随机常微分方程的几种数值求解方法及其应用

Some Numerical Methods and Applications of Stochastic Ordinary Differential Equations

随机微分方程是概率论与确定性微分方程相结合的产物,与确定性微分方程精确解的求解相比,随机微分方程精确解的求解是十分困难的。于是针对近几十年来兴起的热门边缘学科——随机微分方程的求解方法,提出了求随机微分方程数值解的方法应用及比较。讨论了求解随机微分方程数值解的方法,即Euler-Maruyama方法、Milstein方法和Runge-Kutta方法,并应用几个实例比较了在不同布朗运动影响下随机微分方程的精确解与确定性微分方程的精确解的不同之处,还比较了不同数值方法的求解结果及数值解与精确解的误差;编程图示结果表明:Milstein方法和Runge-Kutta方法的数值解比Euler-Maruyama方法更接近真解,这些与理论分析是一致的,该结论对随机常微分方程数值求解理论方法的应用具有一定的指导意义。

Stochastic ordinary differential equation(SODE)is the product of the combination of probability theory and ordinary differential equation(ODE).It is more difficult to solve the exact solutions for stochastic differential equations than to solve the exact solutions of deterministic differential equations.In view of a popular interdisciplinary subject——sovliving stochastic differential equations,the application and comparison of the numerical solutions of stochastic differential equations are discussed in this paper.So we discussed numerical methods of stochastic differential equations,including Euler-Maruyama method,Milstein method and Runge-Kutta method.The differences between the exact solutions of stochastic differential equations and the exact solutions of deterministic differential equations under the influence of different Brownian motion are compared by several examples,and the results of different numerical methods and the errors between the numerical solutions and the exact solutions are also compared.The results show that the numerical solutions of the Milstein method and Runge-Kutta method are closer to the true solution than the Euler-Maruyama method,which are consistent with the theoretical analysis.This conclusion has some guiding significance for the theoretical methods and applications of numerical solutions of stochastic ordinary differential equations.