Stochastic Linear-Quadratic Optimal Control Problems with Markovian Regime Switching and $H_\infty$ Constraint under Partial Information

TL;DR

This paper addresses a complex stochastic control problem involving Markovian regime switching, partial information, and $H_ abla$ constraints, providing a theoretical solution and practical application to stock market investment.

Contribution

It develops a novel approach combining filtering, Riccati equations, and orthogonal decomposition to solve a zero-sum stochastic differential game under partial information with $H_ abla$ constraints.

Findings

Derived the closed-loop saddle point for the game.

Proved the $H_ abla$ performance criterion is satisfied.

Applied results to a stock market investment model.

Abstract

This paper is concerned with a stochastic linear-quadratic optimal control problem of Markovian regime switching system with model uncertainty and partial information, where the information available to the control is based on a sub-$\sigma$-algebra of the filtration generated by the underlying Brownian motion and the Markov chain. Based on $H_\infty$ control theory, we turn to deal with a soft-constrained zero-sum linear-quadratic stochastic differential game with Markov chain and partial information. By virtue of the filtering technique, the Riccati equation approach, the method of orthogonal decomposition, and the completion-of-squares method, we obtain the closed-loop saddle point of the zero-sum game via the optimal feedback control-strategy pair. Subsequently, we prove that the corresponding outcome of the closed-loop saddle point satisfies the $H_\infty$ performance criterion. Finally, the obtained theoretical results are applied to a stock market investment problem to further illustrate the practical significance and effectiveness.