Discrete-Time LQ Stochastic Two-Person Nonzero-Sum Difference Games with Random Coefficients:~Open-Loop Nash Equilibrium
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| Format: | Preprint |
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2024
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| _version_ | 1866909654944055296 |
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| author | Wu, Yiwei Li, Xun Meng, Qingxin |
| author_facet | Wu, Yiwei Li, Xun Meng, Qingxin |
| contents | 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$Δ$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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01741 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete-Time LQ Stochastic Two-Person Nonzero-Sum Difference Games with Random Coefficients:~Open-Loop Nash Equilibrium Wu, Yiwei Li, Xun Meng, Qingxin Optimization and Control 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$Δ$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. |
| title | Discrete-Time LQ Stochastic Two-Person Nonzero-Sum Difference Games with Random Coefficients:~Open-Loop Nash Equilibrium |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2410.01741 |