The Annals of Probability
- Ann. Probab.
- Volume 45, Number 4 (2017), 2707-2765.
General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula
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 multi-dimensional 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.
Ann. Probab., Volume 45, Number 4 (2017), 2707-2765.
Received: January 2015
Revised: June 2016
First available in Project Euclid: 11 August 2017
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Primary: 60H99: None of the above, but in this section
Friz, Peter K.; Shekhar, Atul. General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula. Ann. Probab. 45 (2017), no. 4, 2707--2765. doi:10.1214/16-AOP1123. https://projecteuclid.org/euclid.aop/1502438438