Electronic Journal of Probability

The stochastic Cauchy problem driven by a cylindrical Lévy process

Umesh Kumar and Markus Riedle

Full-text: Open access

Abstract

In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Lévy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.

Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 10, 26 pp.

Dates
Received: 14 February 2019
Accepted: 22 December 2019
First available in Project Euclid: 29 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1580267010

Digital Object Identifier
doi:10.1214/19-EJP407

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60G51: Processes with independent increments; Lévy processes 60G20: Generalized stochastic processes 60H05: Stochastic integrals

Keywords
cylindrical Lévy process Cauchy problem stochastic Fubini theorem cylindrical infinitely divisible

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kumar, Umesh; Riedle, Markus. The stochastic Cauchy problem driven by a cylindrical Lévy process. Electron. J. Probab. 25 (2020), paper no. 10, 26 pp. doi:10.1214/19-EJP407. https://projecteuclid.org/euclid.ejp/1580267010


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