Continuity properties and the support of killed exponential functionals

Anita Behme, Alexander Lindner, Jana Reker and Victor Rivero

Stochastic Processes and their Applications, 2021, vol. 140, issue C, 115-146

Abstract: For two independent Lévy processes ξ and η and an exponentially distributed random variable τ with parameter q>0, independent of ξ and η, the killed exponential functional is given by Vq,ξ,η≔∫0τe−ξs−dηs. Interpreting the case q=0 as τ=∞, the random variable Vq,ξ,η is a natural generalisation of the exponential functional ∫0∞e−ξs−dηs, the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein–Uhlenbeck process. In this paper we show that also the law of the killed exponential functional Vq,ξ,η arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein–Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of ∫0te−ξs−dηs for fixed t≥0, as well as for integrals of the form ∫0∞f(s)dηs for deterministic functions f. Furthermore, applying the same techniques to the case q=0, new results on the absolute continuity of the improper integral ∫0∞e−ξs−dηs are derived.

Keywords: Generalised Ornstein–Uhlenbeck process; Exponential functional; Lévy processes; Killing; Absolute continuity; Support (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414921000958
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:140:y:2021:i:c:p:115-146

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2021.06.002

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().