Stability Analysis Of Solution For Two Classes Of Stochastic Differential Equations

The study of stochastic differential equations have been sixty years old historysince the Japanese mathematician It pioneered stochastic calculus theories in1950s. Stochastic differential equations have been the rapid development until rightnow.Stochastic differential equations are widely used in biology, physics, economics,control and other fields. For a long time, because the lack of effective numericalmethods for solving SDEs and computer resources available so that when you createmathematical models tend to ignore the influence of random factors. Recently, somescholars have achieved certain useful results in the numerical solution of stochasticdifferential equations, which will also mean some random questions can be studiedby means of mathematical software.Firstly, This paper introduces some used definitions and theorems of the basictheory of probability and stochastic processes. And then this paper describes thebackground knowledge of SDEs and some properties of their analytic solutions,existence and uniqueness theorem and their expression. Due to the complexstochastic systems, it is usually difficult to obtain the analytical expression of thetheoretical solution of the equation, so the construction of the numerical methodbecomes extremely important. In this thesis, it mainly gives the conditions ofp-moment stability and mean square stability of two types of SDEs(one is generalSDE, another is SDDE), and gives their corresponding numerical simulations. TheEuler-Maruyama method is applied to SDEs, and numerical methods to prove this ismean-square stable, and gives the method satisfies mean square stable condition. Weuse numerical examples prove the error between the Euler-Maruyama method forobtaining the solution with real solutions is very small, and prove the superiority ofEuler-Maruyama method. At the end of this paper, we present the selected range ofthe step when a solution of stochastic differential equations is stable, and givesspecific numerical examples, to verify the validity of the Euler-Maruyama method,furthermore validate the conclusions in a practical point.