Risk Theory with the Gamma Process

Abstract

The aggregate claims process is modelled by a process with independent, stationary and nonnegative increments. Such a process is either compound Poisson or else a process with an infinite number of claims in each time interval, for example a gamma process. It is shown how classical risk theory, and in particular ruin theory, can be adapted to this model. A detailed analysis is given for the gamma process, for which tabulated values of the probability of ruin are provided.

An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density

“…We first prove (i) by showing that v a * (S) satisfies the conditions of the verification lemma. Using (9), all the conditions of the verification lemma can be proved following the same arguments as in the proofs of Lemma 5 and Theorem 2 of [20], with the exception being the condition that v a * (S) (0) ≥ S. (Note that in deducing the analogue of Equation (4) of [20], we also use the fact that (exp (−q…”

“…Within this problem we assume that the underlying dynamics of the risk process are described by a spectrally negative Lévy process, which is now widely accepted and used as a replacement for the classical Cramér-Lundberg process (cf. [1], [3], [8], [9], [11], [14], [16], [19], [20], and [23]). Recall that a Cramér-Lundberg risk process {X t : t ≥ 0} corresponds to…”

On the link between credibility and frequency premium

“…All the policyholders of our sample stay in the portfolio at the second period. 5 On our data, the average of the second component of (12) derived from the power link estimated in (9) is equal to 0.99.…”

“…The gamma process can also be seen as a limit of compound Poisson processes (see Dufresne et al, 1991, for definitions and applications to ruin theory).…”

“…First, credibility is derived from the usual formula with a 4 A thorough analysis of stochastic migration between risk levels during different periods is given in Brouhns et al (2003). 5 They are actually present during seven years. See Bolancé et al (2003) for an investigation of the whole panel data set.…”