Processus de Markov, de Levy, Files d'attente, Actuariat et Fiabilité ...

Chapitre 4
The Cramér-Lundberg and Levy risk
models
4.1 Introduction
The problem of approximating the distribution of sums of independent random variables
and the related ”gambler’s ruin” problem, first studied by Bernoulli, DeMoivre, Fermat,
Huyghens, Laplace, Lagrange, Pascal, etc., are masterpieces of applied probability. The second
problem, also known as ”first passage problem”, is central to numerous applications in
actuarial science, telecommunications, mathematical finance, etc., and has stimulated many
illustrious mathematicians, among which are Thiele (differential equation, interpolation formula),
Gram (Gram-Charlier expansion), Lundberg, Cantelli, Cramér, Erlang, de Finetti,
Segerdahl, Wald and Thorin.
For extensive references, see e.g. Borovkov (1976), Prabhu (1997), Rolski et al. (1999),
and Asmussen (2000), (2003).
The Cramer-Lundberg risk model. The ”bird’s eye’s view” of the ”actuarial ruin
problem” is captured by the so called ”Cramér Lundberg risk model” (?) Lun :
N(t)
∑
X(t) = u + c t − C(t) := u + c t − C i where : (4.1) CLmod
– X(0) = u denotes the initial reserves of the risk process
– c is the premium rate per unit of time.
– The IID nonnegative RVs C i with CDF B(u) model the ”claims”, N(t) is an independent
Poisson process with intensity λ > 0 counting the number of claims, and the
compound Poisson process C(t) = ∑ N(t)
i=1 C i models the ”cumulative claims”.
Remarque 4.1.1 The case of practical interest is when the net profit rate p = c − EC(1) is
positive. The relative profit with respect to the expenses
i=1
θ = c − EC(1)
EC(1)
= p
EC(1)
is called relative safety loading. The premium rate may be decomposed as
c = EC(1) + p = (1 + θ) EC(1).
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