Families of efficient second order Runge-Kutta methods for the weak approximation of It\^o stochastic differential equations

TL;DR

This paper classifies and extends efficient second order Runge-Kutta methods for simulating Itô stochastic differential equations, improving computational efficiency and demonstrating promising numerical results.

Contribution

It provides a complete classification of explicit second order Runge-Kutta methods with minimal stages for Itô SDEs and extends a known method with minimized error constant.

Findings

The new class has linear dependence on Wiener processes for function evaluations.

The extended RK32 method shows promising numerical performance.

The classification enables efficient method design for stochastic simulations.

Abstract

Recently, a new class of second order Runge-Kutta methods for It\^o stochastic differential equations with a multidimensional Wiener process was introduced by R\"o{\ss}ler. In contrast to second order methods earlier proposed by other authors, this class has the advantage that the number of function evaluations depends only linearly on the number of Wiener processes and not quadratically. In this paper, we give a full classification of the coefficients of all explicit methods with minimal stage number. Based on this classification, we calculate the coefficients of an extension with minimized error constant of the well-known RK32 method to the stochastic case. For three examples, this method is compared numerically with known order two methods and yields very promising results.