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 this paper we consider a discrete time insurance risk model with interest income. Using the recursive calculation method of De Vylder & Goovaerts (1988), recursive equations for the finite time ruin probabilities and the distribution of the time of ruin are derived. Fredholm type integral equations for the ultimate ruin probability, the distribution of the severity of ruin, the joint distribution of surplus before and after ruin, and the probability of absolute ruin are obtained. Numerical results are included.

We consider a classical surplus process modified by the payment of dividends when the insurer's surplus exceeds a threshold. We use a probabilistic argument to obtain general expressions for the expected present value of dividend payments, and show how these expressions can be applied for certain individual claim amount distributions. We then consider the question of maximising the expected present value of dividend payments subject to a constraint on the insurer's ruin probability.

We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.

In this paper we use Fourier/Laplace transforms to evaluate numerically relevant probabilities in ruin theory as an application to insurance. The transform of a function is split in two: the real and the imaginary parts. We use an inversion formula based on the real part only, to get the original function.

By using an appropriate algorithm to compute integrals and making use of the properties of these transforms we are able to compute numerically important quantities either in classical or non-classical ruin theory. As far as the classical model is concerned the problems considered have been widely studied. In what concerns the non-classical model, in particular models based on more general renewal risk processes, there is still a long way to go. In either case the approach presented is an easy method giving good approximations for reasonable values of the initial surplus.

To show this we compute numerically ruin probabilities in the classical model and in a renewal risk process in which claim inter-arrival times have an Erlang(2) distribution and compare to exact figures where available. We also consider the computation of the probability and severity of ruin in the classical model.

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.

In this paper we present a stable recursive algorithm for the calculation of the probability of ultimate ruin in the classical risk model. We also present stable recursive algorithms for the calculation of the joint and marginal distributions of the surplus prior to ruin and the severity of ruin. In addition we present bounds for these distributions.

In this paper we present algorithms to calculate the probability and severity of ruin in both finite and infinite time for a discrete time risk model. We show how the algorithms can be applied to give approximate values for the same quantities in the classical continuous time risk model.

The compound binomial model is a discrete time analogue (or approximation) of the compound Poisson model of classical risk theory. In this paper, several results are derived for the probability of ruin as well as for the joint distribution of the surpluses immediately before and at ruin. The starting point of the probabilistic arguments are two series of random variables with a surprisingly simple expectation (Theorem 1) and a more classical result of the theory of random walks (Theorem 2) that is best proved by a martingale argument.

In the usual model of the collective risk theory, we are interested in the severity of ruin, as well as its probability. As a quantitative measure, we propose G(u, y), the probability that for given initial surplus u ruin will occur and that the deficit at the time of ruin will be less than y, and the corresponding density g(u, y). First a general answer in terms of the transform is obtained. Then, assuming that the claim amount distribution is a combination of exponential distributions, we determine g; here the roots of the equation that defines the adjustment coefficient play a central role. An explicit answer is also given in the case in which all claims are of constant size.