Stochastic expansions and Hopf algebras

TL;DR

This paper demonstrates that the sinh-log series outperforms the exponential Lie series and stochastic Taylor methods in mean-square error for simulating nonlinear stochastic differential systems without drift, leveraging Hopf algebra structures.

Contribution

It proves the superiority of the sinh-log series over the exponential Lie series in mean-square error for non-commuting diffusion vector fields without drift, using Hopf algebra techniques.

Findings

Sinh-log series has smaller mean-square error than stochastic Taylor and exponential Lie series.

The Hopf algebra structure underpins the analysis and comparison of series.

Numerical studies confirm the theoretical advantages of the sinh-log series.

Abstract

We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell--Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that at low orders this method is more accurate in the mean-square error than corresponding stochastic Taylor methods. However it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here we prove that when there is no drift, and the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series, is always smaller than the corresponding stochastic Taylor error, in fact to all orders. Our proof utilizes the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations. We illustrate the benefits of the proposed series in numerical studies.