François Dufresne

Risk theory for the compound Poisson process that is perturbed by diffusion

The classical model of collective risk theory is extended in that a diffusion process is added to the compound Poisson process. It is shown that the probabilities of ruin (by oscillation or by a claim) satisfy certain defective renewal equations. The convolution formula for the probability of ruin is derived and interpreted in terms of the record highs of the aggregate loss process. If the distribution of the individual claim amounts are combinations of exponentials, the probabilities of ruin can be calculated in a transparent fashion. Finally, the role of the adjustment coefficient (for example, for the asymptotic formulas) is explained.

Risk Theory with the Gamma Process

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.

The surpluses immediately before and at ruin, and the amount of the claim causing ruin

Abstract In the classical compound Poisson model of the collective risk theory we consider X, the surplus before the claim that causes ruin, and Y, the deficit at the time of ruin. We denote by f(u; x, y) their joint density (u initial surplus) which is a defective probability density (since X and Y are only defined, if ruin takes place). For an arbitrary claim amount distribution we find that f(0; x, y) = ap(x + y), where p(z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f(u; x, y) can be calculated explicitly, if the claim amount distribution is exponential or, more generally, a combination of exponential distributions. We are also interested in X + Y, the amount of the claim that causes ruin. Its density h(u; z) can be obtained from f(u; x, y). One finds, for example, that h(0; z) = azp(z).

Three Methods to Calculate the Probability of Ruin

The first method, essentially due to GOOVAERTS and DE VYLDER, uses the connection between the probability of ruin and the maximal aggregate loss random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distributions. Then the probability of ruin can be calculated in a transparent fashion; the main problem is to determine the nontrivial roots of the equation that defines the adjustment coefficient. For the third method one observes that the probability, of ruin is related to the stationary distribution of a certain associated process. Thus it can be determined by a single simulation of the latter. For the second and third methods the assumption of only proper (positive) claims is not needed.

The probability and severity of ruin for combinations of exponential claim amount distributions and their translations

Abstract In the classical compound Poisson model of the collective theory of risk let ψ(u, y) denote the probability that ruin occurs and that the negative surplus at the time of ruin is less than − y. It is shown how this function, which also measures the severity of ruin, can be calculated if the claim amount distribution is a translation of a combination of exponential distributions. Furthermore, these results can be applied to a certain discrete time model.

The probability of ruin for the Inverse Gaussian and related processes

Abstract We consider a family of aggregate claims processes that contains the gamma process, the Inverse Gaussian process, and the compound Poisson process with gamma or degenerate claim amount distribution as special cases. This is a one-parameter family of stochastic processes. It is shown how the probability of ruin can be calculated for this family. Extensive numerical results are given and the role of the parameter is discussed.

Rational ruin problems—a note for the teacher

Abstract It is shown how user friendly examples of ruin theory problems can be constructed, i.e., examples where all the parameters are integers or rational numbers. Tables containing more than 200 such examples are provided.

An extension of Kornya's method with application to pension funds.

A simple extension of the method of Kornya is derived. The extended method applies to the convolution of triatomic distributions with nonnegative support while the original method is restricted to diatomic distributions. This way, the algorithm can be applied in the calculation of the distribution of the total claims of a pension fund where only death and disability of active members are considered.

Risk theory with the gamma process

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.