General Rough integration, Levy Rough paths and a Levy--Kintchine type formula

TL;DR

This paper extends rough path theory to include jumps, introducing Levy rough paths and a Levy--Kintchine type formula, providing a pathwise approach to stochastic integration with jump processes.

Contribution

It develops a theory of rough paths with jumps, including an extension theorem, integration, and characterization of Levy rough paths, with explicit formulas for expected signatures.

Findings

Established Lyons' extension theorem for jump rough paths

Introduced Levy rough paths with specific conditions

Provided explicit formulas for expected signatures

Abstract

We consider rough paths with jumps. In particular, the analogue of Lyons' extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against cadlag processes. A class of Levy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and and a certain integrability property of the Carnot--Caratheodory norm with respect to the Levy measure on the group, using Hunt's framework of Lie group valued Levy processes. Examples of Levy rough paths include standard multi-dimensional Levy process enhanced with stochastic area as constructed by D. Williams, the pure area Poisson process and Brownian motion in a magnetic field. An explicit formula for the expected signature is given.