Stochastic linear-quadratic differential game with Markovian jumps in an infinite horizon
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909294070333440 |
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| author | Wu, Fan Li, Xun Xiong, Jie Zhang, Xin |
| author_facet | Wu, Fan Li, Xun Xiong, Jie Zhang, Xin |
| contents | This paper investigates a two-person non-homogeneous linear-quadratic stochastic differential game (LQ-SDG, for short) in an infinite horizon for a system regulated by a time-invariant Markov chain. Both non-zero-sum and zero-sum LQ-SDG problems are studied. It is shown that the zero-sum LQ-SDG problem can be considered a special non-zero-sum LQ-SDG problem. The open-loop Nash equilibrium point of non-zero-sum (zero-sum, respectively) LQ-SDG problem is characterized by the solvability of a system of constrained forward-backward stochastic differential equations (FBSDEs, for short) in an infinite horizon and the convexity (convexity-concavity, respectively) of the performance functional and their corresponding closed-loop Nash equilibrium strategy are characterized by the solvability of a system of constrained coupled algebra Riccati equations (CAREs, for short) with certain stabilizing conditions. In addition, the closed-loop representation of open-loop Nash equilibrium point for non-zero-sum (zero-sum, respectively) LQ-SDG is provided by the non-symmetric (symmetric, respectively) solution to a system CAREs. At the end of this paper, we provide three concrete examples and solve their open-loop/closed-loop Nash equilibrium strategy based on the obtained results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_12818 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic linear-quadratic differential game with Markovian jumps in an infinite horizon Wu, Fan Li, Xun Xiong, Jie Zhang, Xin Optimization and Control 93E03, 93E20 This paper investigates a two-person non-homogeneous linear-quadratic stochastic differential game (LQ-SDG, for short) in an infinite horizon for a system regulated by a time-invariant Markov chain. Both non-zero-sum and zero-sum LQ-SDG problems are studied. It is shown that the zero-sum LQ-SDG problem can be considered a special non-zero-sum LQ-SDG problem. The open-loop Nash equilibrium point of non-zero-sum (zero-sum, respectively) LQ-SDG problem is characterized by the solvability of a system of constrained forward-backward stochastic differential equations (FBSDEs, for short) in an infinite horizon and the convexity (convexity-concavity, respectively) of the performance functional and their corresponding closed-loop Nash equilibrium strategy are characterized by the solvability of a system of constrained coupled algebra Riccati equations (CAREs, for short) with certain stabilizing conditions. In addition, the closed-loop representation of open-loop Nash equilibrium point for non-zero-sum (zero-sum, respectively) LQ-SDG is provided by the non-symmetric (symmetric, respectively) solution to a system CAREs. At the end of this paper, we provide three concrete examples and solve their open-loop/closed-loop Nash equilibrium strategy based on the obtained results. |
| title | Stochastic linear-quadratic differential game with Markovian jumps in an infinite horizon |
| topic | Optimization and Control 93E03, 93E20 |
| url | https://arxiv.org/abs/2408.12818 |