Continuous stage stochastic Runge–Kutta methods

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 CS...

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Main Authors:	Xuan Xin, Wendi Qin, Xiaohua Ding
Format:	Article
Language:	English
Published:	SpringerOpen 2021-01-01
Series:	Advances in Difference Equations
Subjects:	Stochastic differential equations Continuous stage stochastic Runge–Kutta methods B-series
Online Access:	https://doi.org/10.1186/s13662-021-03221-2

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record_format	Article
spelling	doaj-1c6e730e247a44638532317100881e392021-01-24T12:39:50ZengSpringerOpenAdvances in Difference Equations1687-18472021-01-012021112210.1186/s13662-021-03221-2Continuous stage stochastic Runge–Kutta methodsXuan Xin0Wendi Qin1Xiaohua Ding2Department of Mathematics, Harbin Institute of Technology at WeihaiDepartment of Mathematics, Harbin Institute of Technology at WeihaiDepartment of Mathematics, Harbin Institute of Technology at WeihaiAbstract 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.https://doi.org/10.1186/s13662-021-03221-2Stochastic differential equationsContinuous stage stochastic Runge–Kutta methodsB-series
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Xuan Xin Wendi Qin Xiaohua Ding
spellingShingle	Xuan Xin Wendi Qin Xiaohua Ding Continuous stage stochastic Runge–Kutta methods Advances in Difference Equations Stochastic differential equations Continuous stage stochastic Runge–Kutta methods B-series
author_facet	Xuan Xin Wendi Qin Xiaohua Ding
author_sort	Xuan Xin
title	Continuous stage stochastic Runge–Kutta methods
title_short	Continuous stage stochastic Runge–Kutta methods
title_full	Continuous stage stochastic Runge–Kutta methods
title_fullStr	Continuous stage stochastic Runge–Kutta methods
title_full_unstemmed	Continuous stage stochastic Runge–Kutta methods
title_sort	continuous stage stochastic runge–kutta methods
publisher	SpringerOpen
series	Advances in Difference Equations
issn	1687-1847
publishDate	2021-01-01
description	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.
topic	Stochastic differential equations Continuous stage stochastic Runge–Kutta methods B-series
url	https://doi.org/10.1186/s13662-021-03221-2
work_keys_str_mv	AT xuanxin continuousstagestochasticrungekuttamethods AT wendiqin continuousstagestochasticrungekuttamethods AT xiaohuading continuousstagestochasticrungekuttamethods
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Continuous stage stochastic Runge–Kutta methods

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