The Annals of Probability

Stochastic integration with respect to cylindrical Lévy processes

Adam Jakubowski and Markus Riedle

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A cylindrical Lévy process does not enjoy a cylindrical version of the semimartingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Lévy processes in Hilbert spaces. The space of admissible integrands consists of càglàd, adapted stochastic processes with values in the space of Hilbert–Schmidt operators. Neither the integrands nor the integrator is required to satisfy any moment or boundedness condition. The integral process is characterised as an adapted, Hilbert space valued semimartingale with càdlàg trajectories.

Article information

Ann. Probab. Volume 45, Number 6B (2017), 4273-4306.

Received: December 2015
Revised: October 2016
First available in Project Euclid: 12 December 2017

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Digital Object Identifier

Primary: 60H05: Stochastic integrals
Secondary: 60B11: Probability theory on linear topological spaces [See also 28C20] 60G20: Generalized stochastic processes 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Cylindrical Lévy processes stochastic integration decoupled tangent sequence cylindrical Brownian motion random measures


Jakubowski, Adam; Riedle, Markus. Stochastic integration with respect to cylindrical Lévy processes. Ann. Probab. 45 (2017), no. 6B, 4273--4306. doi:10.1214/16-AOP1164.

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