Ruin-related Quantities Research Articles

Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples.

Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples.

On the Gerber–Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution

In this paper, we consider the Sparre Andersen risk model with an arbitrary interclaim time distribution and a fairly general class of distributions for the claim sizes. Via a two-step procedure which involves a combination of a probabilitic and an analytic argument, an explicit expression is derived for the Gerber–Shiu discounted penalty function, subject to some restrictions on its form. A special case of Sparre Andersen risk models is then further analyzed, whereby the claim sizes’ distribution is assumed to be a mixture of exponentials. Finally, a numerical example follows to determine the impact on various ruin related quantities of assuming a heavy-tail distribution for the interclaim times.

Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time risk models

We consider a class of compound renewal (Sparre Andersen) risk process with claim inter-arrival times having a discrete K m distribution, i.e., the probability generating function (p.g.f.) of its distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. Li [16] gives a recursive formula for the Gerber-Shiu function. In this paper, an explicit formula for the Gerber-Shiu function is given in terms of a compound geometric distribution function, through which many ruin related quantities are analyzed, e.g., ruin probability, the p.g.f. of the time of ruin, joint and marginal distributions of the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support.