General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula

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 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 a standard multi-dimensional Levy process enhanced with a 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