Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations |
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Authors: | Andreas Rößler |
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Institution: | 1. Darmstadt University of Technology, Fachbereich Mathematik , Darmstadt, Germany roessler@mathematik.tu-darmstadt.de |
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Abstract: | Abstract A general class of stochastic Runge-Kutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic Runge-Kutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic Runge-Kutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic Runge-Kutta methods converging with an arbitrarily high order. |
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Keywords: | Order condition Rooted tree analysis Stochastic differential equation Stochastic Runge-Kutta method Weak approximation |
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