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

TL;DR

This work addresses a finite-horizon stochastic linear-quadratic control problem for systems with Markovian regime switching, model uncertainty, and partial information. It casts the problem as a soft-constrained zero-sum $LQ$ stochastic differential game under $H_\infty$ constraints and develops a Riccati-equation–based, filter-driven synthesis to produce a closed-loop saddle point whose outcome achieves the $H_\infty$ performance. The key contributions are the two equivalent Riccati formulations, the orthogonal-decomposition and completion-of-squares techniques to derive optimal feedback strategies, and a rigorous demonstration that the resulting control and disturbance satisfy the disturbance-attenuation criterion. A stock-market investment example demonstrates practical applicability and behavior under bear/bull regimes and varying $\gamma$, highlighting the method's relevance for robust decision-making under regime-switching uncertainty.

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-$σ$-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.

Figures (10)

• Figure 1: The numerical solutions of Riccati equations $\Pi(\cdot,1)$, $\Pi(\cdot,2)$, $P(\cdot,1)$ and $P(\cdot,2)$
• Figure 2: The closed-loop saddle point $(\hat{\Theta}^*(\cdot,\alpha(\cdot)),\tilde{\Theta}^*(\cdot,\alpha(\cdot)))$
• Figure 3: The state process $x^*(\cdot)$, the filtering state process $\hat{x}^*(\cdot)$ and the difference $\tilde{x}^*(\cdot)$
• Figure 4: The robust $H_\infty$ optimal control $u^*(\cdot)$ and the worst-case disturbance $v^*(\cdot)$
• Figure 5: The closed-loop saddle point $(\hat{\Theta}^*(\cdot,\alpha(\cdot)),\tilde{\Theta}^*(\cdot,\alpha(\cdot)))$ ($\gamma=2$)

Theorems & Definitions (30)

• Definition 2.1: finite-horizon stochastic $H_\infty$ closed-loop optimal control
• Remark 2.1
• Definition 2.2
• Remark 2.2
• Proposition 3.1
• proof
• Corollary 3.1
• proof
• Proposition 3.2
• proof