Discrete-Time LQ Stochastic Two Person Nonzero Sum Difference Games With Random Coefficients:~Closed-Loop Nash Equilibrium

TL;DR

This paper develops a comprehensive framework for finding closed-loop Nash equilibria in discrete-time stochastic LQ nonzero-sum games with random coefficients, addressing complex coupled stochastic Riccati equations and extending solution methods to uncertain environments.

Contribution

It introduces a novel approach to characterize Nash equilibria via decoupled stochastic Hamiltonian systems and solves the associated coupled Riccati equations under minimal regularity conditions.

Findings

Established necessary and sufficient conditions for equilibrium existence.

Linked equilibrium strategies to coupled Lyapunov-type equations.

Addressed randomness in both dynamics and costs, expanding applicability.

Abstract

This paper investigates closed-loop Nash equilibria for discrete-time linear-quadratic (LQ) stochastic nonzero-sum difference games with random coefficients. Unlike existing works, we consider randomness in both state dynamics and cost functionals, leading to a complex structure of fully coupled cross-coupled stochastic Riccati equations (CCREs). The key contributions lie in characterizing the equilibrium via state-feedback strategies derived by decoupling stochastic Hamiltonian systems governed by two symmetric CCREs-these random coefficients induce a higher-order nonlinear backward stochastic difference equation (BS$\triangle$E) system, fundamentally differing from deterministic counterparts. Under minimal regularity conditions, we establish necessary and sufficient conditions for closed-loop Nash equilibrium existence, contingent on the regular solvability of CCREs without requiring strong assumptions. Solutions are constructed using a dynamic programming principle (DPP), linking equilibrium strategies to coupled Lyapunov-type equations. Our analysis resolves critical challenges in modeling inherent randomness and provides a unified framework for dynamic decision-making under uncertainty.