Asymptotic error distributions for diffusion-implicit schemes via appurtenant methods

Abstract

We develop an appurtenant-method framework for analyzing asymptotic error distributions and implement it in two diffusion-implicit settings for stochastic differential equations. The argument rests on an explicit appurtenant method which has the same strong order as the implicit method and approximates it with a higher strong order. A local-to-global strong convergence result for two Markov chains turns the one-step expansion of the implicit equation into the required appurtenant method; the limiting distribution can then be obtained from an adapted continuization of the appurtenant method. We first illustrate the framework with the fully implicit stochastic midpoint method for a multidimensional Stratonovich equation, without imposing commutativity on the diffusion fields. This method has strong order $1/2$, and its normalized error is governed by the Lévy areas omitted by the numerical method. We then turn to the main application, namely, a class of fully implicit stochastic Runge--Kutta (SRK) methods of strong order $1$ for Stratonovich equations with scalar multiplicative noise. We derive their asymptotic error distributions and identify a parameter, depending only on the coefficients of the Butcher tableau, that reflects the growth rate of the root-mean-square error of the SRK method.