On the law of killed exponential functionals

For two independent L\'{e}vy processes $\xi$ and $\eta$ an exponentially distributed random variable $\tau$ with parameter $q>0$ that is of $\eta$, the killed exponential functional given by $V_{q,\xi,\eta} := \int_0^\tau \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$. With arising as stationary distribution a Markov process, we calculate infinitesimal generator process use it to derive different distributional equations describing law $V_{q,\xi,\eta}$, well for its Lebesgue density in absolutely continuous case. Various special cases examples are considered, yielding more explicit information on illustrating applications obtained. Interpreting case $q=0$ $\tau=\infty$ leads classical $\int_0^\infty \eta_s$, allowing extend many previous results include killing.