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

## TL;DR

This paper develops a maximum principle for stochastic optimal control problems involving finite state forward-backward stochastic difference systems, extending control theory to discrete-time, finite state models with new adjoint equations.

## Contribution

It introduces a maximum principle for finite state FBS{ extunderscore}Ss, including both partially and fully coupled systems, with new adjoint difference equations and control domain considerations.

## Key findings

- Derived the adjoint difference equation for the systems.
- Established the maximum principle for convex control domains.
- Extended stochastic control theory to finite state, discrete-time systems.

## Abstract

In this paper, we study the maximum principle for stochastic optimal control problems of forward-backward stochastic difference systems (FBS{\Delta}Ss) where the uncertainty is modeled by a discrete time, finite state process, rather than white noises. 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.