Fluctuations of Omega-killed spectrally negative L\'evy processes

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