Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations

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.

Keywords:

• Order condition
• Rooted tree analysis
• Stochastic differential equation
• Stochastic Runge-Kutta method
• Weak approximation

Mathematics Subject Classification:

• 65C30
• 60H35
• 65L05
• 60H10
• 34F05

ACKNOWLEDGMENT

The author is very grateful to the referee for his comments and fruitful suggestions.

Notes

¹Then Y(t) = A(t ₀, Y(t ₀), Θ(t − t ₀)) and Y(t ₀) = A(t ₀, Y(t ₀),0,…, 0).