Two person non-zero-sum linear-quadratic differential game with Markovian jumps in infinite horizon
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916249339953152 |
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| author | Wu, Fan Li, Xun Zhang, Xin |
| author_facet | Wu, Fan Li, Xun Zhang, Xin |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10083 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two person non-zero-sum linear-quadratic differential game with Markovian jumps in infinite horizon Wu, Fan Li, Xun Zhang, Xin Optimization and Control 93E03, 93E15 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 |
| title | Two person non-zero-sum linear-quadratic differential game with Markovian jumps in infinite horizon |
| topic | Optimization and Control 93E03, 93E15 |
| url | https://arxiv.org/abs/2405.10083 |