Classification of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations

TL;DR

This paper classifies stochastic Runge-Kutta methods for weak approximation of Itô SDEs, providing a comprehensive analysis of coefficients for schemes of order one and two with minimal stages, including optimal schemes.

Contribution

It offers a complete classification of explicit stochastic Runge-Kutta schemes of orders one and two, expanding the understanding of their coefficient structures and optimal configurations.

Findings

Classified all coefficients for order one and two schemes with minimal stages.

Identified optimal schemes satisfying higher order conditions.

Provided a framework for constructing efficient stochastic Runge-Kutta methods.

Abstract

In the present paper, a class of stochastic Runge-Kutta methods containing the second order stochastic Runge-Kutta scheme due to E. Platen for the weak approximation of It\^o stochastic differential equation systems with a multi-dimensional Wiener process is considered. Order one and order two conditions for the coefficients of explicit stochastic Runge-Kutta methods are solved and the solution space of the possible coefficients is analyzed. A full classification of the coefficients for such stochastic Runge-Kutta schemes of order one and two with minimal stage numbers is calculated. Further, within the considered class of stochastic Runge-Kutta schemes coefficients for optimal schemes in the sense that additionally some higher order conditions are fulfilled are presented.