Stochastic linear-quadratic differential game with Markovian jumps in an infinite horizon

TL;DR

This paper studies stochastic linear-quadratic differential games with Markovian jumps over an infinite horizon, characterizing Nash equilibria via solutions to FBSDEs and Riccati equations, and provides concrete examples.

Contribution

It develops a comprehensive framework for analyzing both zero-sum and non-zero-sum LQ stochastic differential games with Markovian jumps, including equilibrium characterizations and solution methods.

Findings

Nash equilibria characterized by FBSDE solvability

Closed-loop strategies via Riccati equations

Concrete examples with explicit solutions

Abstract

This paper investigates a two-person non-homogeneous linear-quadratic stochastic differential game (LQ-SDG, for short) in an infinite horizon for a system regulated by a time-invariant Markov chain. Both non-zero-sum and zero-sum LQ-SDG problems are studied. It is shown that the zero-sum LQ-SDG problem can be considered a special non-zero-sum LQ-SDG problem. The open-loop Nash equilibrium point of non-zero-sum (zero-sum, respectively) LQ-SDG problem is characterized by the solvability of a system of constrained forward-backward stochastic differential equations (FBSDEs, for short) in an infinite horizon and the convexity (convexity-concavity, respectively) of the performance functional and their corresponding closed-loop Nash equilibrium strategy are characterized by the solvability of a system of constrained coupled algebra Riccati equations (CAREs, for short) with certain stabilizing conditions. In addition, the closed-loop representation of open-loop Nash equilibrium point for non-zero-sum (zero-sum, respectively) LQ-SDG is provided by the non-symmetric (symmetric, respectively) solution to a system CAREs. At the end of this paper, we provide three concrete examples and solve their open-loop/closed-loop Nash equilibrium strategy based on the obtained results.