Weak Convergence Analysis and Improved Error Estimates for Decoupled Forward-Backward Stochastic Differential Equations

: In this paper, we mainly investigate the weak convergence analysis about the error terms which are determined by the discretization for solving the stochastic differential equation (SDE, for short) in forward-backward stochastic differential equations (FBSDEs, for short), which is on the basis of Itô Taylor expansion, the numerical SDE theory, and numerical FBSDEs theory. Under the weak convergence analysis of FBSDEs, we further establish better error estimates of recent numerical schemes for solving FBSDEs.


Introduction
This work is concerned with the forward-backward stochastic differential equations (FBSDEs) on (Ω, F , F, P): where t ∈ [0, T], with T > 0 being the deterministic terminal time; (Ω, F , F, P) is a complete, filtered probability space with filtration F = (F t ) 0≤t≤T being the natural filtration of the standard d-dimensional Brownian motion W = (W t ) 0≤t≤T ; X 0 ∈ F 0 is the initial condition for the forward stochastic differential equation (FSDE); ξ = ϕ(X T ) ∈ F T is the terminal condition for the backward stochastic differential equation (BSDE); the coefficients b: [0, T] × R q → R q and σ: [0, T] × R q → R q×d , the generator f : We point out that b(·, x, y, z), σ(·, x, y, z, ), and f (·, x, y, z) are all F t -adapted for any fixed numbers x, y and z, and that the two stochastic integrals with respect to W s are of the Itô type. A triple (X t , Y t , Z t ) is called an L 2 -adapted solution of the decoupled FBSDEs (1) and (2) if, in the probability space (Ω, F , F, P), it is {F t }-adapted, square integrable, and satisfies the Equations (1) and (2).
FBSDEs have important applications in many fields, including mathematical finance, partial differential equations, stochastic controls, risk measure, and so on [1][2][3][4][5]. Thus, it is interesting and also important to find solutions of FBSDEs. Usually, it is difficult to get the analytical solutions in an explicit closed form. Thus, numerical methods for solving FBSDEs are desired, especially accurate, effective, and efficient ones. Many numerical schemes for solving BSDE and decoupled FBSDEs have been developed, among which some are numerical methods with low-order convergence rates [6][7][8][9][10] under lower regularity assumptions, while others are high-order numerical methods [11][12][13][14][15][16][17]. It is notable that most of the numerical methods are designed for BSDE, or decoupled FBSDEs. In those methods, the authors place more attention on the truncated error terms, thereby ignoring the errors which are caused by the discretization scheme for solving the SDE (1).
The main purpose of this work is to prove that the weak convergence analysis of the four error terms is determined by the discretization for solving the SDE (1). Our analysis invokes an Ito Taylor expansion, the numerical SDE theory, and the numerical FBSDEs theory. In Section 2, the high-order numerical scheme introduced in [16] is briefly reviewed. Section 3 firstly gives the stability analysis of the proposed Scheme in Section 2; then our main result, weak convergence analysis, together with some useful lemmas is presented; finally, under the weak convergence analysis of FBSDEs, we improve the error estimates about the Schemes [16,17]. Some concluding remarks are made in Section 4.
For a simple representation, let us first introduce the following notations: • | · |: the standard Euclidean norm in the Euclidean space R q , R m , and R m×d .
x ∂ k 2 y φ for l 1 ≤ l and k 1 + k 2 ≤ k. • C k p : the set of k times continuously differentiable functions φ : x ∈ R q → R for which φ and all of its partial derivatives of orders up to and including k have polynomial growth.

Discretization
For the time interval [0, T], we introduce the following partitions: Let ∆t n = t n+1 − t n , n = 0, 1, . . . , N − 1 and ∆t = max 0≤n≤N−1 ∆t n . We also assume that the time partitions have the following regularity: where c 0 ≥ 1 is a constant. Let (X t,x , Y t,x , Z t,x ) be the solution of (1) and (2) with date (t, x); that is,

Reference Equations
To derive the reference equations, we first define the stochastic process The process ∆W s = (∆W 1 s , ∆W 2 s , · · · , ∆W d s ) is a martingale with the properties E X n t n [∆W s ] = 0, E X n t n [∆W i s ∆W j s ] = 0 for i = j, and Then when s = t n+1 , we have E X n t n [∆W t n+1 ] = 0 and E X n t n [(∆W t n+1 for n = 0, 1, · · · , N − 1. The forward SDE (6) can be solved independently by existing numerical methods in the above decoupled case, and in this article, we use numerical schemes of the general form where β represents the convergence rate in the strong or weak sense and A β denotes the corresponding hierarchical set; for more details on the notations, readers can be referred to [18]. Our main goal therefore turns to deducing the reference equations for Y t n ,X n Under the filtration F t n the integrands in (9) and (10) are deterministic functions of s ∈ [t n , t n+1 ], so some numerical integration methods for deterministic integration can be used to approximate these integrals.
First, we apply the trapezoidal rule to obtain with the truncation error Inserting (11) into (9), we obtain the following reference equation for solving Y t n ,X n t n : Y t n ,X n t n = E X n t n [Y t n ,X n t n+1 ] + 1 2 ∆t n f t n ,X n t n Next, for the first integral term on the right-hand side of (10), we easily get the identity where R n z = R n 1 + R n 2 .

Numerical Scheme
Now we propose a new numerical scheme for solving the decoupled FBSDEs (4). Let (X n , Y n , Z n ) denote an approximation to the analytic solution (X t , Y t , Z t ) of (4) at time t = t n , n = N, N − 1, . . . , 0. To simplify the presentation, we let f n = f (t n , X n , Y n , Z n ) for n = N, N − 1, . . . , 0. Based on (13) and (16), we propose a numerical scheme, for solving the FBSDEs (4) as follows. Scheme 1. Given random variables X 0 , Y N and Z N . Let ∆W t n+1 (0 ≤ n ≤ N − 1) be defined by (5) with s = t n+1 . For n = N − 1, N − 2, · · · , 0, solve random variables Y n and Z n by

Stability Analysis
The following result on decoupled FBSDEs is well-known by now; for its proof, see Theorem 4.1 in [16] or Theorem 4.1 in [17].
and (X n , Y n , Z n ), n = 0, 1, · · · , N, be the exact solution of the decoupled FBSDEs (1) and (2) and the approximate solution obtained by Scheme 1, respectively. Assume that the function f (t, x, y, z) is Lipschitz-continuous with respect to x, y, and z, and the Lipschitz constant is L. Let c 0 be the time partitions regularity parameter defined in (3). Then, for the sufficiently small time-step ∆t n satisfying (3), it holds that for n = N − 1, . . . , 1, 0, where C is a positive constant depending on c 0 and L; C is also a positive constant depending on c 0 , T and L; and R i y and R i z are defined in (12) and (16), respectively.
Remark 1. Theorem 1 implies that Scheme 1 is stable, and its solution continuously depends on terminal conditions; that is, for any given positive number , there exists a positive integer δ,

Remark 2.
The terms R n y and R n z in (12) and (16) come from the approximations in their reference equations for Y t n and Z t n . The four terms R n y 1 , R n y 2 , R n z 1 , and R n z 2 are determined by the discretizations (8) for solving the SDE in (6), which reflects the local errors (in the weak sense) of the numerical scheme for SDE. Under certain regularity conditions on b, σ, f , and ϕ, if we get estimates of these terms, a convergence result for the Scheme 1 can be obtained.
In the following lemmas, we will present estimations for R n y , R n z , R n y 1 , R n y 2 , R n z 1 , and R n z 2 under certain regularity conditions on b, σ, f and ϕ. For the sake of presentation simplicity, we only consider the case q = d = 1, and the results obtained also hold true for general positive integers q and d. In order to get the estimates of R n y 1 , R n y 2 , R n z 1 and R n z 2 , we need the weak error estimate about the SDE in (16).

Weak Convergence Analysis
For the convenience, we use the following notions which appeared in [18].
The next lemma provides an estimate for the higher moments of a multiple Itô integral.
In order to give the estimates of R n z 1 and R n z 2 , we need the following lemmas. Due to the complexity of the Itô Taylor expansion, we give some useful notations.
To give an important proposition, first we need to prove the following lemma.
By Lemma 1, only the terms with α + = (i) in the sum on the right-hand side of the above inequality are not zero. For α + = (i), it is easy to check ω(α, β) = l(α). By Lemma 1 and the Hölder inequality, we have the estimatê The inequality (26) in Lemma 3 and the polynomial growth bound on f α give us the estimateK From the definition of the P K z (s), we also haveK i 2 ≤ C. Combining the above estimates, we obtain Step 2. The case 2 ≤ l ≤ 2(β + 1). We take the deterministic Taylor expansion of the function F p at the point X n+1 − X n to obtain whereF k 1 ,...,k r (t) = E X n t n (X t n ,X n ,k 1 t n+1 − X n+1,k 1 ) × · · · × (X t n ,X n ,k r t n+1 − X n+1,k r ) (1) First we estimateF k 1 ,...,k r for r = 1, that is,F k for k = 1, . . . , q. From the definition of F p there exists a p ∈ P l−1 and a q ∈ {1, 2, . . . , l}, such that ∂ ∂x k F p (X n+1 − X n ) = qF p (X n+1 − X n ).
Thus, using the upper equality and Hölder inequality to (33), we obtain