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2021 On the law of killed exponential functionals
Anita Behme, Alexander Lindner, Jana Reker
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Electron. J. Probab. 26: 1-35 (2021). DOI: 10.1214/21-EJP616


For two independent Lévy processes ξ and η and an exponentially distributed random variable τ with parameter q>0 that is independent of ξ and η, the killed exponential functional is given by Vq,ξ,η:=0τeξsdηs. With the killed exponential functional arising as the stationary distribution of a Markov process, we calculate the infinitesimal generator of the process and use it to derive different distributional equations describing the law of Vq,ξ,η, as well as functional equations for its Lebesgue density in the absolutely continuous case. Various special cases and examples are considered, yielding more explicit information on the law of the killed exponential functional and illustrating the applications of the equations obtained. Interpreting the case q=0 as τ= leads to the classical exponential functional 0eξsdηs, allowing to extend many previous results to include killing.


The authors would like to thank Mladen Savov and the anonymous referee for their careful reading and valuable suggestions.


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Anita Behme. Alexander Lindner. Jana Reker. "On the law of killed exponential functionals." Electron. J. Probab. 26 1 - 35, 2021.


Received: 4 March 2020; Accepted: 29 March 2021; Published: 2021
First available in Project Euclid: 3 May 2021

arXiv: 2003.02073
Digital Object Identifier: 10.1214/21-EJP616

Primary: 60E07
Secondary: 46N30, 60E10, 60J35


Vol.26 • 2021
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