L\'evy processes under level-dependent Poissonian switching

Abstract

In this paper, we derive identities for the upward and downward exit problems and resolvents for a process whose motion changes between two L\'evy processes if it is above (or below) a barrier $b$ and coincides with a Poissonian arrival time. This can be expressed in the form of a (hybrid) stochastic differential equation, for which the existence of its solution is also discussed. All identities are given in terms of new generalisations of scale functions (counterparts of the scale functions from the theory of L\'evy processes). To illustrate the applicability of our results, the probability of ruin is obtained for a risk process with delays in the dividend payments.