# Maximum principle for stochastic optimal control problem of forward-backward stochastic difference systems

## TL;DR

This paper establishes a maximum principle for stochastic optimal control problems involving forward-backward stochastic difference systems, covering both partially and fully coupled equations, with a focus on convex control domains.

## Contribution

It introduces a novel maximum principle for FBS{ extDelta}Ss, including new adjoint difference equations and applicable to both partially and fully coupled systems.

## Key findings

- Derived the adjoint difference equation using a product rule representation.
- Established the maximum principle for convex control domains.
- Applied the framework to both partially and fully coupled FBS{ extDelta}Ss.

## Abstract

In this paper, we study the maximum principle for stochastic optimal control problems of forward-backward stochastic difference systems (FBS{\Delta}Ss). Two types of FBS{\Delta}Ss are investigated. The first one is described by a partially coupled forward-backward stochastic difference equation (FBS{\Delta}E) and the second one is described by a fully coupled FBS{\Delta}E. By adopting an appropriate representation of the product rule and an appropriate formulation of the backward stochastic difference equation (BS{\Delta}E), we deduce the adjoint difference equation. Finally, the maximum principle for this optimal control problem with the control domain being convex is established.