Expansions for Markov-modulated systems and approximations of ruin probability

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

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