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
for z ∈ C.
For large value of z and fixed a, from page 179 of Olver et al. (2010), we have the following
asymptotics
Γ(a, z) ∼ z
z>>a a−1 e −z
a − 1
1 +
z
un
+ · · · + + . . . ,
zn where un = (a − 1)(a − 2) . . . (a − n) for | arg(z)| < 3π/2. For large value of z, no expansion
is needed for γ(a, z) since we have γ(a, z) ∼ Γ(a).
Beta function Using the Stirling formula, the beta function β(x, y) can be approximated
for large values x and y fixed. We recall β(x, y) = Γ(x)Γ(y)/Γ(x + y). We have
Γ(x) ∼
x→+∞ e−xx x
2π gk
and Γ(x + y) ∼
x xk x→+∞
k≥0
e−x−y (x + y) x+y
2π gk
,
x + y xk k≥0
where the term e−x−y (x + y) x+y
2π
x+y ∼ +∞ e−xxx+y
2π
x . Therefore we obtain
where the coefficients dk are given by
Γ(y)
β(x, y) ∼
x→+∞ xy dk
,
xk k≥0
d0 = 1 and dk = gk −
k−1
m=0
dmgk−m.
From Olver et al. (2010), page 184, we have asymptotics for the incomplete beta ratio
function
Iβ(a, b, x) =
β(a, b, x)
β(a, b)
∼
a→+∞ Γ(a + b)xa
b−1
(1 − x)
k≥0
k 1
x
,
Γ(a + k + 1)Γ(b + k) 1 − x
for large values a and fixed values of b > 0, 0 < x < 1. Multplying by β(a, b), we get
β(a, b, x) ∼
a→+∞ xa
k b−1 Γ(a)Γ(b) x
(1 − x) ,
Γ(a + k + 1)Γ(b + k) 1 − x
k≥0
with x ≤ 1 and a → +∞.
Finally, to get the asymptotic of ¯ β(a, b, x) for large values of b, we use ¯ β(a, b, x) = β(b, a, 1−
x). We have
206
¯β(a, b, x) ∼
b→+∞ (1 − x)b−1
a Γ(b)Γ(a) 1 − x
x
Γ(b + k + 1)Γ(a + k) x
k≥0
k
,