基于g-期望的部分可观测非零和随机微分博弈（英文）
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摘要
本文研究了g-期望下的部分可观测非零和随机微分博弈系统,该系统的状态方程由It?-Lévy过程驱动,成本函数由g-期望描述.根据Girsanov定理和凸变分技巧,本文得到了最大值原理和验证定理.为对所获结果进行说明,本文讨论了关于资产负债管理的博弈问题.
This paper is concerned with a partially observed nonzero-sum stochastic differential game system under g-expectation, where the state is governed by a It?-Lévy process and the cost functionals are described by g-expectations.Based on Girsanov's theorem and convex variation techniques, we derive a maximum principle and a verification theorem.An asset-liability management game problem is discussed to illustrate the results.
This paper is concerned with a partially observed nonzero-sum stochastic differential game system under g-expectation, where the state is governed by a It?-Lévy process and the cost functionals are described by g-expectations.Based on Girsanov's theorem and convex variation techniques, we derive a maximum principle and a verification theorem.An asset-liability management game problem is discussed to illustrate the results.
引文
[1]HO Y.Optimal terminal maneuver and evasion strategy.SIAM Journal on Control and Optimization,1966,4(3):421-428.
[2]AN T,?KSENDAL B.Maximum principle for stochastic differential games with partial information.Journal of Optimization Theory and Applications,2008,139(3):463-483.
[3]WANG G,YU Z.A Pontryagin’s maximum principle for non-zero sum differential games of BSDEs with applications.IEEE Transactions on Automatic Control,2010,55(7):1742-1747.
[4]ZHU H,ZHANG C.Infinite time horizon nonzero-sum linear quadratic stochastic differential games with state and controldependent noise.Control Theory and Technology,2013,11(4):629-633.
[5]WU J,LIU Z.Maximum principle for mean-field zero-sum stochastic differential game with partial information and its application to finance.European Journal of Control,2017,37(5):8-15.
[6]PENG S.Backward SDE and related g-expectations//El Karoui Mazliak(Eds.).Backward Stochastic Differential Equations,Pitman Research Notes in Mathematics Series,1997,364:141-159.
[7]?KSENDAL B,SULEM A.Risk minimization in financial markets modeled by It?o-L′evy processes.Afrika Matematika,2015,26:939-979.
[8]PENG S.Nonlinear Expectations,Nonlinear Evaluations and Risk Measures.Berlin Heidelberg:Springer Verlag,2004,1856:165-253.
[9]?KSENDAL B,SULEM A.Forward-backward stochastic differential games and stochastic control under model uncertainty.Journal of Optimization Theory and Applications,2014,161(1):22-55.
[10]YU Z.Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system.Asian Journal of Control,2012,14(1):173-185.
[11]HUI E,XIAO H.Maximum principle for differential games of forward-backward stochastic systems with applications.Journal of Mathematical Analysis and Applications,2012,386(1):412-427.
[12]AN T,?KSENDAL B.A maximum principle for stochastic differential games with g-expectations and partial information.Stochastics,2012,84(2/3):137-155.
[13]WANG G,WU Z,XIONG J.Maximum principles for forwardbackward stochastic control systems with correlated state and observation noises.SIAM Journal on Control and Optimization,2013,51(1):491-524.
[14]WU Z.A maximum principle for partially observed optimal control of forward-backward stochastic control systems.Science China,2010,53(11):2205-2214.
[15]XIAO H.The maximum principle for partially observed optimal control of forward-backward stochastic systems with random jumps.Journal of Systems Science&Complexity,2011,24:1083-1099.
[16]XIONG J,ZHANG S,ZHUANG Y.A partially observed nonzero sum differential game of forward-backward stochastic differential equations and its application in finance.arXiv:1601.00538v1[math.OC],2016.
[17]TANG S,LI X.Necessary conditions for optimal control of stochastic systems with random jumps.SIAM Journal on Control and Optimization,1994,32(5):1447-1475.
[18]CHEN Z,EPSTEIN L.Ambiguity,risk,and asset returns in continuous time.Econometrica,2002,70(4):1403-1443.
[19]WU Z.Forward-backward stochastic differential equations with Brownian motion and Poisson process.Acta Mathematicae Applicatae Sinica,1999,15(4):433-443.
[20]?KSENDAL B.Stochastic Differential Equations.Sixth Edition.New York:Springer,2003.
[21]HUANG J,WANG G,WU Z.Optimal premium policy of an insurance firm:full and partial information.Insurance:Mathematics and Economics,2010,47:208-215.
[22]XIONG J,ZHOU X.Mean-variance portfolio selection under partial information.SIAM Journal on Control and Optimization,2007,46(1):156-175.
[2]AN T,?KSENDAL B.Maximum principle for stochastic differential games with partial information.Journal of Optimization Theory and Applications,2008,139(3):463-483.
[3]WANG G,YU Z.A Pontryagin’s maximum principle for non-zero sum differential games of BSDEs with applications.IEEE Transactions on Automatic Control,2010,55(7):1742-1747.
[4]ZHU H,ZHANG C.Infinite time horizon nonzero-sum linear quadratic stochastic differential games with state and controldependent noise.Control Theory and Technology,2013,11(4):629-633.
[5]WU J,LIU Z.Maximum principle for mean-field zero-sum stochastic differential game with partial information and its application to finance.European Journal of Control,2017,37(5):8-15.
[6]PENG S.Backward SDE and related g-expectations//El Karoui Mazliak(Eds.).Backward Stochastic Differential Equations,Pitman Research Notes in Mathematics Series,1997,364:141-159.
[7]?KSENDAL B,SULEM A.Risk minimization in financial markets modeled by It?o-L′evy processes.Afrika Matematika,2015,26:939-979.
[8]PENG S.Nonlinear Expectations,Nonlinear Evaluations and Risk Measures.Berlin Heidelberg:Springer Verlag,2004,1856:165-253.
[9]?KSENDAL B,SULEM A.Forward-backward stochastic differential games and stochastic control under model uncertainty.Journal of Optimization Theory and Applications,2014,161(1):22-55.
[10]YU Z.Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system.Asian Journal of Control,2012,14(1):173-185.
[11]HUI E,XIAO H.Maximum principle for differential games of forward-backward stochastic systems with applications.Journal of Mathematical Analysis and Applications,2012,386(1):412-427.
[12]AN T,?KSENDAL B.A maximum principle for stochastic differential games with g-expectations and partial information.Stochastics,2012,84(2/3):137-155.
[13]WANG G,WU Z,XIONG J.Maximum principles for forwardbackward stochastic control systems with correlated state and observation noises.SIAM Journal on Control and Optimization,2013,51(1):491-524.
[14]WU Z.A maximum principle for partially observed optimal control of forward-backward stochastic control systems.Science China,2010,53(11):2205-2214.
[15]XIAO H.The maximum principle for partially observed optimal control of forward-backward stochastic systems with random jumps.Journal of Systems Science&Complexity,2011,24:1083-1099.
[16]XIONG J,ZHANG S,ZHUANG Y.A partially observed nonzero sum differential game of forward-backward stochastic differential equations and its application in finance.arXiv:1601.00538v1[math.OC],2016.
[17]TANG S,LI X.Necessary conditions for optimal control of stochastic systems with random jumps.SIAM Journal on Control and Optimization,1994,32(5):1447-1475.
[18]CHEN Z,EPSTEIN L.Ambiguity,risk,and asset returns in continuous time.Econometrica,2002,70(4):1403-1443.
[19]WU Z.Forward-backward stochastic differential equations with Brownian motion and Poisson process.Acta Mathematicae Applicatae Sinica,1999,15(4):433-443.
[20]?KSENDAL B.Stochastic Differential Equations.Sixth Edition.New York:Springer,2003.
[21]HUANG J,WANG G,WU Z.Optimal premium policy of an insurance firm:full and partial information.Insurance:Mathematics and Economics,2010,47:208-215.
[22]XIONG J,ZHOU X.Mean-variance portfolio selection under partial information.SIAM Journal on Control and Optimization,2007,46(1):156-175.