P-th Almost Periodic Type Mild Solutions To A Class Of Neutral Stochastic Differential Equations

Stochastic differential equations are important from the viewpoint of applications since they incorporate natural randomness into the mathematical description of the phenomena.In particular,the qualitative properties of p-th almost periodic type solutions of various stochastic differential equations(such as the existence and uniqueness of solutions,the global exponential stability and the asymptotically stability of the solutions)become a hot topic in research.In this paper,not only is the existence of square mean asymptotically almost periodic mild solutions to a class of neutral stochastic differential equations studied,but also the uniqueness of the solutions is discussed.In addition,the existence of p-th pseudo almost periodic mild solution to this kind of equations is studied,and the uniqueness of the solutions is also studied.The main contents of this paper are as follows:Firstly,it is aimed at the research that the existence and uniqueness of square mean asymptotically almost periodic solutions to a class of neutral stochastic differential equations.Specifically,the knowledge of operator theory and the concept and properties of square mean asymptotically almost periodic stochastic processes are expounded.And mainly utilizing equivalent definitions of square mean asymptotically almost periodic stochastic process,the composition of square mean asymptotically stochastic process is studied.Then,by using Cauchy-Schwarz inequality,the theory of anomalous integral and Banach fixed point theorem,it is discussed that there is a unique square mean asymptotically almost periodic mild solutions to this kind of equations under what assumptions.Secondly,based on the first research content,the sufficient conditions that ensure the existence and uniqueness of p-th pseudo almost periodic mild solutions to this kind of neutral stochastic differential equations are investigated.First of all,the theoretical knowledge of p-th pseudo almost periodic stochastic processes is introduced.Then four assumptions are listed.Finally,using the Holder(5)(5)inequality,Lebesgue control convergence theorem,?Ito isometry identity and Banach fixed point theorem,the existence and uniqueness of p-th pseudo almost periodic mildsolution to this class of equations are discussed.