Research On Central Limit Theorem Under Sublinear Expectations And Two-Person Zero-sum Stochastic Differential Game

In recent decades,the financial market in the world has been in frequent turbu-lence.People found that they can get both fortune and serious risks from financial markets.Meanwhile,they have been aware of the importance of financial risk mnan-agement.Facts show that the previous methods of assessing and managing financial risks are not competent.So,how to evaluate effectively,prevent and resolve the fi-nancial system risk has become a common concern.A large number of research shows that most of risks in financial are nonlinear,but the theory of measuring and man-aging nonlinear financial risks is still in exploration and discussion,it is meaningful and urgent to research financial risks.In 1973,the French mathematician Bismut[2]introduced the linear backward stochastic differential equation for the first time when studying stochastic optimal control.In 1990,Pardox and Peng[39]introduced the general nonlinear backward s-tochastic differential equation(BSDE).Delbaen(1998)[13],Artzner et al.[1]proposed the consistent risk measurement theory in 1999.In 1997 Peng introduced a nonlinear expectation g-expectation based on backward stochastic differential equation.We found that g-expectation could construct a consistent risk measure([52]).Howev-er,g-expectation is still a quasilinear mathematical expectation,which cannot cover completely nonlinear situations.For example,g-expectation is powerless for volatility uncertainty,but volatility uncertainty is common in many fields.In order to study the uncertainty of volatility in financial market,Peng put forward the sublinear expecta-tion theory in 2006,and put forward G-normal distribution,maximum distribution,etc.The law of large number and the central limit theorem under sublinear expec-tation are proposed.Peng proved the law of large numbers and the central limit theorem under the sublinear expectation by using partial differential equation.Since Peng founded the central limit theorem under sublinear expectation,some problems have not been solved.Peng once raised an open question:can we prove the central limit theorem under sublinear expectation by method of probability?Can the(2+?)-moment be reduced to Lindeberg condition similar to the classical case?Inspired by the results achieved in the deterministic case,Fleming and Sougani-dis studied two-person zero-sum stochastic differential game in 1989[21],they proved that the lower value function and the upper value function of the game satisfy the dynamic programming principle under appropriate conditions,and the lower value function and the upper value function are the only viscous solution of the corre-sponding Bellman-Isaacs equation.After the establishment of Backward Stochastic Differential equation,the theory of stochastic control was developed(Peng[40]),Hamadene,Lepeltier[25],Hamadene,Lepeltier and Peng[29]were respectively ap-plied to stochastic differential game.In 2008,Buckdahn and Li studied two-person zero-sum stochastic differential game with backward stochastic differential equation in[5].They defined the cost function by the solution of stochastic differential equa-tion with control.In 2011,Buckdahn and Li continued to study two-person zero-sum stochastic differential game in[6].The cost function discussed is a.double-reflection backward stochastic differential equation.In 2012 Biswas[3]studied the case where the state is jump diffusion driven.In 2015 Yu[61]studied two-person zero-sum linear quadratic stochastic differential games.Inspired by the above work,In view of Peng's open problem of central limit theorem under sublinear expectation,the following works are done:(1)The thesis proves the central limit theorem under sublinear expectation by method of probability;(2)(2+?)-moment is reduced to(2.2.3).Meanwhile,The thesis studies the state equation is a two-person zero-sum stochastic differ-ential game problem of reflection stochastic differential equation.We prove that the lower value function satisfying the principle of dynamic programming,and the lower value function and the upper value function are viscous solutions of cor-responding Isaacs equation.The following is the structure and main content of this paper:(?)In Chapter 1,We briefly introduces the background of the problem to be discussed and the basic concepts of backward stochastic differential equation and sublinear expectation space.(?)In Chapter 2,We introduce the research background of the central limit theorem under sublinear expectation.It includes the main results of the central limit theorem under sublinear expectation and the methods of proving the theorem,as well as some lemmas used in our proof.Suppose([?,F,P)is a probability space.We construct two sets of sequence of auxiliary function Hm,n,Lm,n and Fm,n Km,n,in this probability space,and it is proved that the sequence of auxiliary function have good properties.After established the theory of sublinear expectation space,Peng proved the central limit theorem under sublinear expectation by using a partial differential e-quation(PDE)method,his work has been regarded as a milestone in the field of central limit theorem under sublinear expectation and motivated many extensions along this line.Many works have been done to prove central limit theorem under sublinear expectation by using a PDEs method.Our goal in this paper is to prove the central limit theorem using a different method.Theorem 0.2.1 Assume that{Xi}i=1n is a sequence of independent random variables under sublinear expectation E with zero means E[Xi]=?[Xi]?0 and finite common variances(?)and(?)such that Condition(2.2.3)holds.Then,for any function ? ?C[-?,+?],where {Bt} is a standard Brownian motion under probability measure P.Instead of using partial differential equations,we used a basic probabilistic tech-nique to prove the central limit theorem for sublinear expectations.We weaken the assumption of(2+?)-moment in the central limit theorem of Peng([43]),and be-cause we do not use the method of partial differential equation,this also makes the assumption of(?)?0 in Peng's proof no longer necessary.(?)In Chapter 3,We mainly studies a kind of special zero-sum two-person stochastic differential game.For given admissible control u(·)?u and v(·)E V,the corresponding state process starting from ??L2(Ft:G)at the initial time t is governed by the following reflected SDEGiven the control process u(·)?ut,T and v(·)?Vt,T,we introduce the following cost functional:where,Yt,x;u,v is the solution of the following generalized BSDE:For any u(·)?u and(·)? V and ? ?L2(Ft;G),there exists a unique solution to the following generalized BSDE:where(Xst,?;u,v,Kst,?;u,v)is the unique solution of reflected SDE(3.3.1)Definition 0.3.1 the lower value function of the stochastic differential game the upper value function of the stochastic differential game The lower value function W(t,x)has the following property:Theorem 0.3.2(Deterministic Property)For any(t,x)?[0,T]× G,we have W(t,x)?E[W(t,x)],a.s.By identifying W(t,x)with its deterministic version E[W(t,x)],we can consider W:[0,T]×G?R as a deterministic function.Theorem 0.3.3(Continuity on x)There exists a constant C?0 such that,for all andTheorem 0.3.4(DPP)Suppose(H1)-(H4)hold,the lower value function W(t,x)satisfied the following DPP:for any 0?t?t+?,x ?G,Theorem 0.3.5 Suppose(H1)-(H4)hold,the lower value function W(t.x)is contin-uous in t.We prove that the lower value function and the upper value function of the stochastic differential game problem are the viscous solutions of the corresponding Isaacs equat,ion with nonlinear Neumann boundary condition.Theorem 0.3.6 Suppose(H1)-(H4)hold,the function W(t,x)defined in(3.3.12)is a viscosity solution of(3.4.1).In a similar way,it can be proved that the upper value function U(t,x)is the viscous solution of the equation(3.4.2).