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
The usual way to achieve this is to take a series expansion of f around x0 as g. f is said to
take a series expansion at x0 if for all N ∈ N, we have
f(x) = x0
N
anφn(x) + o(φN(x)) ,
n=0
where (φn)n is a sequence of so-called gauge functions such that ∀n ∈ N, φn+1(x) = o(φn(x))
around x0. This condition is equivalent to
N−1
f(x) − anφn(x) =
x0
n=0
O(φN(x)) ,
for all N ∈ N. In this paper, we also use the following notation for a series expansion of f at
x0
f(x) ∼ x0
+∞
n=0
anφn(x).
When x0 ∈ R, we generally choose φn(x) = (x − x0) n , whereas for x0 = +∞ we use
φn(x) = x −n . Integration by part is a standard tool to study integral asymptotics and derive
asymptotics, as pointed in Olver et al. (2010). Integration by part will be extensively used in
the next two subsections. In Appendix 4.6.5, we recall two integration by part theorems.
4.3.2 Continuous time framework
In this subsection, we present and show the A + B/u rule for the continuous time model.
Theorem 4.3.1. Let us consider the continuous time framework of Subsection 4.2.1 with a
positive latent variable Θ and θ0 = λ/c.
(i) For all u > 0, the ruin probability is bounded
182
ψ(u) ≤ FΘ(θ0) + 1
u
FΘ(θ0)
× .
θ0
(ii) If Θ has a continuous distribution with density fΘ such that fΘ is almost everywhere
differentiable on [θ0, +∞[ and f ′ Θ being a Lebesgue-integrable, then we have
ψ(u) = FΘ(θ0) + fΘ(θ0)
1
+ o .
u u
(iii) If in addition fΘ is Ck-1 almost everywhere on [θ0, +∞[ and f (k)
Θ
and bounded on [θ0, +∞[, then we have
k−1
ψ(u) = FΘ(θ0) +
i=0
where h(x) = θ0fΘ(x + θ0)/(x + θ0), so that
h (i) (0) =
i
(−1) j
j=0
h (i)
(0) 1
+ o
ui+1 uk
,
i!
(i − j)!θ j f
0
(i−j)
Θ (θ0).
is Lebesgue integrable