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

Yiwei Wu; Xun Li; Qingxin Meng

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arXiv · 2410.01741

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

Yiwei Wu·Xun Li·Qingxin Meng

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Abstract

This paper presents a pioneering investigation into discrete-time two-person non-zero-sum linear quadratic (LQ) stochastic games with random coefficients. We derive necessary and sufficient conditions for the existence of open-loop Nash equilibria using convex variational calculus. To obtain explicit expressions for the Nash equilibria, we introduce fully coupled forward-backward stochastic difference equations (FBS$\Delta$E, for short), which provide a dual characterization of these Nash equilibria. Additionally, we develop non-symmetric stochastic Riccati equations that decouple the stochastic Hamiltonian system for each player, enabling the derivation of closed-loop feedback forms for open-loop Nash equilibrium strategies. A notable aspect of this research is the complete randomness of the coefficients, which results in the corresponding Riccati equations becoming fully nonlinear higher-order backward stochastic difference equations. It distinguishes our non-zero-sum difference game from the deterministic case, where the Riccati equations reduce to algebraic forms.

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BibTeX RIS

Yiwei Wu, Xun Li, Qingxin Meng. 2024-10-02. Discrete-Time LQ Stochastic Two-Person Nonzero-Sum Difference Games with Random Coefficients:~Open-Loop Nash Equilibrium. https://arxiv.org/abs/2410.01741

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