Approximation scheme for solutions of backward stochastic diﬀerential equations via the representation theorem

We are interested in the approximation and simulation of solutions for the backward stochastic diﬀerential equations. We suggest two approximation schemes, and we study the L 2 induced error.


Introduction
Backward Stochastic Differential Equations (BSDEs in short) have been introduced by Pardoux and Peng [16].One of the main motivations for studying the BSDE's has been to use it to solve diverse problems in mathematical finance [8], in optimal stochastic control [11] and stochastic games [6].In this paper, we propose to simulate the solution (X, Y, Z) of the coupled Forward-Backward SDE defined for all 0 ≤ t ≤ T by (F BSDE) X t = x + t 0 b(s, X s )ds + t 0 σ(s, X s )dW s F orward Y t = g(X T ) + T t f (s, X s , Y s , Z s )ds − T t Z s dW s Backward where x ∈ R and W is a Brownian motion defined on some complete filtered probability space (Ω, F, P, (F t ) 0≤t≤T ).The forward component X will be approximated by the Euler or the Milshtein Scheme.The simulation of the couple (Y, Z) is more delicate.Douglas et al presented in [7] a numerical method for a class of Forward-Backward SDEs based on the finite difference approximation of the associated PDE and the four steps scheme developed in [13].In [5], Chevance proposed a numerical method for BSDE's where the generator f is not dependent on the control gradient Z.The difficulty is in the approximation of the process Z that comes only from the martingale representation Theorem.In the case where f depends only on Z, Bally has developed in [1] a discretization scheme considers a random time partition.His method is more difficult to implement.The work of Zhang [17] is more interesting but doesn't answer the question of approximation of Z. Bouchard and Touzi [3] invested this work to give an implicit approximation scheme.Recently, Gobet et al [10] propose a new numerical scheme based on iterative regression function basis which coefficients are evaluated using Monte Carlo simulation.
Our approach is to give an approximation scheme via the representation for the solution (Y, Z) of the Backward SDE.We suggest a pseudo approximation scheme then an explicit discretization scheme, and we calculate the L 2 error induced to these approximations.
We denote by ∆W t i+1 = W t i+1 − W t i and E i = E[./Ft i ] for all i.Finally, C will denote a generic constant independent of π that may take different values from line to line.
We shall make use of the following assumptions: ), and all derivatives (respect to x) are uniformly bounded by a common constant K > 0. Further, there exists a constant κ > 0 such that for any x ∈ R and t Under these assumptions, there exists a unique adapted process (X, Y, Z) solution of the (F BSDE) such that The following estimates are standard [17] max We shall cite main representation of the solution (X, Y, Z).
To begin with, let us define for 0 < t < r < T the process where ∇X is the unique solution of the following Linear SDE It is known that ∇X is invertible and As a consequence of these notations, one has the following estimates [15]: . (1.4) The representation formula can be written P-almost surely [14]: Remark 1.2.A direct consequence of the representation formula of the matringale integrand Z that: If condition 1 and condition 2 hold; for all p ≥ 2, there exists a constant C p > 0 depending only on T, K and p such that (see [15] Lemma 2.5).
Now, we are ready to give the approximation schemes.

Pseudo-discretization
In this part, we consider that the forward component X will be approximated by the classical Euler scheme We set for all and we have the following estimates for all p ≥ 2 To approximate the solution of the Backward SDE, we consider the natural discrete time scheme , and for i = n − 1, ..., 0 By a backward induction argument, we can see that ( Ȳt i , Zt i ) ∈ L 2 for any i, and they are deterministic functions of Xt i .For later use, we need to introduce a continuous time approximation for (Y, Z).Since, from the classical martingale representation Theorem, there exists a predictable process Zs dW s .
We then define for all t ∈ (t i , t i+1 ) Zs dW s Zt = Zt i . (2. 2) The following Theorem provides the error estimates due to this approximation Theorem 2.1.There exists a constant C > 0 depending only on T and K such that Proof.We denote Θ s = (X s , Y s , Z s ) and Θs = ( Xs , Ȳs , Zs ).From the Itô's formula and the Lipschitz property of f , we get for every t ∈ (t i , t i+1 ) and α > 0. Applying the Gronwall Lemma, we obtain by (1.2) where In particular for t = t i , we have Iterating the last inequality to get by the condition 2, (1.3) and (2.1).On the other hand, from the expression of Z and Z, if we denote by we can write Since (2.2) and (1.4) we have |π|.
The Lipschitz property of σ and the condition 1 insure that Taking the L 2 estimates and replacing by (2.5), then we have Now, we plug (2.6) in (2.4) to obtain for i = 0 For α sufficiently larger to C we have n−1 j=0 E|Z t j − Zt j | 2 ≤ C, and we conclude by (2.4) and (2.3) that Together, with (1.3) we conclude that This completes the proof of the Theorem.
Remark 2.2.We can obtain a similar result if we replace in the approximation of the forward component X the Euler scheme by the Milshtein scheme.
In this scheme, the process N is only dependent on the forward component X, we henceforth call it a pseudo-approximation.The following backward scheme raises this handicap.

Explicit Approximation scheme
In this section we consider that the processus X will be approximated by the Milshtein scheme described by For t ∈ (t i , t i+1 ), let us put where K is an increasing function.Let us define the processes ∇X π and (∇X π ) −1 by ∇X π t 0 = (∇X π t 0 ) −1 = 1, and for every t ∈ (t i , t i+1 ] We denote Θ π s = (X π s , Y π s , Z π s ), and let β > 0 be a constant to be chosen later on.From the Lipschitz property of f we get Using Gronwall Lemma and the classical iteration to obtain On the other hand, σ is bounded by K, then we can write .