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

Abstract

In this paper, we solve exit problems for a level-dependent L\'evy process which is exponentially killed with a killing intensity that depends on the present state of the process. Moreover, we analyse the respective resolvents. All identities are given in terms of new generalisations of scale functions (counterparts of the scale function from the theory of L\'evy processes), which are solutions of Volterra integral equations. Furthermore, we obtain similar results for the reflected level-dependent L\'evy processes. The existence of the solution of the stochastic differential equation for reflected level-dependent L\'evy processes is also discussed. Finally, to illustrate our result, the probability of bankruptcy is obtained for an insurance risk process.