Two Numerical Methods Of Solving The Stochastic Differential Equations

Stochastic differential equation originated in the20th century. In the past100years not only the relative theory has been through rapid development, but it isalso widely used in real life. In the past, many scholars applied deterministicmathematical model to study phenomenon appears in the physical, biological,economic, and contrology and other areas. However, with the rapid developmentof science, they found that only the certainty factors are not enough to completelyreflect the actual situation. There are many external factors that need to beconsidered into the phenomenon we study. Therefore, it is necessary for us to thinkover uncertainty factors when we are study mathematical models. Due to all thereasons mentioned above, stochastic differential equation gets attention and rapiddevelopment. Although it is hard to obtain the exact solutions of the equation, westill treat it as a goal. Therefore, whether the numerical method that we used tosolve the stochastic differential equation is effective or not becomes important. Ifwe want to obtain effective numerical method, it is necessary to discuss theconvergence of the numerical method.This paper discusses the stochastic differential equation (SDEs) andconstructs a split-stepθ method for solving SDEs, namely the split-stepθ methodand do research into the convergence and stability of this numerical method. Weobtain the mean-square convergence of the split-step θ method. And themean-square order is0.5. Besides, we also build up another method for solvingSDEs, called a split-step balanced θ method. After that, we still consider theconvergence and stability of this method. Finally, we do the numericalexperimentation in order to verify our conclusion.