Fluctuation theory for level-dependent Lévy risk processes

Abstract

A level-dependent Lévy process solves the stochastic differential equation dU(t)=dX(t)−ϕ(U(t))dt, where X is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with ϕk(x)=∑j=1kδj1{x≥bj}. A general rate function ϕ that is non-decreasing and locally Lipschitz continuous is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of Lévy processes. We show how fluctuation identities for U can be expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations.