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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Lévy processes with respect to the Whittaker convolution


Authors: Rúben Sousa, Manuel Guerra and Semyon Yakubovich
Journal: Trans. Amer. Math. Soc. 374 (2021), 2383-2419
MSC (2020): Primary 60G51, 60B15, 47D07, 33C15
DOI: https://doi.org/10.1090/tran/8294
Published electronically: January 21, 2021
MathSciNet review: 4223020
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Abstract:

It is natural to ask whether it is possible to construct Lévy-like processes where actions by random elements of a given semigroup play the role of increments. Such semigroups induce a convolution-like algebra structure in the space of finite measures.

In this paper, we show that the Whittaker convolution operator, related with the Shiryaev process, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. We then introduce the class of Lévy processes with respect to the Whittaker convolution and study their basic properties. We obtain a martingale characterization of the Shiryaev process analogous to Lévy’s characterization of Brownian motion.

Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed for differential operators whose associated convolution does not satisfy the usual compactness assumption on the support of the convolution.


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Additional Information

Rúben Sousa
Affiliation: CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
ORCID: 0000-0003-4956-9620
Email: rubensousa@fc.up.pt

Manuel Guerra
Affiliation: CEMAPRE and ISEG (School of Economics and Management), Universidade de Lisboa, Rua do Quelhas, 1200-781 Lisbon, Portugal
MR Author ID: 660730
Email: mguerra@iseg.ulisboa.pt

Semyon Yakubovich
Affiliation: CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
MR Author ID: 216522
Email: syakubov@fc.up.pt

Keywords: Lévy process, index Whittaker transform, generalized convolution, Shiryaev process, martingale characterization, infinitely divisible distributions
Received by editor(s): June 25, 2018
Received by editor(s) in revised form: January 24, 2020
Published electronically: January 21, 2021
Additional Notes: The first and third authors were partly supported by CMUP (UID/MAT/00144/2019), which is funded by Fundação para a Ciência e a Tecnologia (FCT) (Portugal) with national (MEC), European structural funds through the programmes FEDER under the partnership agreement PT2020, and Project STRIDE – NORTE-01-0145-FEDER-000033, funded by ERDF – NORTE 2020. The first author was also supported by the grant PD/BD/135281/2017, under the FCT PhD Programme UC|UP MATH PhD Program. The second author was partly supported by FCT/MEC through the project CEMAPRE – UID/MULTI/00491/2013.
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