Level-Dependent Dynamics

Abstract

This chapter considers a variant of the Cramér-Lundberg model which has the special feature that the claim arrival rate $\lambda (x)$ and the premium rate $r(x)$ are functions of the current surplus level x. If only the premium rate depends on that surplus level, then duality holds with a so-called shot-noise M/G/1 queue, which can be used to derive the ruin probability from a known queueing result. When both rates are level-dependent, we exploit the fact that the integro-differential equation for the survival probability only involves those two rates as a fraction $r(x)/\lambda (x)$. This reduces the determination of the ruin probability to that for the case in which only the premium rate is level-dependent. Then we consider a model in which the claim interarrival times depend on the current surplus level in a specific way: they equal an exponentially distributed quantity minus a fraction of the current surplus level, truncated at 0. The chapter is concluded by the analysis of a model in which tax payments are deducted from the premium income whenever the surplus process is at a running maximum, leading to the so-called tax identity.