Existence of Game Value and Approximating Nash Equilibrium for Path-dependent Stochastic Differential Game
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- 英文论文题名：Existence of Game Value and Approximating Nash Equilibrium for Path-dependent Stochastic Differential Game
- 论文作者：Fu Zhang
- 英文论文作者：Fu Zhang ; College of Science ; University of Shanghai for Science and Technology ; School of Mathematical Sciences Fudan University
- 年：2017
- 作者机构：College of Science,University of Shanghai for Science and Technology;School of Mathematical Sciences Fudan University;
- 英文论文关键词：stochastic differential game ; path-dependent ; game value ; Nash equilibrium
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（A）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（A）
- 语种：英文
- 分类号：O211.6
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：6
- 文件大小：273k
- 原文格式：D
- 会议级别：全国
摘要
In this paper we study a two-player zero-sum stochastic differential game for a path-dependent stochastic system.Two basic theoretic problems in SDG are addressed: existence of game value and Nash equilibrium. Due to the typical nonMarkovian structure, the game value is a random field. Dividing the time horizontal into small intervals, we approximate the path-dependent game by a series of state-dependent games. We utilize the state-dependent viscosity solution theory to prove that,under Isaacs' condition the game value exists. In our model, coefficients of diffusion of the system contain control and strategy,are path-dependent, and could be degenerate. The dimension of the state space is high. The existence of approximating Nash equilibrium is given under the formula about nonanticipative strategy with delay.
In this paper we study a two-player zero-sum stochastic differential game for a path-dependent stochastic system.Two basic theoretic problems in SDG are addressed: existence of game value and Nash equilibrium. Due to the typical nonMarkovian structure, the game value is a random field. Dividing the time horizontal into small intervals, we approximate the path-dependent game by a series of state-dependent games. We utilize the state-dependent viscosity solution theory to prove that,under Isaacs' condition the game value exists. In our model, coefficients of diffusion of the system contain control and strategy,are path-dependent, and could be degenerate. The dimension of the state space is high. The existence of approximating Nash equilibrium is given under the formula about nonanticipative strategy with delay.
In this paper we study a two-player zero-sum stochastic differential game for a path-dependent stochastic system.Two basic theoretic problems in SDG are addressed: existence of game value and Nash equilibrium. Due to the typical nonMarkovian structure, the game value is a random field. Dividing the time horizontal into small intervals, we approximate the path-dependent game by a series of state-dependent games. We utilize the state-dependent viscosity solution theory to prove that,under Isaacs' condition the game value exists. In our model, coefficients of diffusion of the system contain control and strategy,are path-dependent, and could be degenerate. The dimension of the state space is high. The existence of approximating Nash equilibrium is given under the formula about nonanticipative strategy with delay.
引文
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[3]I.Ekren,N.Touzi,and J.Zhang.Viscosity solutions of fully nonlinear parabolic path dependent PDEs:part II.Ann.Probab.,44(4):2507–2553,2016.
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[11]S.G.Peng and F.L.Wang.BSDE,path-dependent PDE and nonlinear Feynman-Kac formula.Sci.China Math.,59(1):19–36,2016.
[12]T.Pham and J.Zhang.Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation.SIAM J.Control Optim.,52(4):2090–2121,2014.
[13]S.Tang and F.Zhang.Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations.Discrete Contin.Dyn.Syst.,35(11):5521–5553,2015.
[2]R.Buckdahn and J.Li.Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations.SIAM J.Control Optim.,47(1):444–475,2008.
[3]I.Ekren,N.Touzi,and J.Zhang.Viscosity solutions of fully nonlinear parabolic path dependent PDEs:part II.Ann.Probab.,44(4):2507–2553,2016.
[4]R.J.Elliott.The existence of value in stochastic differential games.SIAM J.Control Optimization,14(1):85–94,1976.
[5]R.J.Elliott and M.H.A.Davis.Optimal play in a stochastic differential game.SIAM J.Control Optim.,19(4):543–554,1981.
[6]W.H.Fleming and P.E.Souganidis.On the existence of value functions of two-player,zero-sum stochastic differential games.Indiana Univ.Math.J.,38(2):293–314,1989.
[7]A.Friedman.Stochastic differential games.J.Differential Equations,11:79–108,1972.
[8]A.Friedman.Differential games.Number 18 in Regional Conference Series in Mathematical.American Mathematical Sciences,Providence,Rhode Island,1974.
[9]S.Hamad`ene and J.P.Lepeltier.Backward equations,stochastic control and zero-sum stochastic differential games.Stochastics Stochastics Rep.,54(3-4):221–231,1995.
[10]S.Hamad`ene,J.-P.Lepeltier,and S.Peng.BSDEs with continuous coefficients and stochastic differential games.In Backward stochastic differential equations(Paris,1995–1996),volume 364 of Pitman Res.Notes Math.Ser.,pages115–128.Longman,Harlow,1997.
[11]S.G.Peng and F.L.Wang.BSDE,path-dependent PDE and nonlinear Feynman-Kac formula.Sci.China Math.,59(1):19–36,2016.
[12]T.Pham and J.Zhang.Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation.SIAM J.Control Optim.,52(4):2090–2121,2014.
[13]S.Tang and F.Zhang.Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations.Discrete Contin.Dyn.Syst.,35(11):5521–5553,2015.