The exponential Lie series for continuous semimartingales

TL;DR

This paper establishes that the logarithm of the flowmap for stochastic differential systems driven by continuous semimartingales can be expressed as an exponential Lie series, facilitating advanced integration schemes.

Contribution

It introduces a natural basis change for quadratic covariation processes and provides a self-contained proof of the Chen-Strichartz formula for the Lie series coefficients.

Findings

Proves the exponential Lie series representation for flowmaps.

Derives an explicit formula for Lie series coefficients.

Enhances Lie group integration methods for stochastic systems.

Abstract

We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an exponential Lie series. This relies on a natural change of basis to vector fields for the associated quadratic covariation processes, analogous to Stratonovich corrections. The flowmap can then be expanded as a series in compositional powers of vector fields and the logaritm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct self-contained proof of the corresponding Chen-Strichartz formula which provides an explicit formula for the Lie series coefficients. Such exponential Lie series are important in the development of strong Lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves.