In this work, we are concerned with the high-order numerical methods for coupled forward-backward stochastic differential equations (FBSDEs). Based on the FBSDEs theory, we derive two reference ordinary differential equations (ODEs) from the backward SDE, which contain the conditional expectations and their derivatives. Then, our high-order multistep schemes are obtained by carefully approximating the conditional expectations and the derivatives, in the reference ODEs. Motivated by the local property of the generator of diffusion processes, the Euler method is used to solve the forward SDE; however, it is noticed that the numerical solution of the backward SDE is still of high-order accuracy. Such results are obviously promising: on one hand, the use of the Euler method (for the forward SDE) can dramatically simplify the entire computational scheme, and on the other hand, one might be only interested in the solution of the backward SDE in many real applications such as option pricing. Several numerical experiments are presented to demonstrate the effectiveness of the numerical method.

In this work, we propose a simple yet effective gradient projection algorithm for a class of stochastic optimal control problems. We first reduce the optimal control problem to an optimization problem for a convex functional by means of a projection operator. Then we propose a convergent iterative scheme for the optimization problem. The key issue in our iterative scheme is to compute the gradient of the objective functional by solving the adjoint equations that are given by backward stochastic differential equations (BSDEs). The Euler method is used to solve the resulting BSDEs. Rigorous convergence analysis is presented, and it is shown that the entire numerical algorithm admits a first order rate of convergence. Several numerical examples are carried out to support the theoretical finding.

In this paper, we study the error estimate of the $\theta$-scheme for the backward stochastic differential equation $y_t=\varphi(W_T)+\int_t^Tf(s,y_s)ds-\int_t^Tz_sdW_s$. We show that this scheme is of first-order convergence in $y$ for general $\theta$. In particular, for the case of $\theta=\frac{1}{2}$ (i.e., the Crank-Nicolson scheme), we prove that this scheme is of second-order convergence in $y$ and first-order in $z$. Some numerical examples are also given to validate our theoretical results.

A Numerical Method and its Error Estimates for the Decoupled Forward-Backward Stochastic Differential Equations - Volume 15 Issue 3 - Weidong Zhao, Wei Zhang, Lili Ju

High Order Numerical Schemes for Second-Order FBSDEs with Applications to Stochastic Optimal Control - Volume 21 Issue 3 - Weidong Zhao, Tao Zhou, Tao Kong