We consider the Sparre Andersen risk process with interclaim times that belong to the class of distributions with rational Laplace transform. We construct error bounds for the ruin probability based on the Pollaczek–Khintchine formula, and develop an efficient algorithm to approximate the ruin probability for completely monotone claim size distributions. Our algorithm improves earlier results and can be tailored towards achieving a predetermined accuracy of the approximation.
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.
We construct a risk process, where the law of the next jump time or jump size can depend on the past through earlier jump times and jump sizes. Some distributional properties of this process are established. The compensator is found and some martingale properties are discussed.
We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.
Ruin Probability Approximations in Sparre Andersen Models with Completely Monotone Claims