Zero-sum stochastic linear-quadratic Stackelberg differential games of Markovian regime-switching system

TL;DR

This paper studies a zero-sum stochastic linear-quadratic Stackelberg differential game with Markovian regime switching, deriving explicit solutions and optimal controls through Riccati equations.

Contribution

It introduces a novel approach to solve the leader's problem using coupled differential Riccati equations in a Markovian regime-switching setting.

Findings

Explicit feedback representation for the follower's response

Unique solvability of the optimality system in one-dimensional case

Constructed explicit optimal control for the leader

Abstract

This paper investigates a zero-sum stochastic linear-quadratic (SLQ, for short) Stackelberg differential game problem, where the coefficients of the state equation and the weighting matrices in the performance functional are regulated by a Markov chain. By utilizing the findings in \citet{Zhang.X.2021_ILQM}, we directly present the feedback representation to the rational reaction of the follower. For the leader's problem, we derive the optimality system through the variational method and study its unique solvability from the Hilbert space point of view. We construct the explicit optimal control for the leader based on the solution to coupled differential Riccati equations (CDREs, for short) and obtain the solvability of CDREs under the one-dimensional framework. Finally, we provide two concrete examples to illustrate the results developed in this paper.