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July 2017 General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula
Peter K. Friz, Atul Shekhar
Ann. Probab. 45(4): 2707-2765 (July 2017). DOI: 10.1214/16-AOP1123

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.

Citation

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Peter K. Friz. Atul Shekhar. "General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula." Ann. Probab. 45 (4) 2707 - 2765, July 2017. https://doi.org/10.1214/16-AOP1123

Information

Received: 1 January 2015; Revised: 1 June 2016; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 06786092
MathSciNet: MR3693973
Digital Object Identifier: 10.1214/16-AOP1123

Subjects:
Primary: 60H99

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 4 • July 2017
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