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

TL;DR

This paper develops new mathematical tools called generalized scale functions to analyze exit problems and resolvents for level-dependent spectrally negative Lévy processes that are exponentially killed, with applications to insurance risk.

Contribution

It introduces a novel class of scale functions satisfying Volterra integral equations for level-dependent Lévy processes, extending classical Lévy process theory.

Findings

Derived explicit formulas for exit probabilities and resolvents.

Established existence of solutions for reflected level-dependent Lévy SDEs.

Applied results to compute bankruptcy probabilities in insurance models.

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.