The Exponential Lie Series and a Chen-Strichartz Formula for Levy Processes

TL;DR

This paper develops a series expansion for the flowmap of stochastic differential equations driven by Levy processes, extending deterministic and continuous stochastic results to include jumps and stochastic coefficients.

Contribution

It derives a Chen-Strichartz formula for Levy-driven SDEs, providing explicit series components and showing the flowmap's logarithm is a Lie series, extending previous deterministic and continuous stochastic results.

Findings

Derived a series expansion for Levy-driven SDEs

Explicit formula for series components

Extended Chen-Strichartz formula to jump processes

Abstract

In this paper, we derive a Chen-Strichartz formula for stochastic differential equations driven by Levy processes, that is, we derive a series expansion of the logarithm of the flowmap of the stochastic differential equation in terms of commutators of vector fields with stochastic coefficients, and we provide an explicit formula for the components in this series. The stochastic components are generated by the Levy processes that drive the stochastic differential equation and their quadratic variation and power jumps; the vector fields are given as linear combinations of commutators of elements in the pre-Lie Magnus expansion generated by the original vector fields governing our stochastic differential equation. In particular, we show the logarithm of the flowmap is a Lie series. These results extend previous results for deterministic differential equations and continuous stochastic differential equations. For these, the Chen-Strichartz series has shown to play a pivotal role in the design of numerical integration schemes that preserve qualitative properties of the solution such as the construction of geometric numerical schemes and in the context of efficient numerical schemes.