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
where γ(., .) denotes the incomplete lower gamma function. When x tends to infinity, only
the term γ(b + 1, ˜x) changes and tends to Γ(b + 1). With the integral I2(λ, k − 1, α − 1, +∞),
the resulting claim distribution has mass probability function
k−1
P (X = k) = qδk0 + (1 − δk0)(1 − q)
j=0
C j
k−1 (−1)j λα .
(λ + j) α
Similarly with I2(λ, k, α − 1, +∞), the survival function is given by
P (X > k) = (1 − q)
k
j=0
k
(−1)
j
j
λα .
(λ + j) α
Using I2(λ − 1, u + 1, α − 1, θ0), the ruin probability can be deduced.
Proposition 4.2.2. Let us consider the discrete time framework of Subsection 4.2.2 with a
latent variable Θ gamma distributed Ga(α, λ).
ψ(u) =
Γ(α, λθ0)
Γ(α)
1 − q
+
qu+1 u+1
u + 1
(−1)
j
j γ(α, θ0(λ
α + j − 1)) λ
,
Γ(α) λ + j − 1
j=0
with λ > 1, θ0 = − log(1 − q) and for u ≥ 0.
Finally, consider Θ is Lévy distributed Le(α). We use the integral
x
I3(a, n, b, x) = e −θ a
1 − e −θ n
θ −3/2 b
−
e θ dθ.
Using the change of variable, we have
n
n
I3(a, n, b, x) = (−1)
j
n−j 2
j=0
0
∞
˜x
a+n−j
−
e y2 e −by2
dy.
with ˜x = x−1/2 . This integral is linked to the generalized incomplete upper gamma function.
Using Appendix 4.6.4, we get
n
n
I3(a, n, b, x) = (−1)
j
j=0
j
√
π
2 √
e
b
2√ √
b(a+j) b
erfc √x + a + j √
x
√
b
√x − a + j √
x .
When x tends to infinity, we have
I3(a, n, b) =
n
j=0
+e −2√ b(a+j) erfc
n
(−1)
j
j
√
π
√b e −2√b √ a+j
.
Using I3(0, k − 1, α 2 /4) and I3(0, k, α 2 /4), the mass probability and survival functions are
given by
k−1
k − 1
P (X = k) = (1 − q)
(−1)
j
j e −α√ k
j k
and P (X > k) = (1 − q) (−1)
j
j e −α√j .
180
j=0
j=0