Continuity properties and the support of killed exponential functionals

For two independent L\'evy processes $\xi$ and $\eta$ an exponentially distributed random variable $\tau$ with parameter $q>0$, of $\eta$, the killed exponential functional is given by $V_{q,\xi,\eta} := \int_0^\tau \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$. Interpreting case $q=0$ as $\tau=\infty$, $V_{q,\xi,\eta}$ a natural generalization $\int_0^\infty \eta_s$, law which well-studied in literature it stationary distribution generalised Ornstein-Uhlenbeck process. In this paper we show that also arises solution to stochastic differential equation, thus establishing close connection processes. Moreover, support continuity functionals characterised, many sufficient conditions for absolute are derived. We obtain various new $\smash{\int_0^t\mathrm{e}^{-\xi_{s-}}\mathrm{d}\eta_s}$ fixed $t\geq0$, well integrals form $\smash{\int_0^\infty f(s) \mathrm{d}\eta_s}$ deterministic functions $f$. Furthermore, applying same techniques $q=0$, results on improper integral \eta_s$