  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.
