The Annals of Probability

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

Peter K. Friz and Atul Shekhar

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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 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.

Article information

Source
Ann. Probab., Volume 45, Number 4 (2017), 2707-2765.

Dates
Received: January 2015
Revised: June 2016
First available in Project Euclid: 11 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1502438438

Digital Object Identifier
doi:10.1214/16-AOP1123

Mathematical Reviews number (MathSciNet)
MR3693973

Zentralblatt MATH identifier
06786092

Subjects
Primary: 60H99: None of the above, but in this section

Keywords
Young integration rough paths Lévy processes general theory of processes

Citation

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


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