Open Access
November 2015 Lévy processes and stochastic integrals in the sense of generalized convolutions
M. Borowiecka-Olszewska, B.H. Jasiulis-Gołdyn, J.K. Misiewicz, J. Rosiński
Bernoulli 21(4): 2513-2551 (November 2015). DOI: 10.3150/14-BEJ653


In this paper, we present a comprehensive theory of generalized and weak generalized convolutions, illustrate it by a large number of examples, and discuss the related infinitely divisible distributions. We consider Lévy and additive process with respect to generalized and weak generalized convolutions as certain Markov processes, and then study stochastic integrals with respect to such processes. We introduce the representability property of weak generalized convolutions. Under this property and the related weak summability, a stochastic integral with respect to random measures related to such convolutions is constructed.


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M. Borowiecka-Olszewska. B.H. Jasiulis-Gołdyn. J.K. Misiewicz. J. Rosiński. "Lévy processes and stochastic integrals in the sense of generalized convolutions." Bernoulli 21 (4) 2513 - 2551, November 2015.


Received: 1 December 2013; Revised: 1 March 2014; Published: November 2015
First available in Project Euclid: 5 August 2015

zbMATH: 1333.60091
MathSciNet: MR3378476
Digital Object Identifier: 10.3150/14-BEJ653

Keywords: Lévy process , scale mixture , stochastic integral , symmetric stable distribution , weakly stable distribution

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

Vol.21 • No. 4 • November 2015
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