Expansions for Markov-modulated systems and approximations of ruin probability

Let N be a stationary Markov-modulated marked point process on R with intensity beta* and consider a real-valued functional psi(N). In this paper we study expansions of the form E psi(N) = a(0) + beta*a(1) + ... + (beta*)(n)a(n) + o((beta*)(n)) for beta* --> 0. Formulas for the coefficients a(1) are derived in terms of factorial moment measures of N. We compute a(1) and a(2) for the probability of ruin phi(u) with initial capital u for the risk process in the Markov-modulated environment; a(0)=0. Moreover, we give a sufficient condition for phi(u) to be an analytic function of beta*. We allow the premium rate function p(x) to depend on the actual risk reserve.