Algebraic structure of stochastic expansions and efficient simulation

TL;DR

This paper explores the algebraic foundations of stochastic Taylor expansions to develop more accurate and efficient numerical integrators for stochastic differential systems, introducing a new class of optimal integrators.

Contribution

It establishes the convolution shuffle algebra as the framework for stochastic integrators, introduces a new class of efficient integrators, and proves the sinhlog integrator's optimality at all orders.

Findings

The algebraic structure of stochastic expansions is characterized by the convolution shuffle algebra.

A new class of efficient stochastic integrators is constructed.

The sinhlog integrator is proven to be optimal within this class at all orders.

Abstract

We investigate the algebraic structure underlying the stochastic Taylor solution expansion for stochastic differential systems.Our motivation is to construct efficient integrators. These are approximations that generate strong numerical integration schemes that are more accurate than the corresponding stochastic Taylor approximation, independent of the governing vector fields and to all orders. The sinhlog integrator introduced by Malham & Wiese (2009) is one example. Herein we: show that the natural context to study stochastic integrators and their properties is the convolution shuffle algebra of endomorphisms; establish a new whole class of efficient integrators; and then prove that, within this class, the sinhlog integrator generates the optimal efficient stochastic integrator at all orders.