Existence Of Value Functions For A Class Of Differential Game

In this paper,two kinds problems of two-person zero-sum differential game with asymmetric information and related information are studied in the absence of Isaacs condition: the equivalent form of the upper conjugate of the upper-valued function and the existence of the value function.First,we discuss the equivalent form of the upper conjugate of the upper-valued function of a two-person zero-sum differential game with asymmetric information and related information.Because we are studying the problem of two-person zero-sum differential game in the case that Isaacs conditions are not provided.To this end,we first define the value function of the game by defining a special random strategy-a non-anticipative random strategy with delay.Then,according to the definition of the upper conjugate,the upper conjugate function of the upper-valued function is obtained,and the equivalent form of the upper conjugate function is established.Compared with the related results in the existing literature,our proof method is simpler and the process is clearer.Secondly,the principle of sub-dynamic programming is discussed according to the above equivalent form.Finally,it is proved that under the given random strategy,when the mesh of the time partition tends to zero,both the upper-valued function and the lowervalued function converge to a limit function which is the value of the game and we give a description of the mathematical characteristics of the value.Our results extend the type of information in [21] from independent information to the situation of related information,making it more practical and more widely used.The full text is divided into five chapters:The first chapter is the introduction,which summarizes the historical background of game theory and the research status of differential game,and introduces the main research contents of this paper.The second chapter is the preparatory knowledge.This chapter introduces the basic knowledge of game theory,the description of the differential game model,some concepts and lemmas which are necessary to be used in the following chapters.In first section of the third chapter,the model of two-person zero-sum differential game,the definition of non-anticipative random strategy with delay and value functionsof this game are supplied.The second section gives the decomposition of probability measure μ,and the definition of the upper conjugate.Finally the equivalent form of the upper conjugate of the upper-valued function of the two-person zero-sum differential game with asymmetric information and related information is verified.In first section of the fourth chapter,the regularity of the value function is given,and the principle of sub-dynamic programming is proved by using the main results proved in the previous chapter.The second section proves the existence of the value function of such differential game and gives the description of the value function.The fifth chapter is a summarizing for the full text.Moreover,we look forward to the direction for further research.