For two independent L\'evy processes ξ and η and an exponentially
distributed random variable τ with parameter q>0, independent of ξ
and η, the killed exponential functional is given by Vq,ξ,η​:=∫0τ​e−ξs−​dηs​. Interpreting the case
q=0 as τ=∞, the random variable Vq,ξ,η​ is a natural
generalization of the exponential functional ∫0∞​e−ξs−​dηs​, 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, the support and continuity of the
law of killed exponential functionals is characterized, and many sufficient
conditions for absolute continuity are derived. We also obtain various new
sufficient conditions for absolute continuity of
∫0t​e−ξs−​dηs​ for fixed t≥0, as well as
for integrals of the form ∫0∞​f(s)dηs​ for
deterministic functions f. Furthermore, applying the same techniques to the
case q=0, new results on the absolute continuity of the improper integral
∫0∞​e−ξs−​dηs​ are derived. We
also show that the law of the killed exponential functional Vq,ξ,η​
arises as a stationary distribution of a solution to a certain stochastic
differential equation, thus establishing a close connection to generalised
Ornstein-Uhlenbeck processes