Finite Element Approximation To Periodic Solution Of Stochastic Differential Equation

Stochastic phenomena are widespread in nature and daily life.The related theories of stochastic differential equations provide basic ideas for the modeling,analysis,and prediction of stochastic phenomena.The periodicity of the movement of things is one of the most basic phenomena in nature.Thus,the study of periodic solutions of stochastic differential equations is an important problem in the field of stochastic differential equations.This paper mainly studies the finite element approximation of periodic solutions of stochastic differential equations according to the distribution.This paper consists of five parts.In the introduction part,we introduce the research background and research status of periodic solutions of stochastic differential equations and the main research content of this paper.In the second part,we introduce the relevant content of stochastic differential equations and stochastic periodic solutions in distribution.In the third part,we deduce the Fokker-Planck equation that the density function of the solution of the one-dimensional stochastic differential equation satisfies.In the fourth part,we give the specific method of solving the periodic solution of the Fokker-Planck equation with the finite element method and prove that the solution of the Fokker-Planck equation over finite interval approximately satisfies the conservation of the space integral in the process of numerical calculation.In the fifth part,we calculate four numerical examples to verify the effectiveness of the method.In the sixth part,we summarize the research work of the whole paper.