For two independent Lévy processes ξ and η and an exponentially distributed random variable τ with parameter that is independent of ξ and η, the killed exponential functional is given by . 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 , 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 as leads to the classical exponential functional , 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.
"On the law of killed exponential functionals." Electron. J. Probab. 26 1 - 35, 2021. https://doi.org/10.1214/21-EJP616