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This paper is concerned with the first crossing of an order statistic point process through general moving boundaries. Our purpose is to determine exact boundary crossing probabilities in both one and two boundary cases. Simple recursive methods are obtained that exploit an underlying algebraic structure of polynomial type. This structure is a direct consequence of the order statistic property. The proposed approach is easy to implement and efficient. Perspectives in statistics are also announced. [ABSTRACT FROM AUTHOR]

In risk management, the distribution of underlying random variables is not always entirely known. Sometimes, only the mean value and some shape information on the distribution (decreasingness, convexity …) are available. The present paper provides discrete convex extrema for several situations of this type. The starting point is the class of discrete distributions whose probability mass functions are nonincreasing on a support D n ≡ { 0 , 1 , … , n } . Convex extrema in that class of distributions are well-known. Our purpose is to point out how additional shape constraints of convexity type modify these extrema. Two cases are considered: the p.m.f. is globally convex on N , or it is convex only from a given point m ≥ 1 . The corresponding convex extrema are derived by using simple crossing properties between two distributions. Several applications to insurance problems are briefly presented. The results provide a complement to recent works by Lefèvre and Loisel [9,10] . [ABSTRACT FROM AUTHOR]

Recently, Lefèvre and Picard (Insur Math Econ 49:512-519, ) revisited a non-standard risk model defined on a fixed time interval [0, t]. The key assumption is that, if n claims occur during [0, t], their arrival times are distributed as the order statistics of n i.i.d. random variables with distribution function F( s), 0 ≤ s ≤ t. The present paper is concerned with two particular cases of that model, namely when F( s) is of linear form (as for a (mixed) Poisson process), or of exponential form (as for a linear birth process with immigration or a linear death-counting process). Our main purpose is to obtain, in these cases, an expression for the non-ruin probabilities over [0, t]. This is done by exploiting properties of an underlying family of Appell polynomials. The ultimate non-ruin probabilities are then derived as a limit. [ABSTRACT FROM AUTHOR]

Appell polynomials are known to play a key role in certain first-crossing problems. The present paper considers a rather general insurance risk model where the claim interarrival times are independent and exponentially distributed with different parameters, the successive claim amounts may be dependent and the premium income is an arbitrary deterministic function. It is shown that the non-ruin (or survival) probability over a finite horizon may be expressed in terms of a remarkable family of functions, named pseudopolynomials, that generalize the classical Appell polynomials. The presence of that underlying algebraic structure is exploited to provide a closed formula, almost explicit, for the non-ruin probability. [ABSTRACT FROM AUTHOR]

In the compound Poisson risk model, several strong hypotheses may be found too restrictive to describe accurately the evolution of the reserves of an insurance company. This is especially true for a company that faces natural disaster risks like earthquake or flooding. For such risks, claim amounts are often inter-dependent and they may also depend on the history of the natural phenomenon. The present paper is concerned with a situation of this kind, where each claim amount depends on the previous claim inter-arrival time, or on past claim inter-arrival times in a more complex way. Our main purpose is to evaluate, for large initial reserves, the asymptotic finite-time ruin probabilities of the company when the claim sizes have a heavy-tailed distribution. The approach is based more particularly on the analysis of spacings in a conditioned Poisson process. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the classical problem of how to evaluate the joint rank statistics distributions for two independent i.i.d. samples from a common continuous distribution. It is pointed that these distributions rely on an underlying polynomial structure of negative binomial type. That property is exploited to obtain, in a systematic and unified way, closed forms and simple recursions, some well established, for computing the joint tail and rectangular probabilities of interest. [Copyright &y& Elsevier]

An important question in insurance is how to evaluate the probabilities of (non-) ruin of a company over any given horizon of finite length. This paper aims to present some (not all) useful methods that have been proposed so far for computing, or approximating, these probabilities in the case of discrete claim severities. The starting model is the classical compound Poisson risk model with constant premium and independent and identically distributed claim severities. Two generalized versions of the model are then examined. The former incorporates a non-constant premium function and a non-stationary claim process. The latter takes into account a possible interdependence between the successive claim severities. Special attention will be paid to a recursive computational method that enables us to tackle, in a simple and unified way, the different models under consideration. The approach, still relatively little known, relies on the use of remarkable families of polynomials which are of Appell or generalized Appell (Sheffer) types. The case with dependent claim severities will be revisited accordingly. [ABSTRACT FROM AUTHOR]

In the context of the classical Poisson ruin model Gerber (1988a,b) and Shiu (1987, 1989) have obtained two formulae for the ruin and non ruin probabilities in infinite time. Here these two formulae are generalized to the case of an arbitrary premium process when all claims are integer-valued, as in Picard & Lefèvre (1997). Moreover, this generalization throws a new light on the two known formulae and it then leads very simply to a third new formula. [ABSTRACT FROM AUTHOR]

The present note first discusses the concept of s-convex pain functions in decision theory. Then, the economic behavior of an agent with such a pain function is represented through the comparison of some recursive lotteries. [ABSTRACT FROM AUTHOR]

This article introduces several classes of discrete stochastic orderings for comparing discrete random variables that are valued in an arbitrary ordered finite grid of non-negative points. These order relations correspond to particular cases of integral stochastic orderings which are generated by different classes of functions of convex/concave-type defined on the grid. The main question addressed in the article is how an extension of the grid of points can affect such stochastic orderings. It will be shown that a crucial factor is the location of the additional point that is inserted in the grid. Stochastic orderings have numerous uses in many areas of probability, statistics and decision theory. Orderings proper to discrete random variables X and Y have received much less attention. Now, a more general situation is when the random variables X and Y take on values in an arbitrary ordered finite grid of nonnegative points. Stochastic orderings specific for comparing such random variables have not yet been investigated so far. These orderings, however, are of direct interest in various fields of applications, especially for problems of risky decision making, of portfolio selection and of insurance premium evaluation.

A quantum mechanics approach is proposed to model non-life insurance risks and to compute the future reserve amounts and the ruin probabilities. The claim data, historical or simulated, are treated as coming from quantum observables and analyzed with traditional machine learning tools. They can then be used to forecast the evolution of the reserves of an insurance company. The following methodology relies on the Dirac matrix formalism and the Feynman path-integral method. [ABSTRACT FROM AUTHOR]