Asymptotic error distribution for stochastic Runge--Kutta methods of strong order one

TL;DR

This paper derives the asymptotic error distribution for stochastic Runge--Kutta methods of strong order 1, introduces a framework for implicit methods, and analyzes their long-term mean-square error behavior.

Contribution

It presents the first derivation of asymptotic error distribution for fully implicit stochastic differential equation methods and compares their efficiency.

Findings

SRK methods of strong order 1 have a specific asymptotic error distribution.

Methods with weak order 2 share the same asymptotic error distribution and minimal long-term errors.

The parameter ta_1 determines the growth rate of the mean-square error.

Abstract

This work gives the asymptotic error distribution of the stochastic Runge--Kutta (SRK) method of strong order $1$ applied to Stratonovich-type stochastic differential equations. For dealing with the implicitness introduced in the diffusion term, we propose a framework to derive the asymptotic error distribution of diffusion-implicit or fully implicit numerical methods, which enables us to construct a fully explicit numerical method sharing the same asymptotic error distribution as the SRK method. Further, we show that the limit distribution $U(T)$ satisfies $\mathbf E|U(T)|^2\le e^{L_1T}(1+\eta_1)T^3$ for some $\eta_1\ge0$ only depending on the coefficients of the SRK method. Thus, we infer that $\eta_1$ is the key parameter reflecting the growth rate of the mean-square error of the SRK method. Especially, among the SRK methods of strong order $1$, those of weak order $2$ correspond to $\eta_1=0$, sharing the unified asymptotic error distribution, and have the smallest mean-square errors after a long time. This property is also found for the case of additive noise. It seems that we are the first to give the asymptotic error distribution of fully implicit numerical methods for stochastic differential equations.