Linear-Convex partially observed optimal control problem with Markov chain and input constraint

¹ Suzhou Research Institute, Shandong University, Jiangsu 215123, Suzhou, PR China
² School of Mathematics, Shandong University, Shandong 250100, Jinan, PR China

^* Corresponding author: huangzy@sdu.edu.cn

Received: 23 May 2023
Accepted: 28 February 2025

Abstract

In this paper, we study a linear–convex problem for partially observed forward–backward stochastic control system with Markov chain and input constraints. The observation is assumed to be controlled and follows a regime-switching stochastic differential equation, whose drift term is linear with respect to the state process x and control strategy u. Firstly, for the general case, by using the backward separate approach to decompose the state and observation, we obtain the optimal control strategy by virtue of stochastic maximum principle. We then prove the well-posedness of the stochastic Hamiltonian system using the method of continuity. Secondly, for the linear-quadratic case under linear subspace constraints, we present the feedback representation of the optimal control strategy. Finally, we apply our theoretical results to an asset-liability problem to demonstrate their practical significance.