General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula
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Peter K. Friz [1] ; Atul Shekhar [2]
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[1] Weierstrass Institute for Applied Analysis and Stochastics
Weierstrass Institute for Applied Analysis and Stochastics
Berlin, Stadt, Alemania
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[2] Indian Statistical Institute
Indian Statistical Institute
India
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 4, 2017, págs. 2707-2765
- Idioma: inglés
- Enlaces
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Resumen
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
