The Basic Theory Of Stochastic Functional Differential Equations With Infinite Delay

This paper mainly discusses the basic theory of stochastic functional differential equations with infinite delay in several different phase spaces, such as, boundary continuous function space BC((—∞, 0];R~d), space C_h, space C_g and space B. The existence-and-uniqueness theorem of solution for stochastic functional differential equations are obtained, respectively, then the estimate of error between approximate solution and accurate solution are given. It shows that the approximate solution is a powerful tool to compute the accurate solution.It is divided into five chapters.As the beginning of this paper, Chapter 1 offers the research background for this paper, some refereed notations, necessary definitions and lemmas, and some known results made by former authors for stochastic differential equations and stochastic functional differential equations, respectively.The basic theory of stochastic functional differential equations is discussed in phase space BC((—∞, 0];R~d) in Chapter 2. Under the uniform Lipschitz condition, we weaken the linear growth condition to obtain the moment estimate of solution for stochastic functional differential equations. Further, we derive the existence-and-uniqueness theorem of solution for stochastic functional differential equations, and give the estimate for the error between approximate solution and accurate solution. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence-and-uniqueness theorem is valid for stochastic functional differential equations on finite interval. Finally, the existence-and-uniqueness theorem holds on the entire interval for stochastic functional differential equations.Chapter 3 deals with the basic theory of stochastic functional differential equations in phase space C_h. We weaken the linear growth condition, together with the uniform Lipschitz condition, to find the moment estimate of solution for stochastic functional differential equations. Moreover, the existence-and-uniqueness theorem and the estimate for the error between approximate solution and accurate one are obtained for stochastic functional differential equations .Additionally,we assume that the linear growth condition and the local Lipschitz condition hold, then the existence-and-uniqueness is also derived on finite interval for stochastic functional differential equations. Finally, the existence-and-uniqueness theorem holds for stochastic functional differential equations on the entire interval.The author investigates the basic theory of stochastic functional differential equations in phase space Cg in Chapter 4. We make use of weakening the linear growth condition and the uniform Lipschitz condition, to obtain the moment estimate of solution for stochastic functional differential equations. Furthermore, the existence-and-uniqueness theorem and the estimate for the error between approximate solution and accurate one are deduced for stochastic functional differential equations. In addition, if the linear growth condition and the local Lipschitz condition hold, then the existence-and-uniqueness theorem can be found on finite interval for stochastic functional differential equations. Finally, we give the existence-and-uniqueness theorem on the entire interval.Chapter 5 considers the basic theory of stochastic functional differential equations in general phase space B. We impose the uniform Lipschitz condition together with the weaken linear growth condition, to derive the moment estimate of solution for stochastic functional differential equations. Consequently, for stochastic functional differential equations, the existence-and-uniqueness theorem can be shown and the estimate for the error between approximate solution and accurate solution is given. On the other hand, if the linear growth condition and the local Lipschitz condition are fulfilled, then for stochastic functional differential equations, the existence-and-uniqueness theorem can be built on finite interval. Finally, for stochastic functional differential equations, the existence-and-uniqueness theorem is discussed on the entire interval.At the end of this paper, the author summarizes the innovations of my paper and proposes the direction of future work. Finally, related literatures are listed.