In a classical continuous time-surplus process
when the insurer’s initial surplus is given by u
, an explicit expression for the probability of ultimate ruin exists, as is already well-known, for a limited number of claim-size distributions such as the exponential and mixed exponential distributions. When the claim-size distribution is exponential, simple analytic results for the ruin probability in infinite time may be possible. Nevertheless, as Grandell
) has pointed out, there is really no reason to believe that the exponential distribution is a realistic description of the claim behavior.
Although for most of the general claim amount distributions, e.g., heavy-tailed, the Laplace transform technique does not work, explicit expressions under other assumptions, such as Pareto distributions, have been obtained but they are too complicated and require large computation to calculate the values of the ultimate ruin probability. For example, Garcia
) derived complicated exact solutions under series representation and Seal
) and Wei and Yang
) under integral representations. Grandell and Segerdahl (1971) showed that for the gamma claim amount distribution under some restrictions on the parameters, the exact value of ruin probability can be computed via a formula which involves a complicated integral. In Ramsay
), an expression based on numerical integration was derived for the probability of ultimate ruin under the classical compound Poisson risk model, given an initial reserve of u
in the case of Pareto individual claim amount distributions. Furthermore, Albrecher et al.
) have obtained closed-form expressions for ruin probability functions under some kind of dependence assumption also using the mixing representation. In this regard, as Asmussen and Albrecher
) pointed out, the ideal situation is to come up with closed-form solutions for the ruin probabilities; however, these are limited. More recently, Tamturk and Utev
) computed ruin probabilities via a quantum mechanics approach and Sarabia et al.
) obtained ruin mixtures function in an aggregation of dependent risk model using mixtures of exponential distributions and finally Gómez-Déniz et al.
) has obtained closed-form expressions for the probability and severity of ruin when the claim size is assumed to follow a Lindley distribution. More recent results in the calculation of the ruin probabilities can be found in Kyprianou
) where the survival probability is defined for a general spectrally negative Lévy process. To compute these probabilities, it is necessary to invert the Laplace transform (see Kuznetzov et al.
)). In addition, when the Laplace exponent is a rational function, it admits an explicit expression written in terms of the sum of complex exponentials. A relevant example is the case where the jump size is phase-type distributed, which can in theory approximate any Lévy process by phase-type fitting (Egami and Yamazaki
)). The joint density of ruin time and overshoot can also be simply computed once the scale function, i.e., the survival probability, is known (see Yamazaki
In this paper, new ruin probability functions and severity of ruin are simply derived by a mixture mechanism, which is based on the use of appropriate loss functions. This procedure has resulted very useful to obtain new and adequate probability functions to fit insurance claim data and to derive new credibility expressions.
The structure of the paper is as follows. In Section 2
we revise some basic elements of the classical ruin theory. The mixture mechanism proposed in this work is given in Section 3
. The main results are provided in Section 4
. Here, closed-form expressions for mixture ruin probabilities and mixture severity of ruin when the mixing distribution belongs to the exponential family of distributions are given. Additionally, an expression for the upper bound of mixture ruin probability function is illustrated. Next numerical applications are shown in Section 5
and some final comments are presented in the last Section.