Square-mean Asymptotically Almost Automorphic Mild Solutions To Two Classes Of Stochastic Differential Equations

The stochastic differential equation plays an import part in artificial intelligence,space,and other advanced technology,because the penomenon of nature can be well described by it.For example,the filrate,Dirichlet,optimal stopping and normal stochastic control problems can be solved by using stochastic differential equations.The applied range of square-mean almost automorphic type functions is larger than square-mean almost periodic type functions'.The combination of square-mean almost automorphic type functions and square-mean almost automorphic type stochastic process can supply more theoretical foundations and solutions to problems in reality.It's aimed at disscussing whether there exist unique square-mean asymptotically almost automorphic mild solutions to two classes of stochastic differential equations in this paper.Firstly,it's disscussed whether there exists a unique square-mean asymptotically almost automorphic mild solution for a class of abstract semilinear stochastic differential evolution equations,which are in a real separable Hilbert space.For this reason,the concepts of square-mean asymptotically almost automorphic functions are introduced,and followingly the concepts of square-mean asymptotically almost automorphic stochastic process.According to the composibility of the two introduced above,the existence and uniqueness of the solutions,for the stochastic differential equations,are discussed by using,with the combination of lipschitz condition and some assumptions,the infinitesimal generator of a uniformly exponentially stable C0-semigroup,the Banach fixed point theorem,It(?) isometry and Cauchy-Schwarz inequality.Secondly,it's disscussed whether there exists a unique square-mean asymptotically almost automorphic mild solution for another class of stochastic differential equations.The relation between the norm of fractional power space constructed by the infinitesimal generator of analytic semigroup and the two norm of Hillbert space is introduced at the basis of the concepts above.Combining with composibility lemma,the relation and other relative properties,the existence of the solution is disscussed.The uniquess of the solution is later disscussed by using the Banach fixed point theorem,It(?) isometry and Lipschtiz condition.