The Study Of Two Numerical Methods For ITO Stochastic Differential Equation

Stochastic differential equations, a new subject in the last century, a branch ofprobability theory, which broaden the practical application field in the probabilitytheory and solve a lot of problems in real systems. The application of stochasticdifferential equations mainly concentrated on some deterministic differentialsystems with obvious instability, such as financial system and control system, alsostochastic differential equations more widely used in practice. However, unlikeordinary differential equations and partial differential equations, the solution ofstochastic differential equations is not easy to obtain, which is an important problemtroubling many researchers. Optimistically, with the development of computertechnology in recent years, more and more people study its numerical solution, andalso made a lot of good results.In this paper, we will first introduce the research background, research statusand some basic theory of stochastic differential equation. In this section, we mainlyintroduce the basic concepts of stochastic processes, which emphasis on twocommon processes: Wiener processes and Markov processes, and give an analogpath of Brownian motion. Then, we introduce some concepts of stochastic calculusand give the existence and Markov property of the solution of the stochasticdifferential equation.In the main part of this paper, we will study two numerical methods, Eulermethod and Milstein method. To illustrate the validity of the numerical method, wefirst introduce the concept of convergence and stability. Because we mainly focuson numerical simulation of the solution process, so we will study the strongconvergence and mean-square stability. For the Euler method, we first prove that itsstrong convergence order is0.5in theory, and then we give numerical validation.For Milstein methods, we will give a detailed derivation of it, and then give thenumerical validation of Ito formula. For the two methods, we have calculated themean square stability region of them, and then give the numerical verification. Atthe same time, we give the numerical simulation of experimental equation, use thenumerical methods to approximate analytical solution Finally, as a method ofapplication, we will use the Milstein methods to solve the population migrationproblems in practical.