Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise

TL;DR

This paper introduces a new class of third order Runge-Kutta methods for weak approximation of SDEs with additive noise, reducing computational effort and applicable to multidimensional noise, showing promising numerical results.

Contribution

The paper presents a novel third order Runge-Kutta scheme that requires fewer evaluations and is applicable to multidimensional noise, improving upon existing methods like Platen's.

Findings

The new method has deterministic order four.

It requires fewer function evaluations than Platen's method.

Numerical tests show promising accuracy and efficiency.

Abstract

A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen's method, which to the knowledge of the author has been up to now the only known third order Runge-Kutta scheme for weak approximation, the new class of methods affords less random variable evaluations and is also applicable to SDEs with multidimensional noise. Order conditions up to order three are calculated and coefficients of a four stage third order method are given. This method has deterministic order four and minimized error constants, and needs in addition less function evaluations than the method of Platen. Applied to some examples, the new method is compared numerically with Platen's method and some well known second order methods and yields very promising results.