Square-mean Asymptotically Almost Periodic Solutions Fortwo Classes Of Stochastic Differential Equations

Stochastic differential equations are established in solving some stochasticphenomena. Many properties of stochastic differential equations have been usedwidely. The existence and uniqueness of square-mean almost periodic type solutionsof the equation which base on almost periodic type function theory and stochasticprocess theory have more research significance. In this paper, the existence anduniqueness of square-mean asymptotically almost periodic solutions for two classes ofstochastic differential equations are discussed.The contents of this paper are as follow:In the first section, some research and the background of almost periodic typefunctions theory and stochastic process theory are introduced at present. Thedevelopment tendency of these equations is also introduced.The second section, the existence and uniqueness of square-mean asymptoticallyalmost periodic solutions for a class of nonlinear stochastic differential equation isdiscussed. At first some theoretical knowledges about almost periodic process andsome theories ofC0group are introduced, uniformly exponentially stable andstrongly continuous family of operators. Then the asymptotically almost periodicproperties of the jointly continuous functions are discussed. Finally, the existence andthe uniqueness of square-mean asymptotically almost periodic solution for a class ofstochastic differential equation is discussed by using the above property, Fubini theory,Holder inequality, and Banach fixed point theory.The third section, according to the second section, the existence and theuniqueness of square-mean asymptotically almost periodic solution for a class ofnonautonomous stochastic differential equation are discussed. First of all theoreticalknowledges of solving this kind of differential equation are given. After this theconcept of the operator ‘Acquistapace-Terreni’ conditions is introduced. Finally, theexistence and the uniqueness of square-mean asymptotically almost periodic solutionfor this kind of differential equations are discussed which similar to the differentialequations of the second section by using the Fubini theory, Holder inequality, andBanach fixed point theory.