Risk-sensitive Zero-sum Stochastic Differential Games And Related Backward Stochastic Differential Equations

Pardoux and Peng first proposed the nonlinear backward stochastic differential equation(BSDE)in 1990.They proved the existence and uniqueness of the adaptive solution of this type of equation.Since then,BSDE theory has been developed rapidly and has been widely used in many fields,such as mathematical finance,stochastic games and optimal control,partial differential equations,etc.With the development of BSDE theory,many new forms have been proposed.In 2000,Kobylanski[21]first introduced the BSDE of the generator with respect to the growth of z-squared,which explained the existence of this type of BSDE,and gave the comparison theorem.In 2010,Delong and Imkeller[5]introduced a backward stochastic differential equation with time-delayed generator.Shi and Wang[37]used a special form of delayed BSDE to study the problem of nonzerosum stochastic differential games.On this basis,Lu[23]proved the comparison theorem of delayed BSDE without the generator of Zt-? and studied its applica.tion in zero-sum stochastic differential games.El-Karoui and Hamadene[9]uses the BSDE theory with quadratic growth to find the optimal control and saddle point strategies when the risk sensitivity coefficient is consta.nt.In this paper,we mainly study two different forms of risk-sensitive zero-sum stochastic differential game problems.We give the related backward stochastic differential equation and prove the existence and uniqueness of its solution,and then use the BSDE theory to get the saddle point of the game problem.The first type we study is the zero-sum stochastic differential game problem with timedependent risk sensitivity coefficients.We consider the risk sensitivity coefficient as a time function F(t),and use the BSDE theory with quadratic growth to obtain the saddle point of the game problem.This situation is an extension of the situation discussed by El-Karoui and Hamadene.The second type we study is the risksensitive zero-sum stochastic differential game problem in time-delayed systems.Its cost function depends not only on(xt)t?[0,T],but also on(xt)t?[-?,0].For this kind of game problem,we have obtained its saddle point by studying the delayed BSDE theory with quadratic growth.