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

tel-00703797, version 2 - 7 Jun 2012
when k ∈ N. Using the Kronecker product δij, it can be rewritten as
P (X = k) = qδ0,k + (1 − q)(1 − δ0,k)ρ(1 − ρ) k−1 .
4.6. Appendix
The expectation and variance are given by E(X) = (1 − q)/ρ and V ar(X) = (1 − q)(1 − ρ +
q)/ρ 2 . We also have
P (X > k) = (1 − q)(1 − ρ) k .
We get back to the classic geometric distribution with ρ = q.
Secondly, we present the 0,1-modified geometric distribution: X ∼ G(q, ρ, 1 − α)
which is stricly equivalent to
⎧
⎨
q if k = 0,
P (X = k) =
(1 − q)ρ
⎩
(1 − q)(1 − ρ)(1 − α)α
if k = 1,
k−2 otherwise.
P (X = 0) = q = 1 − p and P (X = k/X > 0) = ρδk,1 + (1 − ρ)(1 − α)α k−2 (1 − δk,1).
The mean and the variance are given by
1 − ρ
E(X) = p 1 +
1 − α
3 − α
and V ar(X) = qρ + (1 − q)(1 − ρ)
(1 − α)
We get back to the 0-modified geometric distribution with α = 1 − ρ.
4.6.4 Error function linked terms
2 − p2
We want to compute the following integral, linked to the error function
J(a, b, x) =
∞
x
e −ay2 −b/y 2
dy,
,
2 1 − ρ
1 + .
1 − α
where a, b, x > 0. The SAGE mathematical software (Stein et al. (2011)) suggests to do a
change of variable in order to get e−t2dt. Since the equation t2 = ay2 + b/y2 does not have
a unique solution, we consider
t = ± √ ay + √ b/y.
This leads to split the integral J(a, b, x). With algebric manipulations, we get
Therefore,
with ˜x1 = √ ax +
J(a, b, x) =
2 √ ady = √ ady +
2 √ aJ(a, b, x) = e 2√ ∞
ab
√
b
−y2 dy + √ √
b
ady − dy.
−y2 ˜x1
√
b
x and ˜x2 = √ √
b ax − x . Hence
√ π
4 √ a
e 2√ ab erfc
√ax +
e −t2
dt + e −2√ab √
b
x
∞
˜x2
+ e −2√ ab erfc
e −t2
dt,
√
√ax b
− .
x
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