Stochastic B-series and order conditions for exponential integrators

TL;DR

This paper develops a comprehensive order theory for stochastic exponential integrators applied to stiff stochastic differential equations, using B-series and rooted trees to unify Itô and Stratonovich cases.

Contribution

It introduces a general order condition framework for stochastic exponential integrators based on B-series, applicable to both Itô and Stratonovich integrals.

Findings

Derived order conditions for stochastic exponential integrators.

Unified treatment of Itô and Stratonovich integrals.

Provides a basis for designing higher-order stochastic integrators.

Abstract

We discuss stochastic differential equations with a stiff linear part and their approximation by stochastic exponential integrators. Representing the exact and approximate solutions using B-series and rooted trees, we derive the order conditions for stochastic exponential integrators. The resulting general order theory covers both It\^{o} and Stratonovich integration.