Two person non-zero-sum linear-quadratic differential game with Markovian jumps in infinite horizon

TL;DR

This paper studies a non-zero-sum linear-quadratic differential game with Markovian jumps in an infinite horizon, providing conditions for stability, characterizing optimal controls via coupled Riccati equations, and deriving Nash equilibria.

Contribution

It introduces a novel framework for analyzing non-zero-sum LQ differential games with Markovian jumps, including stability conditions and explicit solutions for Nash equilibria.

Findings

Characterized closed-loop optimal control via coupled Riccati equations.

Established sufficient conditions for L^2-stability of the state process.

Derived Nash equilibria using solutions to coupled BSDEs and Riccati equations.

Abstract

This paper investigates an inhomogeneous non-zero-sum linear-quadratic (LQ, for short) differential game problem whose state process and cost functional are regulated by a Markov chain. Under the $L^2$ stabilizability framework, we first provide a sufficient condition to ensure the $L^2$-integrability of the state process and study a class of linear backward stochastic differential equation (BSDE, for short) in infinite horizon. Then, we seriously discuss the LQ problem and show that the closed-loop optimal control is characterized by the solutions to coupled algebra Riccati equations (CAREs, for short) with some stabilizing conditions and a linear BSDE. Based on those results, we further analyze the non-zero-sum stochastic differential game problem and give the closed-loop Nash equilibrium through the solution to a system of two cross-coupled CAREs and two cross-coupled BSDEs. Finally, some related numerical