Word combinatorics for stochastic differential equations: splitting integrators
For researchers in stochastic numerics, this provides a more accessible framework for designing and analyzing splitting integrators, though it is incremental over existing BCH-based methods.
The paper develops a word combinatorics technique to analyze splitting integrators for stochastic differential equations, enabling systematic local error expansion and formulation of order conditions without the BCH formula, revealing a weak order barrier of two.
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.