Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
Chapitre 4. Asymptotiques de la probabilité de ruine
with d+ = √ uθ0 + α/(2 √ θ0) and d− = √ uθ0 − α/(2 √ θ0). We deduce that
I(u, θ0) = θ0
√
u
α euθ0
1 − 1
α √
e
u
α√u erfc (d+)
+ 1 + 1
α √
e
u
−α√u erfc (d−) −
2
√ e
πuθ0
−uθ0−α2
/(4θ0)
.
By reordering the terms, we get the formula of Albrecher et al. (2011), which is
−αc/4λ
I(u, θ0) = e e (cα+2λ√u) 2 /4λc −1 + α √ u erfc (d+)
+e (cα−2λ√u) 2 /4λc 1 + α √ u erfc (d−) − 2α
.
πλ/c
4.6.3 For the discrete time model
Geometric distribution
In this subsection, we study the geometric distribution, its properties and its minor extensions.
Classic geometric distribution The geometric distribution G(p) is characterized by the
following probability (mass) function
P (X = k) = p(1 − p) k ,
where k ∈ N and 0 ≤ p ≤ 1. Note that it takes values in all integers N. Another definition of
the geometric distribution is p(1 − p) k−1 so that the random variable takes values in strictly
positive integers. In this case, we can interpret this distribution as the distribution of the first
moment we observe a specific event occuring with probability p in a serie of indepedent and
identically distributed Bernoulli trials.
The probability generating function is given by
p
GX(z) =
1 − (1 − p)z .
With this characterizition, it is straightforward to see that summing n geometric random
variates G(p) has a negative binomial distribution N B(n, 1−p), see, e.g., Chapter 5 of Johnson
et al. (2005). The first two moments can be derived E(X) = (1−p)/p and V ar(X) = (1−p)/p 2
when p > 0. In the following subsection, a net profit condition will force E(X) < 1 which is
quivalent to p > 1/2. Furthermore, we also have P (X > k) = (1 − p) k+1 .
Modified geometric distributions The geometric distribution will be used to model the
claim severity. It is very restrictive to model claims by a one-parameter distribution. Thus, we
introduce two modified versions of the classic geometric distributions: 0-modified geometric
distribution and 0,1-modified geometric distribution.
The principle is simple, we modify respectively the first and the first two probabilities of
the probability function. Firstly, we introduce the 0-modified geometric distribution. That is
X ∼ G(q, ρ) when
208
P (X = k) =
q if k = 0,
(1 − q)ρ(1 − ρ) k−1 otherwise.
,