We consider a class of Markovian risk models in which the insurer collects premiums at rate c1 (c2) whenever the surplus level is below (above) a constant barrier level b. We derive the Laplace­Stieltjes transform (LST) of the distribution of the time to ruin as well as the LST (with respect to time) of the joint distribution of the time to ruin, the surplus prior to ruin, and the deficit at ru...

We focus on numerical evaluation of some quantities of interest in ruin theory, and on the practical use of the fast Fourier transform algorithm (FFT) in this context. We discuss the general application of the FFT for stochastic models, and we illustrate this by looking again at the probability of ruin in the classical risk model and by extending this approach to evaluation of the first moment ...

This paper studies ruin probabilities in two discrete-time risk models with premiums, claims and rates of interest modelled by three autoregressive moving average processes. Generalized Lundberg inequalities for ruin probabilities are derived by using recursive technique. A numerical example is given to illustrate the applications of these probability inequalities. Keywords—Lundberg inequality,...

In this paper, we consider a dual risk process which can be used to model the surplus of a business that invests money constantly and earns gains randomly in both time and amount. The occurrences of the gains and their amounts are assumed follow a semi-Markovian structure (e.g. Reinhard (1984)). We analyze a quantity resembling the Gerber-Shiu expected discounted penalty function (Gerber and Sh...

The Gerber-Shiu function provides a way of measuring the risk of an insurance company. It is given by the expected value of a function that depends on the ruin time, the deficit at ruin, and the surplus prior to ruin. Its computation boils down to the evaluation of the overshoot/undershoot distributions of the surplus process at ruin. In this paper, we approximate it in a closed form by fitting...

This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, a simple and direct method for determining the finite time (and ultimate) ruin probabilities, the distribution of the ruin severity, the reserve...

We study the probability of ruin before time t for the family of tempered stable Lévy insurance risk processes, which includes the spectrally positive inverse Gaussian processes. Numerical approximations of the ruin time distribution are derived via the Laplace transform of the asymptotic ruin time distribution, for which we have an explicit expression. These are benchmarked against simulations...