Stochastic control for Backward Stochastic Differential Equations with semi-Markov chain noises

TL;DR

This paper develops stochastic control methods for backward stochastic differential equations driven by semi-Markov chain noises, providing existence, comparison, and verification theorems without PDE techniques.

Contribution

It extends existing control results to semi-Markov chain noises and introduces backward stochastic difference equations in discrete time.

Findings

Established existence and comparison theorems for the control problem.

Derived adjoint processes via backward stochastic difference equations.

Provided a verification theorem without PDE methods.

Abstract

In this paper, we extend the results of Elliott and Yang \cite{elliott3} and discuss the control of a stochastic process for which the driving noise is provided by a martingale associated with a semi-Markov Chain. An existence and a comparison theorem are obtained. In our discrete time setting, adjoint processes are provided by backward stochastic difference equations. Technical results from partial differential equation theory to establish a verification theorem are not required.