Two Implicit Runge-Kutta Methods for Stochastic Differential Equation

Fuwen Lu,Zhiyong Wang

doi:10.4236/am.2012.310162

Abstract

In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two fully implicit schemes are presented and their stability qualities are discussed. And the numerical report illustrates the better numerical behavior.

Highlights

In this paper, we want to obtain numerical methods for strong solution of Stochastic Differential Equations of Itô type.dy = f y t dt g y t dW t, y (1.1)Note that f is a slowly varying continuous component function, which is called drift coefficient, g is the rapidly varying continuous function called the diffusion coefficient
I will construct some implicit stochastic Runge-Kutta method (SRK) for SDEs of Itô type
We show general form of Runge-Kutta methods for SDEs of Itô form

Summary

Introduction

We want to obtain numerical methods for strong solution of Stochastic Differential Equations of Itô type. Wang ZY [7] mainly considered the strong order SRKs for the SDEs of Itô form In his PhD thesis he offered us the Colored Rooted tree theory for Itô tpye, and constructed some 2-stage and 3-stage explicit methods. Along this line, I will construct some implicit SRKs for SDEs of Itô type. Burrage [2] presented Colored Rooted Tree theory in her PhD thesis, and Wang [7] did the research especially for Itô SDEs. Similar to the deterministic condition, the definition of the elementary differential can be associated with t T.

F X0 I01 F X0 I111

A B 1 B 2 B p

T I1 2 T

Stability

Numerical Results