Discrete-Time LQ Stochastic Two Person Nonzero Sum Difference Games With Random Coefficients:~Closed-Loop Nash Equilibrium
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909698918187008 |
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| author | Meng, Qingxin Wu, Yiwei |
| author_facet | Meng, Qingxin Wu, Yiwei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete-Time LQ Stochastic Two Person Nonzero Sum Difference Games With Random Coefficients:~Closed-Loop Nash Equilibrium Meng, Qingxin Wu, Yiwei Optimization and Control 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. |
| title | Discrete-Time LQ Stochastic Two Person Nonzero Sum Difference Games With Random Coefficients:~Closed-Loop Nash Equilibrium |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2507.16412 |