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Two person non-zero-sum linear-quadratic differential game with Markovian jumps in infinite horizon (2405.10083v1)
Published 16 May 2024 in math.OC
Abstract: 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 $L2$ stabilizability framework, we first provide a sufficient condition to ensure the $L2$-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
- M. Ait Rami and X. Y. Zhou, Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls, IEEE Transactions on Automatic Control, 45.
- M. Ait Rami, X. Y. Zhou and J. Moore, Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon, Systems & Control Letters, 41 (2000), 123–133.
- B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, Inc., 1990.
- Y. Bai, A hybrid stochastic differential reinsurance and investment game with bounded memory, European Journal of Operational Research, 21.
- T. Başar and G. J. Olsder, Dynamic noncooperative game theory, Mathematics in science and engineering: v. 160, Academic Press, 1982.
- S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM, Philadelphia, PA, 1994.
- L. Chen and Y. Shen, Stochastic Stackelberg differential reinsurance games under time-inconsistent mean–variance framework, Insurance: Mathematics and Economics, 88 (2019), 120–137.
- R. J. Elliott and M. H. A. Davis, Optimal play in a stochastic differential game, SIAM Journal on Control and Optimization, 19 (1981), 543–554.
- W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana University Mathematics Journal, 38 (1989), 293 – 314.
- J. Huang, X. Li and J. Yong, A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon, Mathematical Control and Related Fields, 5 (2015), 97–139.
- Y. Ji and H. J. Chizeck, Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control, 35 (1990), 777–788.
- I. Karatzas and I.-M. Zamfirescu, Martingale approach to stochastic differential games of control and stopping, The Annals of Probability, 36.
- X. Li and X. Y. Zhou, Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon, Communications in Information and Systems, 2 (2002), 265–282.
- X. Li, X. Y. Zhou and M. Ait Rami, Indefinite stochastic linear quadratic control with Markovian jumps in infinite time horizon, Journal of Global Optimization, 27 (2003), 149–175.
- Q. Song, G. G. Yin and Z. Zhang, Numerical solutions for stochastic differential games with regime switching, IEEE Transactions on Automatic Control, 53 (2008), 509–521.
- J. Sun, X. Li and J. Yong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems, SIAM Journal on Control and Optimization, 54 (2016), 2274–2308.
- J. Sun, J. Yong and S. Zhang, Linear quadratic stochastic two-person zero-sum differential games in an infinite horizon, ESAIM: Control, Optimisation and Calculus of Variations, 22 (2016), 743–769.
- Z. Sun, I. Kemajou-Brown and O. Menoukeu-Pamen, A risk-sensitive maximum principle for a Markov regime-switching jump-diffusion system and applications, ESAIM: Control, Optimisation and Calculus of Variations, 24 (2018), 985–1013.
- S. Tang and S. Hou, Switching games of stochastic differential systems, SIAM Journal on Control and Optimization, 46 (2007), 900–929.
- J. Wen, X. Li, J. Xiong and X. Zhang, Stochastic linear-quadratic optimal control problems with random coefficients and Markovian regime switching system, SIAM Journal on Control and Optimization, 61 (2023), 949–979.
- D. W. K. Yeung, L. A. Petrosjan and L. A. Petrosyan, Cooperative stochastic differential games, Springer series in operations research and financial engineering, Springer, 2006.
- X. Zhang, X. Li and J. Xiong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems of Markovian regime switching system, ESAIM: Control, Optimisation and Calculus of Variations, 27 (2021), 69.
- X. Zhang, T. K. Siu and Q. Meng, Portfolio selection in the enlarged markovian regime-switching market, SIAM Journal on Control and Optimization, 48 (2010), 3368–3388.
- H. Zhu, C. Zhang and N. Bin, Infinite horizon linear quadratic stochastic Nash differential games of Markov jump linear systems with its application, International Journal of Systems Science, 45 (2014), 1196–1201.