Claude Lefèvre

The probability of ruin in finite time with discrete claim size distribution

Abstract The ruin time T is considered as the time of first crossing between a compound Poisson trajectory and an upper increasing boundary. Under the assumption that the claim sizes are integer-valued, we show that the distribution of T can be expressed in terms of generalized Appell polynomials. Using the algebraic properties of these polynomials elegant expressions are obtained for P(T > x).

A class of bivariate stochastic orderings, with applications in actuarial sciences

This paper is concerned with the bivariate extension of a wide class of univariate orderings said to be of convex-type. Attention is paid to random vectors valued in a rectangle or an orthant of the real plane. Various orderings used in probability and statistics (such as the stochastic dominance, the upper orthant order, the orthant convex order, the correlation order and the supermodular order) can be seen as particular cases. The practical applications of these orderings seem to be very promising, especially in actuarial sciences.

Extremal generators and extremal distributions for the continuous s-convex stochastic orderings

In the first part of this paper, the extremal generators of the continuous s-convex stochastic orderings introduced by Denuit et al., [Denuit, M., Lefevre, Cl., Shaked, M., 1998. Mathematical Inequalities and Applications, 1, 585–613] are identified. Then, the problem of constructing the extremal distributions with respect to these orderings in moment spaces is reexamined in whole generality using some results given in [De Vylder, F.E., 1996. Advanced Risk Theory. A Self-Contained Introduction. Editions de l’Universite Libre de Bruxelles, Swiss Association of Actuaries, Bruxelles]. An illustration in life insurance enhances the interest of the theory.

Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences

Recently, Fishbum and Lavalle (1995) and Lefevre and Utev (1996) have considered some stochastic order relations specific for arithmetic random variables. The present work is concerned with these orderings, together with two other classes of stochastic order relations closely related. First, attention is paid to characterizations and various properties of all these orderings. Then, sufficient conditions of crossing-type for the two new classes of orderings are derived and extrema among discrete random variables are deduced. This is applied in actuarial sciences to obtain new bounds for the classical single life premiums as well as for the probability of ruin in the compound binomial risk model.

Comparing sums of exchangeable Bernoulli random variables

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,..., n}.

On finite-time ruin probabilities for classical risk models

This paper examines the problem of ruin in the classical compound binomial and compound Poisson risk models. Our primary purpose is to extend to those models an exact formula derived by Picard & Lefèvre (1997) for the probability of (non-)ruin within finite time. First, a standard method based on the ballot theorem and an argument of Seal-type provides an initial (known) formula for that probability. Then, a concept of pseudo-distributions for the cumulated claim amounts, combined with some simple implications of the ballot theorem, leads to the desired formula. Two expressions for the (non-)ruin probability over an infinite horizon are also deduced as corollaries. Finally, an illustration within the framework of Solvency II is briefly presented.

Optimal Control of a Birth and Death Epidemic Process

We employ a birth and death process to describe the spread of an infectious disease through a closed population. Control of the epidemic can be effected at any instant by varying the birth and death rates to represent quarantine and medical care programs. An optimal strategy is one which minimizes the expected discounted losses and costs resulting from the epidemic process and the control programs over an infinite horizon. We formulate the problem as a continuous-time Markov decision model. Then we present conditions ensuring that optimal quarantine and medical care program levels are nonincreasing functions of the number of infectives in the population. We also analyze the dependence of the optimal strategy on the model parameters. Finally, we present an application of the model to the control of a rumor.

First crossing of basic counting processes with lower non-linear boundaries: A unified approach through pseudopolynomials (I)

The paper is concerned with the distribution of the level N of the first crossing of a counting process trajectory with a lower boundary. Compound and simple Poisson or binomial processes, gamma renewal processes, and finally birth processes are considered. In the simple Poisson case, expressing the exact distribution of N requires the use of a classical family of Abel-Gontcharoff polynomials. For other cases convenient extensions of these polynomials into pseudopolynomials with a similar structure are necessary. Such extensions being applicable to other fields of applied probability, the central part of the present paper has been devoted to the building of these pseudopolynomials in a rather general framework.

Distribution of the final extent of a rumour process

A rumour model due to Maki and Thompson (1973) is slightly modified to incorporate a continuous-time random contact process and varying individual behaviours in front of the rumour. Two important measures of the final extent of the rumour are provided by the ultimate number of people who have heard the rumour, and the total personal time units during which the rumour is spread. Our purpose in this note is to derive the exact joint distribution of these two statistics. That will be done by constructing a family of martingales for the rumour process and then using a particular family of Gontcharoff polynomials.

The moments of ruin time in the classical risk model with discrete claim size distribution

Firstly exact simple expressions are given for the moments Mr = E0(Tr 1{T λm and c ≤ λm are treated.