Chen et al. (2014), studied a discrete semi-Markov risk model that covers existing risk models such as the compound binomial model and the compound Markov binomial model. We consider their model and build numerical algorithms that provide approximations to the probability of ultimate ruin and the probability and severity of ruin in a continuous time two-state Markov-modulated risk model. We then study the finite time ruin probability for a discrete m-state model and show how we can approximate the density of the time of ruin in a continuous time Markov-modulated model with more than two states.

In ruin theory, the conjecture given in De Vylder and Goovaerts (2000) is an open problem about the comparison of the finite time ruin probability in a homogeneous risk model and the corresponding ruin probability evaluated in the associated model with equalized claim amounts. In this paper we consider a weaker version of the conjecture and show that the integrals of the ruin probabilities with respect to the initial risk reserve are uniformly comparable.

The probability of ruin in continuous and finite time is numerically evaluated in a classical risk process where the premium can be updated according to credibility models and therefore change from year to year. A major consideration in the development of this approach is that it should be easily applicable to large portfolios. Our method uses as a first tool the model developed by Afonso et al. (2009), which is quite flexible and allows premiums to change annually. We extend that model by introducing a credibility approach to experience rating.

We consider a portfolio of risks which satisfy the assumptions of the Bühlmann (1967, 1969) or Bühlmann and Straub (1970) credibility models. We compute finite time ruin probabilities for different scenarios and compare with those when a fixed premium is considered.

This paper shows how the multivariate finite time ruin probability function, in a phase-type environment, inherits the phase-type structure and can be efficiently approximated with only one Laplace transform inversion.

From a theoretical point of view, we also provide below a generalization of Thorin’s formula (1971) for the double Laplace transform of the finite time ruin probability, by considering also the deficit at ruin; the model is that of a Sparre Andersen (renewal) risk process with phase-type interarrival times.

In the case when the claims distribution is of phase-type as well, we obtain also an alternative formula for the single Laplace transform in time (or “exponentially killed probability’’), in terms of the roots with positive real part of the Lundberg’s equations, which complements Asmussen’s representation (1992) in terms of the roots with negative real part.

In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the finite time ruin probability is well given when the claim amount distribution is a mixed exponential. As its consequence, a well-known result about ultimate ruin probability in the classical risk model is obtained.

The upper bound provided by Lundberg's inequality can be improved for the probability of ruin in finite horizon, as Gerber (1979) has shown. This paper studies this upper bound as a function of the retention limit, for an excess of loss arrangement, and compares it with the probability of ruin.