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2000•SIAM Journal on Numerical AnalysisRequires access

Order Conditions of Stochastic Runge--Kutta Methods by B -Series

Kevin Burrage, Pamela Burrage

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Abstract

In this paper, general order conditions and a global convergence proof are given for stochastic Runge--Kutta methods applied to stochastic ordinary differential equations (SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

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What this paper is about

In this paper, general order conditions and a global convergence proof are given for stochastic Runge--Kutta methods applied to stochastic ordinary differential equations (SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

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OpenAlex reports 163 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

In this paper, general order conditions and a global convergence proof are given for stochastic Runge--Kutta methods applied to stochastic ordinary differential equations (SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

Key concepts: Mathematics, Runge–Kutta methods, Ode, Runge–Kutta method, Ordinary differential equation, Stochastic differential equation, Applied mathematics, Stochastic partial differential equation

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