General rough integration, lévy rough paths and a lévy-kintchine-type formula
Annals of Probability
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 càdlàg processes. A class of Lévy 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 Lévy measure on the group, using Hunt's framework of Lie group valued Lévy processes. Examples of Lévy rough paths include a standard multidimensional Lévy 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.
Friz, Peter K. and Shekhar, Atul, "General rough integration, lévy rough paths and a lévy-kintchine-type formula" (2017). Journal Articles. 2502.