Cheap arbitrary high order methods for single integrand SDEs

TL;DR

This paper demonstrates that applying deterministic Runge-Kutta methods to single integrand Stratonovich SDEs yields high-order convergence in mean-square and weak senses, leveraging the similarity of their B-series to ODE solutions.

Contribution

It establishes a novel approach for high-order numerical solutions of single integrand SDEs using deterministic Runge-Kutta methods, exploiting their B-series structure.

Findings

Methods converge with order loor(p_d/2) in mean-square and weak senses.

B-series of solutions are similar to ODE case due to single integrand structure.

Applicability to specific Stratonovich SDE problems with high efficiency.

Abstract

For a particular class of Stratonovich SDE problems, here denoted as single integrand SDEs, we prove that by applying a deterministic Runge-Kutta method of order $p_d$ we obtain methods converging in the mean-square and weak sense with order $\lfloor p_d/2\rfloor$. The reason is that the B-series of the exact solution and numerical approximation are, due to the single integrand and the usual rules of calculus holding for Stratonovich integration, similar to the ODE case. The only difference is that integration with respect to time is replaced by integration with respect to the measure induced by the single integrand SDE.