Set-valued Approximation

With the wide application of stochastic differential equations in various fields,we have a deeper understanding of random phenomena in nature.However,the development process of objective things is not only random,but also inaccurate.When we try to describe the system more realistically,the traditional stochastic differential equations cannot meet our needs.Set-valued stochastic variables can take both randomness and variability into account.So it is of great significance to discuss set-valued stochastic differential equations.In this dissertation,we focuse on set-valued stochastic differential equations and set-valued functional stochastic differential equations.The equations discussed in chapter 3 are set-valued stochastic Lebesgue integrals with almost everywhere new definition and driven by a square integrable martingale.The equations discussed in chapter 4 are set-valued stochastic Lebesgue integrals and driven by a set-valued square integrable martingale.These are different from referenced papers.The idea of set-valued approximation is used to prove the existence and uniqueness theorem of the equations solution under the linear growth condition,Lipschitz continuous condition and set-valued integral inequality.There are four parts in this dissertation.In the first part,we shall discuss the background of the set-valued stochastic differential equations and the stochastic integral with respect to a set-valued square integrable martingale,and we shall discuss the relevant literature of the set-valued approximation and set-valued stochastic differential equations.In the second part,we shall describe the set-valued integral.Firstly,the background of set-valued research and the basic knowledge of the set-valued random variables and the set-valued stochastic process are briefly introduced.Then,the definitions and propertise of the set-valued Lebesgue integral,the stochastic integral with respect to a square integrable martingale,and the stochastic integral with respect to a set-valued square integrable martingale are given.In the part ?,we shall discuss a special type of set-valued stochastic differential equations.Firstly,when the initial value is given,the existence and uniqueness theorem of the equation solution is proposed.Secondly,we prove that the solution is continuous in a certain sense.Thirdly,the existence of solution is proved by mathematical induction and iterative approximation.Finally,the use of the same proof method to prove that the solution of this equation is unique.In the part ?,we discuss the set-valued functional stochastic differential equations and the set-valued stochastic differential delay equations.Firstly,the effect of time delay on stochastic system is analyzed,and the necessities for studying functional and time delay stochastic differential equations are given.Then,when the initial value is given,we shall prove the existence and uniqueness of solution of the set-valued functional stochastic differential equations driven by a set-valued square integrable martingale.Finally,the existence and uniqueness theorem of solution of the set-valued stochastic differential delay equations is derived through its connection with the set-valued functional stochastic differential equations.