LAUSR

In this paper we solve the exit problems for a (reflected) spectrally negative L\'evy process exponentially killed with killing intensity depending on the present state of the process. We analyze respective resolvents. All identities are given in terms of new generalizations of scale functions. Particular cases concern $\omega(x)=q$ when we derive classical exit problems and $\omega(x)=q \mathbf{1}_{(a,b)}(x)$ producing Laplace transforms of occupation times of intervals until first passage times. Our results can be also applied to find bankruptcy probability in so-called Omega model where bankruptcy occurs at rate $\omega(x)$ when the surplus L\'evy process process is at level $x<0$. Finally, we apply derived results for getting some exit identities for a spectrally positive self-similar Markov processes. The main idea of all proofs relies on classical fluctuation identities for L\'evy process, the Markov property and some basic properties of a Poisson process.