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Necessity of weak subordination for some strongly subordinated Lévy processes

Part of: Stochastic processes

Published online by Cambridge University Press:  22 November 2021

Boris Buchmann  and
Kevin W. Lu Open the ORCID record for Kevin W. Lu [Opens in a new window]
Boris Buchmann*
Affiliation:
Australian National University
Kevin W. Lu*
Affiliation:
Australian National University
*
*Postal address: Research School of Finance, Actuarial Studies & Statistics, Australian National University, ACT 0200, Australia. Email address: boris.buchmann@anu.edu.au
**Current address: Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, USA. Email address: kwlu@uw.edu
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Abstract

Consider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy process even when strong subordination does not. Here we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Lévy process then it is necessarily equal in law to weak subordination in two cases: firstly when the subordinator is deterministic, and secondly when it is pure-jump with finite activity.

Keywords

Lévy processsubordinatormultivariate subordinationweak subordinationPoisson point processPoisson random measure

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60G55: Point processes
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 868 - 879
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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