High order numerical schemes for second-order FBSDEs with applications to stochastic optimal control

TL;DR

This paper develops high-order multistep numerical schemes for second-order FBSDEs, reducing computational complexity and demonstrating high accuracy, with applications to stochastic optimal control problems.

Contribution

It extends existing multistep schemes to second-order FBSDEs using Euler discretization, achieving high accuracy and efficiency in complex stochastic control applications.

Findings

Multistep schemes achieve high-order accuracy for 2FBSDEs.

Euler method reduces computational complexity significantly.

Numerical examples confirm effectiveness and accuracy of the schemes.

Abstract

This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.] to solve the second order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discrete the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple $(Y_t, Z_t, A_t, \Gamma_t)$ in the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effective of the proposed numerical schemes. Applications of our numerical schemes for stochastic optimal control problems are also presented.