Expansions for Markov-modulated systems and approximations of ruin probability

Abstract

LetNbe a stationary Markov-modulated marked point process on ℝ with intensityβ∗and consider a real-valued functionalψ(N). In this paper we study expansions of the formEψ(N) =a0+β∗a1+ ·· ·+ (β∗)nan+o((β∗)n) forβ∗→ 0. Formulas for the coefficientsaiare derived in terms of factorial moment measures ofN. We computea1anda2for the probability of ruinφuwith initial capitalufor the risk process in the Markov-modulated environment;a0= 0. Moreover, we give a sufficient condition forϕuto be an analytic function ofβ∗. We allow the premium rate functionp(x) to depend on the actual risk reserve.