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
Getting back the original definition, we have
∆i,0 = P (W1 ≤ i, W2 ≤ 0) = P (W1 ≤ i, W2 ≤ 0) = 1 − LΘ(i + 1) − LΘ(1) + LΘ(i + 2).
Since LΘ is convex, the sequence LΘ(i + 2) − LΘ(i + 1) is an increasing sequence. We conclude
that (∆i,1)i is increasing, so the maximum is attained at +∞. Therefore,
max
i,j ∆i,j = ∆+∞,0 = P (W1 ≤ +∞, W2 ≤ 0) = P (W2 = 0).
On Figure 4.2, we investigate the numerical differences with a given distribution for the
latent variable. We consider Θ follows a gamma distribution Ga(2, 4). In other words, we
assume
α λ
LΘ(t) = ⇔ L
λ + t
−1
Θ (z) = λ(z−1/α − 1),
with α = 2, λ = 4. We plot the unique continuous copula CY1,Y2 and the distribution function
, left-hand and right-hand graphs, respectively.
DW1,W2
1.0
0.8
C(u,v)
0.6
0.4
0.2
Continuous case - Gamma G(2,4)
0.0
0.0
0.2
0.4
u
0.6
0.8
1.00.0
0.2
(a) Continuous case
0.4
0.6
v
1.0
0.8
Figure 4.2: Comparison of copula CY1,Y2
1.0
0.8
H(u,v)
0.6
0.4
0.2
Discrete case - Gamma G(2,4)
0.0
0.0
0.2
0.4
u
0.6
0.8
1.00.0
(b) Discrete case
0.2
0.4
0.6
v
1.0
0.8
and distribution function DW1,W2
CY1,Y2 passes through all the top-left corners of the stairs described by HW1,W2 . The grid of
points used for the graphs is {(FW1 (n), FW2 (m)), n, m ∈ N}. As n increases, the distribution
function FW1 (n) tends 1, so there infinitely many stairs towards the point (1, 1). Graphically,
the maximal stairstep corresponds to (u, v) = (1, FW2 (0)) or (FW1 (0), 1) and ∆∞,0 = 0.36.
The maximal stairstep is also the maximum difference between copula CY1,Y2 and distribution
function DW1,W2 .
4.4.3 Non-identifiability issues
The function DW1,W2 is not a copula but only a distribution function. The maximal
stairstep leads to the question of the maximal differences between two copulas satisfying
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