^<and use them to study the regularity of(Y^<,Ye^<, Z^<) in Section 4. In Section 5, we propose an Euler scheme type approximation of discretely reflected BSDEs and give our main convergence results. Finally, in Section 6, we provide extensions to the continuously reflected case. The Appendix contains the proofs ofa priori estimates which are used several times in the paper.

Notations: M^n,mis the set of matrix with dimensionn×m, we simply writeM^dif m=n=d. Forz∈M^n,m,z^ij denotes the(ij) component ofz,z^i.thei-th row ofz, z^.j thej-th column and z⁰ its transposed matrix. The spaceL^p, forp≥1, is the set of random variables X satisfying||X||_L^p := E[|X|^p]^1p <∞. The norm|.|represents the canonic norm onR^d or onM^d andh., .idenotes the usual scalar product onR^d. For a functionf ∈C¹,∇_xf denotes the Jacobian Matrix of f with respect to x.

Finally, for ease of notations, we shall sometimes writeE_s[.]forE[.| F_s],s∈[0, T].

2 Discretely reflected BSDE

2.1 Definition

Let T > 0 be a finite time horizon and (Ω,F,P) be a stochastic basis supporting a d-dimensional Brownian motion W. We assume that the filtration F = (F_t)_t≤T generated by W satisfies the usual assumptions and thatF_T =F.

LetX be the solution on [0, T]of X_t=X₀+

Z _t

0

b(X_u)du+ Z _t

0

σ(X_u)dW_u (2.1)

where X₀ ∈ R^d and b : R^d 7→ R^d, σ : R^d 7→ M^d satisfy one of the following assumptions, for some positive constantL:

• (Hx1): b,σ areL-Lipschitz continuous.

• (Hx2): b,σareC_b¹ withL-Lipschitz continuous first derivative bounded byL.

Remark 2.1. Observe that, as in [3] and contrary to [15], we make no uniform ellipticity condition on σ. In particular, the standard results of the PDE literature cannot be used to derive strong regularity properties on the solution of the PDE of the form 1.2 associated to 1.3.

Under (Hx1), we clearly have that X ∈ S²(R^d), where for p ≥ 1 and E = R^d or E = M^d, S^p(E) is the set of E-valued progressively measurable processes U such that||U||_S^p :=||sup_t∈[0,T]U_t||_L^p <∞. In particular,

||X||_S2 ≤ C_L, (2.2)

where, from now on,C_L denotes a generic constant, whose value may change from line to line, but which only depends on L, T, X₀ and d (we write C_L^p if it also depends on some extra parameterp≥1).

We then introduce a family of closed convex domains(O_x)_x∈Rd:

O_x := {y∈R^d| ∀`∈ {1, . . . , d}, l<sup>`</sup>(x)≤y^{`</sup>≤h<sup>`}(x)}, (2.3) where the mapsh, l:R^d→R^d satisfy one of the following regularity assumptions:

• (Hb1): h and lareL-Lipschitz continuous.

• (Hb2): for each`∈ {1, . . . , d},h<sup>`</sup>andl^{`</sup>verify for some(ρ<sup>`}₁, ρ^{`</sup><sub>2</sub>) :R<sup>d</sup>→R<sup>d</sup>×R<sup>d</sup>, ρ<sup>`}₃ :R^d→R⁺,

|ρ^{`</sup><sub>1</sub>(x)|+|ρ<sup>`}₂(x)|+|ρ^`₃(x)| ≤ L(1 +|x|^L)

l^{`</sup>(x)−l<sup>`}(y) ≤ ρ^{`</sup><sub>1</sub>(x)<sup>0</sup>(y−x) +ρ<sup>`}₃(x)|x−y|² , ∀x, y∈R^d. h^{`</sup>(y)−h<sup>`}(x) ≤ ρ^{`</sup><sub>2</sub>(x)<sup>0</sup>(y−x) +ρ<sup>`}₃(x)|x−y|² , ∀x, y∈R^d. This assumption is slightly weaker than the semi-convexity assumption of De- finition 1 in [2].

• (Hb3): handlareC_b² withL-Lipschitz continuous first and second derivatives bounded by L and there is ² ∈ (L⁻¹,∞) such that h^{`</sup> > l<sup>`} +², for each

`∈ {1, . . . , d}.

Observe that (Hb3)⇒(Hb2)⇒(Hb1).

Given a set of reflection times

<:={0 =:r₀< r₁ <· · ·< r_κ−1 < r_κ:=T}, κ≥1,

the solution of the discretely reflected BSDE is a triplet(Y^<,Ye^<, Z^<) satisfying Y_T^<=Ye_T^< := g(X_T) ∈ O_XT

and, forj≤κ−1and t∈[r_j, r_j+1), ( Ye_t^< = Y_r^<_j+1+R_rj+1

t f(Θ^<_u)du−R_rj+1

t (Z_u^<)⁰dW_u , Y_t^< = R

³

t , X_t, Ye_t^<

´

, (2.4)

withΘ^<= (X,Ye^<, Z^<).

Here,g:R^d7→R^d,f :R^d×R^d×M^d7→R^dare L-Lipschitz continuous and R^{`</sup>(t, x, y) :=y<sup>`}+ ([l^{`</sup>(x)−y<sup>`}]⁺−[y^{`</sup>−h<sup>`}(x)]⁺)1_{t∈<},

for (t, x, y)∈[0, T]×R^d×R,`∈ {1, . . . , d}.

Observe that

Y_t^< = Ye_t^< , fort /∈ < \ {T}. (2.5) Remark 2.2. Under (Hx1)-(Hb1), such a solution can be defined by backward induction. At each step the existence and uniqueness in S²(R^d)× H²(M^d) follow from e.g. [11]. Here, for p ≥ 1 and E = M^d or E = M^d2,d, H^p(E) is the set of progressively measurableE-valued processes V satisfying

||V||_H^p := k µZ _T

0

|V_r|²dr

¶¹₂

k_L^p < ∞.

Remark 2.3. The case where(Y^<, X) takes values inRⁿ×R^d with n6=dcan be treated in our framework. Indeed, ifd < n, we can set Xⁱ := 0, i.e. bⁱ = 0,σ^i. = 0 and X₀ⁱ = 0, fori > d. Recall that we make no ellipticity assumption. Ifd > n, we can setgⁱ=fⁱ := 0which impliesYⁱ= 0, fori > n, and work withO_x×[−², ²]^d−n,

² >0,x∈R^d.

We provide in the Appendix useful a priori estimates for “reflected” BSDEs in a somehow abstract setting. In our framework, they read as follows.

Proposition 2.1. Under (Hx1)-(Hb1), the following holds sup

t∈[0,T]

E h

|Ye_t^<|²+|Y_t^<|² i

+||Z^<||²_H2 ≤C_Lκ .

Moreover, if(Hf): f<sup>`</sup>depends onz<sup>.i</sup>only fori=`(i.e. ∇_z.if<sup>`</sup>1<sub>{i6=`}</sub> = 0, iff ∈C¹), holds, then

sup

t∈[0,T]

E h

|Ye_t^<|²+|Y_t^<|² i

≤C_L.

Proof. It suffices to apply Proposition 7.1 in the Appendix, with η_r = |X_r| and ξ_r=|h(X_r)| ∨ |l(X_r)|,r∈ <, recall (2.2). 2 2.2 Dependence on the parameters

We now present some estimates on the variation in the solution of (2.4) induced by a variation in the data. Later on, this will allow us to work with smooth parameters (f, g, etc.) before turning to the general case by an approximation argument (see e.g. Proposition 4.2).

In the rest of this section, we consider two discretely reflected BSDEs constructed as follows.

For i ∈ {1,2}, let Xⁱ be an element of S²(R^d), f_i, g_i be L-Lipschitz continuous functions andh_i,l_i boundaries satisfying(Hb1). We denote by(Y^<,i,Ye^<,i, Z^<,i) the solutions of the discretely reflected BSDE associated to these two sets of data and Θ^<,i:= (Xⁱ,Ye^<,i, Z^<,i). We then define δY^<:=Y^<,1−Y^<,2,δYe^< :=Ye^<,1−Ye^<,2,

δZ^< :=Z^<,1−Z^<,2 andδX :=X¹−X²,δf :=f₁(Θ^<,1)−f₂(Θ^<,1),δg:=g₁(X¹)−

g₂(X¹),δh:=h₁(X¹)−h₂(X¹) andδl:=l₁(X¹)−l₂(X¹).

Proposition 2.2. Under (Hx1)-(Hb1), the following holds sup

t∈[0,T]

E h

|δY_t^<|² i

+||δZ||²_H2 ≤C_L µ

κE

·

maxr∈<(|δX_r|²+|δh_r|²+|δl_r|²)

¸

+||δf||²_H2+kδg_Tk²_L2

¶ . The proof of this result requires the following Lemma whose proof uses a key argu- ment which will be very important below when studying the convergence of Euler scheme’s type approximation of (2.4).

Lemma 2.1. Let(Hx1)-(Hb1)hold. Then for each r∈ < \ {T}and`∈ {1, . . . , d}, there exists S<sub>r</sub><sup>`</sup>,Q^{`</sup><sub>r</sub> in F<sub>r</sub> such that S<sub>r</sub><sup>`}∩Q^`_r =∅ and

|(Y_r^<,1)^{`</sup>−(Y<sub>r</sub><sup><,2</sup>)<sup>`}| ≤ |(Ye_r^<,1)^{`</sup>−(Ye<sub>r</sub><sup><,2</sup>)<sup>`}|1_S` r

+

³

|l₁^{`</sup>(X<sub>r</sub><sup>1</sup>)−l<sup>`}₂(X_r²)|+|h^{`</sup><sub>1</sub>(X<sub>r</sub><sup>1</sup>)−h<sup>`}₂(X_r²)|

´ 1_Q`

r . Proof. For ease of notations, we work withd= 1and omit the exponent`. Appro- priateS_r and Q_r are constructed by considering different disjoint cases, depending on the position ofYe_r^<,1 and Ye_r^<,2.

1.a On {l₁(X_r¹) < Ye_r^<,1 < h₁(X_r¹)}, three different cases may occur depending on the position ofYe^<,2.

(i) On {l₂(X_r²)<Ye_r^<,2 < h₂(X_r²)}, we have Y_r^<,1−Y_r^<,2=Ye_r^<,1−Ye_r^<,2.

(ii) On{Yer^<,2 ≤l₂(X_r²)}, we have Yr^<,2 =P(X_r²,Yer^<,2) =l₂(X_r²). Ifl₂(X_r²) ≤Yer^<,1, then0≤Y_r^<,1−Y_r^<,2=Ye_r^<,1−l₂(X_r²)≤Ye_r^<,1−Ye_r^<,2. Ifl₂(X_r²)>Ye_r^<,1, then 0≤l₂(X_r²)−Ye_r^<,1 =Y_r^<,2−Y_r^<,1 ≤l₂(X_r²)−l₁(X_r¹).

(iii) On {h₂(X_r²) ≤ Ye_r^<,2}, similar arguments based on the comparison between h₂(X_r²) and Ye_r^<,1 lead to |Y_r^<,1−Y_r^<,2| ≤ |Ye_r^<,1 −Ye_r^<,2| on {Ye_r^<,1 ≤ h₂(X_r²)} and

|Y_r^<,1−Y_r^<,2| ≤ |h₂(X_r²)−h₁(X_r¹)|on{h₂(X_r²)<Ye_r^<,1}.

1.b We now study the case{Ye_r^<,1 ≤l₁(X_r¹)}which implies Y_r^<,1 =l₁(X_r¹).

(i) On {Ye_r^<,2≤l₂(X_r²)}, we have Y_r^<,1−Y_r^<,2=l₁(X_r¹)−l₂(X_r²).

(ii) On{l₂(X_r²)<Ye_r^<,2< h₂(X_r²)}, there are two disjoint cases. On{Y_r^<,2 < Y_r^<,1},

0 ≤ Yr^<,1 −Yr^<,2 ≤ l₁(X_r¹)−l₂(X_r²). On {Yr^<,2 ≥ Yr^<,1}, 0 ≤ Yr^<,2 −Yr^<,1 ≤

Ye_r^<,2−Ye_r^<,1.

(iii) Finally on {Ye_r^<,2 ≥ h₂(X_r²)}, we also have two disjoint cases. On {h₂(X_r²) >

Y_r^<,1},0≤Y_r^<,2−Y_r^<,1 ≤Ye_r^<,2−Ye_r^<,1. On {h₂(X_r²)≤Y_r^<,1},0≤Y_r^<,1−h₂(X_r²)≤ h₁(X_r¹)−h₂(X_r²).

1.c By symmetry, the caseYe_r^<,1 ≥h₁(X_r¹) is handled similarly. 2 We now provide the proof of Proposition 2.2.

Proof of Proposition 2.2. The proof of this Proposition relies on the abstract results of Proposition 7.1 in the Appendix. Fort∈[r_j, r_j+1), we have

δYe_t^< =δY_r^<_j+1+ Z _rj+1

t

fˆ(u)du− Z _rj+1

t

(δZ_u^<)⁰dW_u , wherefˆ:=δf+f₂(Θ^<,1)−f₂(Θ^<,2).

Sincef₂ is L-Lipschitz continuous, we have

|fˆ_u|² ≤C_L(|η_u|²+|δYe_u^<|²+|δZ_u^<|²), withη_u :=|δf_u|+|δX|.

Moreover, using Lemma 2.1, we can setξ:= 2L|δX|+|δl|+|δh|, sinceh₂ andl₂ are L-Lipschitz continuous.

The proof is then concluded by appealing to Proposition 7.1 and observing that

|δY_T^<| ≤L|δX_T|+|δg_T|, since g₂ is L-Lipschitz continuous. 2

3 Representation results for Z

In this section, we provide different representations forZ^<. The first two ones are stated in terms of the Malliavin derivatives of(X, Y^<, Z^<), the last one is based on their associated “first variation” processes.

In order ensure that (X, Y^<, Z^<) are “smooth” enough, we shall work under the additional assumption:

• (Hr): h,l,f,band σ are C_b¹.

These representations will allow us to provide regularity results for(Y^<, Z^<) under (Hr). This assumption will then be relieved by using an approximation argument based on Proposition 2.2 above.

3.1 Malliavin differentiability of (X, Y^<, Z^<)

In the sequel, we denote byD^1,2 the space of random variableF which are differen- tiable in the Malliavin sense and such that

kFk²_L2+ Z _T

0

kD_tFk²_L2dt <∞.

Here,D_tF denotes the Malliavin derivative ofF at timet≤T, see e.g. [16].

We also consider the space L^1,2 of adapted processes V such that, after possibly passing to a suitable version,V_s ∈D^1,2 for all s≤T and

||V||_H²+ Z _T

0

||D_tV||_H²dt <∞.

In the following, we shall always work with a suitable version if necessary.

Under(Hr),Xbelongs toL^1,2, see [16]. It follows thatR^`(r, X, F)∈ID^1,2 whenever F ∈ID^1,2 and

D_tR^{`</sup>(r, X, F) =D<sub>t</sub>F<sup>`}+(D_tl^{`</sup>(X<sub>r</sub>)−D<sub>t</sub>F<sup>`})1_{l`(Xr)>F<sup>`</sup>}−(D_tF^{`</sup>−D<sub>t</sub>h<sup>`}(X_r))1_{h`(Xr)<F<sup>`</sup>}. (3.1) Indeed, by a direct adaptation of the proof of Proposition 1.2.3 in [16] we deduce that, forG∈ID^1,2,[G]⁺belongs toID^1,2andD_t[G]⁺=α(D_tG)whereαis a random variable bounded by 1 satisfying 1_{G>0}α =1_{G>0}. Thus Proposition 1.3.7 in [16]

implies thatD_t[G]⁺=D_tG1_{G>0}, if G∈ID^1,2.

Combining (2.4), (3.1), and Proposition 5.3 in [11] with an induction argument, we obtain that(Ye^<, Z^<)belongs toL^1,2and that a version ofD_t((Ye^<)^{`</sup>,(Z<sup><</sup>)<sup>.`})is given by the solution inS²(R^d)× H²(M^d) of

D_t(Ye_s^<)^{`</sup> = D<sub>t</sub>(Y<sub>r</sub><sup><</sup><sub>j+1</sub>)<sup>`}+ Z _rj+1

s

(∇_xf^{`</sup>(Θ<sup><</sup><sub>u</sub>)D<sub>t</sub>X<sub>u</sub>+∇<sub>y</sub>f<sup>`}(Θ^<_u)D_tYe_u^<)du (3.2) +

Z _rj+1

s

Xd i=1

∇_z.if^`(Θ^<_u)D_t(Z_u^<)^.idu− Z _rj+1

s

Xd k=1

D_t(Z_u^<)^k`dW_u^k, for s∈[r_j, r_j+1),j < κ, with the terminal condition

D_t(Ye_T^<)^{`</sup> =∇g<sup>`}(X_T)D_tX_T .

We conclude this section with some a priori estimates that will be used later on.

The first one concerningDXis standard, we therefore omit the proof (see e.g. [16]).

Proposition 3.1. Let (Hr) hold. Then, for all p≥2 sup

s≤u∧t

kD_sX_t−D_sX_uk_L^p+||(D_tX−D_uX)1_[t∨u,T]||_S^p≤C_L^p |t−u|¹² , t, u≤T.(3.3) and

|| sup

s≤T

|D_sX| ||_S^p ≤ C_L^p . (3.4) We now turn to the study of(DY^<, DZ^<). For ease of notations, we will from now on denote byβ aF_T-measurable positive random variable, whose value may change from line to line, but satisfies

E[β^p] ≤ C_L^p , ∀p≥1. Proposition 3.2. Let (Hr) hold. Then, fors≤t≤T,

|D_sYe_t^<|²+|D_sY_t^<|²+E_t

"Z _T

t

Xd

`=1

|D_s(Z_u^<)^.`|²du

#

≤ κE_t[β] . (3.5)

If (Hf) holds, then, forp≥2,

|D_sYe_t^<|^p+|D_sY_t^<|^p+E_t

" _d X

`=1

Z _τ` j

t

|D_s(Z_u^<)^.`|²du

#

≤ E_t[β] , (3.6) and

|D_sYe_t^<|^p+|D_sY_t^<|^p ≤ E_t[β] , (3.7) for j≤κ−1, t∈[r_j, r_j+1), s≤t, where

τ_j^{`</sup> = inf{t∈ < |t≥r<sub>j+1</sub>, (Ye<sub>t</sub><sup><</sup>)<sup>`}∈/ [l^{`</sup>(X<sub>t</sub>), h<sup>`}(X_t)]} ∧T , j ≤κ−1, `≤d . (3.8)

Proof. Recall that for F ∈ D^1,2, DF = (D¹F, . . . , D^dF) where Dⁱ denotes the Malliavin derivatives with respect to Wⁱ. Fix q ∈ {1, . . . , d}, by (3.2), we have for allt≤s∈[r_j, r_j+1) andj < κ

D^q_tYe_s^< = D_t^qY_r^<_j+1− Z _rj+1

s

(D^q_tZ_u^<)⁰dW_u +

Z _rj+1

s

³

∇_xf(Θ^<_u)D^q_tX_u+∇_yf(Θ^<_u)D_t^qYe_u^<+∇_zf(Θ^<_u)D^q_tZ_u^<

´ du .

Since f is C_b¹ under (Hr), (7.2) of the Appendix holds with η = |D_t^qX|. Clearly (Af) holds under (Hf).

Moreover, it follows from (3.1), that (D^q_tY^<, D^q_tYe^<) satisfies (A0) (take S_r^{`</sup> = {(Y<sub>r</sub><sup><</sup>)<sup>`}∈[l^{`</sup>(X<sub>r</sub>), h<sup>`}(X_r)]}, for r∈ < and `∈ {1. . . d}).

The result is then a direct application of Proposition 7.1 and Corollary 7.1. 2 Similar arguments based on Proposition 7.1 also lead to

Proposition 3.3. Under (Hr), we have, for all t≤T and r, s≤t,

|D_sYe_t^<−D_rYe_t^<|² + |D_sY_t^<−D_rY_t^<|² + E_t

"Z _T

t

Xd

`=1

|D_s(Z_u^<)^{.`</sup>−D<sub>r</sub>(Z<sub>u</sub><sup><</sup>)<sup>.`}|²du

#

≤κE_t[β]|s−r|. Under(Hf),

|D_sYe_t^<−D_rYe_t^<|² + |D_sY_t−D_rY_t|² + E_t

"

Xd

`=1

Z _τ` j

t

|D_s(Z_u^<)^{.`</sup>−D<sub>r</sub>(Z<sub>u</sub><sup><</sup>)<sup>.`}|²du

#

≤E_t[β]|s−r|,

for j≤κ−1, t∈[r_j, r_j+1) andr, s≤t.

3.2 Representation in terms of Malliavin derivatives of (X, Y^<, Z^<) It follows from [16] and (2.4), viewed in a forward way, that(D_tY_t^<)_t≤T is a version ofZ^<. Hence, (3.2) implies thatZ^< admits a version satisfying

(Z_t^<)⁰=E_t

"

D_tY_r^<_j+1+ Z _rj+1

t

(∇_xf(Θ^<_u)D_tX_u+∇_yf(Θ^<_u)D_tYe_u^<+ Xd i=1

∇_z.if(Θ^<_u)D_t(Z_u^<)^.i)du

#

(3.9) for eachj ≤κ−1 andt∈[r_j, r_j+1).

Following the arguments of [3], we can get rid of the term D_tY_r^<_j+1 in the above expression.