A New Approach for Solving Delayed Forward and Backward Stochastic Differential Equations

TL;DR

This paper introduces a discretization-based method to explicitly solve delayed forward-backward stochastic differential equations, addressing the challenges posed by delays and infinite-dimensional issues.

Contribution

A novel discretization approach is developed to transform and solve delayed FBSDEs explicitly, extending solutions to more general stochastic control problems with delay.

Findings

Explicit solutions for delayed FBSDEs are obtained via discretization.

The method effectively handles the infinite-dimensional nature of delay.

Applicable to stochastic LQ control problems with delays.

Abstract

This paper is concerned with the decoupling of delayed linear forward-backward stochastic differential equations (D-FBSDEs), which is much more involved than the delay-free case due to the infinite dimension caused by the delay. A new approach of `discretization' is proposed to obtain the explicit solution to the D-FBSDEs. Firstly, we transform the continuous-time D-FBSDEs into the discrete-time form by using discretization. Secondly, we derive the solution of the discrete-time D-FBSDEs by applying backward iterative induction. Finally the explicit solution of the continuous-time D-FBSDEs is obtained by taking the limit to the solution of discrete-time form. The proposed approach can be applied to solve more general FBSDEs with delay, which would provide a complete solution to the stochastic LQ control with time delay.