Continuity properties and the support of killed exponential functionals

TL;DR

This paper studies the properties of killed exponential functionals involving Lévy processes, establishing their connection to Ornstein-Uhlenbeck processes, characterizing their support and continuity, and providing new conditions for their absolute continuity.

Contribution

It introduces a stochastic differential equation framework for killed exponential functionals, linking them to generalized Ornstein-Uhlenbeck processes and deriving new criteria for their absolute continuity.

Findings

The law of killed exponential functionals can be characterized as a stationary distribution of a stochastic differential equation.

Support and continuity properties of the law are fully characterized.

New sufficient conditions for absolute continuity of exponential integrals are established.

Abstract

For two independent L\'evy processes $\xi$ and $\eta$ and an exponentially distributed random variable $\tau$ with parameter $q>0$, independent of $\xi$ and $\eta$, the killed exponential functional is given by $V_{q,\xi,\eta} := \int_0^\tau \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$. Interpreting the case $q=0$ as $\tau=\infty$, the random variable $V_{q,\xi,\eta}$ is a natural generalization of the exponential functional $\int_0^\infty \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$, 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 $V_{q,\xi,\eta}$ 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 $\smash{\int_0^t\mathrm{e}^{-\xi_{s-}}\mathrm{d}\eta_s}$ for fixed $t\geq0$, as well as for integrals of the form $\smash{\int_0^\infty f(s) \, \mathrm{d}\eta_s}$ for deterministic functions $f$. Furthermore, applying the same techniques to the case $q=0$, new results on the absolute continuity of the improper integral $\int_0^\infty \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$ are derived.