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∗
)
nan
+ o((β
∗)
n
) for β
∗→ 0. Formulas for the coefficients ai
are derived in terms of factorial moment measures of N. We compute a
1 and a
2 for the probability of ruin φ u
with initial capital u for the risk process in the Markov-modulated environment; a
0 = 0. Moreover, we give a sufficient condition for ϕu
to be an analytic function of β
∗. We allow the premium rate function p(x) to depend on the actual risk reserve.
Simultaneous Ruin Probability for Two-Dimensional Fractional Brownian Motion Risk Process over Discrete Grid