Word combinatorics for stochastic differential equations: splitting integrators

TL;DR

This paper introduces a word combinatorics approach to analyze splitting integrators for stochastic differential equations, enabling systematic error expansion and order condition formulation without BCH formula, revealing an order barrier of two for weak order.

Contribution

It presents a novel combinatorial method to analyze stochastic integrators, simplifying error analysis and establishing an order barrier for weak convergence.

Findings

Systematic expansion of local error using word combinatorics

Formulation of order conditions for integrators

Identification of a weak order barrier of two

Abstract

We present an analysis based on word combinatorics of splitting integrators for Ito or Stratonovich systems of stochastic differential equations. In particular we present a technique to write down systematically the expansion of the local error; this makes it possible to easily formulate the conditions that guarantee that a given integrator achieves a prescribed strong or weak order. This approach bypasses the need to use the Baker-Campbell-Hausdorff (BCH) formula and shows the existence of an order barrier of two for the attainable weak order. The paper also provides a succinct introduction to the combinatorics of words.