Phase-type distributions and risk processes with state-dependent premiums

Abstract

Abstract Consider a risk reserve process with initial reserve u, Poisson arrivals, premium rule p(r) depending on the current reserve r and claim size distribution which is phase-type in the sense of Neuts. It is shown that the ruin probabilities ψ(u) can be expressed as the solution of a finite set of differential equations, and similar results are obtained for the case where the process evolves in a Markovian environment (e.g., a numerical example of a stochastic interest rate is presented). Further, an explicit formula for ψ(u) is presented for the case where p(r) is a two-step function. By duality, the results apply also to the stationary distribution of storage processes with the same input and release rate p(r) at content r.