Fluctuations of Omega-killed level-dependent spectrally negative Lévy processes
Zbigniew Palmowski, Meral Şimşek, Apostolos D. Papaioannou
TL;DR
This work extends fluctuation theory for spectrally negative Lévy processes to level-dependent dynamics killed at a state-dependent rate $\omega(U_t)$. By introducing omega-killed level-dependent scale functions $\mathpzc{W}^{(\omega)}$ and $\mathpzc{Z}^{(\omega)}$, defined as solutions to renewal/Volterra-type integral equations, the authors derive explicit two-sided and one-sided exit identities and resolvent formulas, both in the non-reflected and reflected settings. They prove existence of solutions for the reflected level-dependent SDE via a robust strong-approximation scheme using multi-refracted rate functions, and demonstrate an insurance application by computing bankruptcy probabilities in an Omega risk model. The results unify and generalize prior omega-killed SNLP identities and provide a versatile toolkit for level-dependent fluctuation theory with state-dependent killing.
Abstract
In this paper, we solve exit problems for a level-dependent Lévy 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 Lévy processes), which are solutions of Volterra integral equations. Furthermore, we obtain similar results for the reflected level-dependent Lévy processes. The existence of the solution of the stochastic differential equation for reflected level-dependent Lévy processes is also discussed. Finally, to illustrate our result, the probability of bankruptcy is obtained for an insurance risk process.