A New Kind of High-Order Multi-step Schemes for Forward Backward Stochastic Differential Equations

TL;DR

This paper introduces high-order multi-step numerical schemes for coupled forward-backward stochastic differential equations, leveraging derivatives of conditional expectations and simplifying the forward SDE solution with Euler's method, demonstrating promising accuracy and efficiency.

Contribution

The paper develops a novel high-order multi-step scheme for FBSDEs that simplifies forward SDE solving and maintains high accuracy for backward SDEs, with practical numerical validation.

Findings

High-order accuracy achieved for backward SDEs.

Euler method effectively simplifies forward SDE computation.

Numerical experiments confirm the method's effectiveness.

Abstract

In this work, we concern 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 multi-step schemes are obtained by carefully approximating the derivatives and the conditional expectations 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 Euler method (for the forward SDE) can dramatically simplifies 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 carried out to demonstrate the effectiveness of the numerical method.