Optimal Error Estimates of the Crank-Nicolson Scheme for Solving a Kind of Decoupled FBSDEs

In this paper, under weak conditions, we theoretically prove the second-order convergence rate of the Crank-Nicolson scheme for solving a kind of decoupled forward-backward stochastic differential equations.


Introduction
The existence and uniqueness of the solution for nonlinear backward stochastic differential equations (BSDEs) were first proved by [1].Since then, BSDEs have been extensively studied by many researchers.At the same time, many applications have been found.In [2], Peng obtained the relation between the backward stochastic differential equation and the parabolic partial differential equation (PDE).By using the relation between the BSDE and PDE, a four step scheme was proposed in [3].In [4], some simple numerical schemes were proposed for BSDEs and half-order convergence error estimates were proved.In [5], Zhao et al. proposed some new kind of high accurate numerical method for BSDEs, which the scheme with second order convergence rate was first proposed and analyzed in [6] [7] and [8].However, In [6] [7] [8], the authors only proved the schemes were of high order convergence for solving y and z with the generator f not depending on z.In [8], the authors proved the errors measured in the ( ) In this paper, we will consider the BSDEs (1.1).Under weaker conditions, we proved the Crank-Nicolson scheme has second-order convergence rate for solving the decoupled FBSDEs.In Section 2, we introduce some preliminaries and notation, and introduce the scheme in Section 3. In Section 4, we prove that the scheme is of second-order convergence in solving y and of first-order convergence in solving Γ for the FBSDEs (1.1).

Preliminaries and Notation
Let T be a fixed positive number and  Throughout this paper, C is a generic positive constant depending only on 0 c , T, and upper bounds of functions h, ϕ , 1 f and their derivatives, moreover C can be different from line to line.

Schemes for BSDE (1.1)
We will give a brief review on the schemes proposed in [10] for solving the BSDE (1.1).
For the time interval [ ] 0,T , we introduce the following time partition: The integrand on the right-hand side of (3.2) is a deterministic smooth function of time s.We may use some numerical integration methods to accurately approximate the integral in (3.2).In particular, we use the trapezoidal rule to approximate the integral on the right hand side of (3.2) and obtain ( ) where  d .
Based on (3.5), we have where ) Based on reference Equations (3.3) and (3.6), for solving the BSDEs (1.1) we introduce the following scheme.Scheme 1 Given a random variable N Y , solve random variables n Y and n Z ( ) where

Error Estimates
In this section, we will estimate the errors L norm, where ( ) , is the solution of the BSDE (1.1) and ( ) is the solution of Scheme 1.For the sake of simplicity, we only consider one-dimensional BSDEs (i.e., 1 m d = = ).However, all error estimates we obtain in the sequel also hold for general multidimensional BSDEs.Let In our error analysis, we will use the constraint on the time partition step t ∆ : Let us first introduce the following lemma.Its proof can be found in the reference.
Lemma 1 Let n Y R and n z R be the truncation errors defined in (3.4) and (3.7), respectively.If Here C is a positive constant depending only on T, and the upper bounds of ϕ and f and their derivatives.Y Z be the solution of scheme 1. Assume Then for sufficiently small time step t ∆ , we have where C is a constant depending on 0 c , T, and upper bounds of functions h, ϕ and f and their derivatives.

Proof. Let
By the Hölder inequality (see [11] for details) we get and ( ) .

Conclusion
In this paper, we study the error estimate of the Crank-Nicolson scheme proposed in [10] for solving a kind of decoupled FBSDEs.Under weaker conditions than that in [9], we rigorously prove the second order convergence rate of the Crank-Nicolson scheme.
and let , pensated Poisson process with mean zero and variance n s t − .Now multiply (3.1) by ,n N ρ Τ ∆ , and take the conditional mathematical expectation [ ] n n X t ⋅  on both sides of the derived equation, we obtain by the Itô isometry formula 2