Continuous stage stochastic Runge\u2013Kutta methods

Xuan Xin,Xiaohua Ding,Wendi Qin

doi:10.1186/s13662-021-03221-2

Xuan Xin, Xiaohua Ding + Show 1 more

Open Access

https://doi.org/10.1186/s13662-021-03221-2

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Abstract

In this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.

Highlights

Stochastic differential equations (SDEs) have wide applications in many disciplines like biology, economics, medicine, engineering and finance
7 Conclusions This paper presents the extension work of deterministic continuous stage Runge–Kutta (DCSRK) methods which are upgraded to the stochastic counterpart
The general order conditions are obtained by the use of colored rooted tree and stochastic B-series

Summary

Introduction

Stochastic differential equations (SDEs) have wide applications in many disciplines like biology, economics, medicine, engineering and finance (see, e.g., [1,2,3]). There has been tremendous interests in developing effective and reliable numerical methods for SDEs during the last few decades, for example see [4,5,6,7,8,9,10,11,12,13,14]. Runge–Kutta (RK) methods with continuous stage were firstly presented by Butcher in 1970s [15], and they have been investigated and discussed by several authors recently because of the great advantages in conserving symplecticity [16], preserving energy [17] and so on. Constructing continuous stage stochastic Runge–Kutta (CSSRK) methods for SDEs is a valuable task. This paper mainly aims to construct CSSRK methods for SDEs which can be written in integral form as d dx(t) = g0 x(t) dt + gk x(t) ◦ dWk(t), x(0) = x0 ∈ Rm, k=1

Objectives

Conclusion