The Convergence Of Two Numerical Methods For Solving Stochastic Differential Equations

Stochastic differential equations (SDES), which is a kind of mathematical modelson stochastic progress, usually use Brownian motion to describe the noise, so it cantruly reflect the actual problem. Now research of stochastic differential equations hasbeen applied to economy, biological, physical and so on. However, analytic solution ofthe stochastic differential equations can hardly be obtained, so constructing appropriatenumerical methods and simulating the analytic solutions by using the computer are verynecessary in application, therefore, the convergence and stability of the numericalmethods are highly required. Based on those practical problems, this paper will presenttwo numerical methods for solving SDEs, one called the Exponential Milstein method,and the other is called the Tamed Runge-Kutta method, and both of their convergence isdiscussed. Furthermore, this article mainly discusses the results of these two parts.Firstly, for a class of semi-linear stochastic differential equation, the ExponentialMilstein method is constructed, and it’s order of convergence is proved to be1.Numerical simulation proves that, the Exponential Milstein method achieves smallerroot mean square error than the traditional Milstein method, therefore it can able tosimulate the analytic solution of original stochastic differential equations accurately.The second part considers the stochastic differential equations with one-sidedLipschitz continuous drift coefficient, and constructs explicit numerical method whichmeets the p-order mean consistency. Generally speaking, explicit methods such as theEuler-Maruyama method and the Milstein method, ask for globally Lipschitzcontinuous right end functions of the SDES to ensure the convergence, while theimplicit Euler-Maruyama method does not need, but it requires a huge amount ofcalculation. So a tamed Euler method has been proposed recently, it saves a lot ofcomputation as explicit numerical method, and achieves0.5as the order of convergence.Based on the tamed Euler method, a Tamed Runge-Kutta method is introduced in thisarticle, whose order of convergence is1. At last, the numerical experiment not onlyverifies the convergence order1, but also indicates that the Tamed Runge-Kutta methodis faster than the tamed Milstein method with the same order, so the Tamed Runge-Kutta method is more practical in simulating the large scale stochastic differentialequations.