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
P(X>x) - log scale
1e-07 1e-05 1e-03 1e-01
Survival function - exp mixing
1 10 100 1000 10000
x - log scale
(a) Exponential
exact
asympto
P(X>x) - log scale
1e-10 1e-08 1e-06 1e-04 1e-02
Survival function - gamma mixing
2 5 10 20 50 200 500 2000
x - log scale
(b) Gamma
asympto
exact
Figure 4.1: Survival functions
P(X>x) - log scale
0.15 0.20 0.25 0.30
Survival function - stable mixing
5 10 50 500 5000
x - log scale
(c) Lévy stable
then, they have a dependence structure due to an Archimedean survival copula with generator
, the inverse Laplace transform of Θ.
φ = L −1
Θ
Therefore, in continuous time, the dependence structure is simply an Archimedean copula.
Regarding the discrete-time setting, things are more complicated: the dependence among
discrete random variables is a complex topic. Genest and Neslehova (2007) present issues
linked to discrete copula. To better understand the problem, we recall the Sklar theorem, see,
e.g., Joe (1997); Nelsen (2006).
Theorem (Sklar). Let H be the bivariate distribution function of a random pair (X, Y ). There
exists a copula C such that for all x, y ∈ R,
exact
asympto
H(x, y) = C(FX(x), FY (y)). (4.9)
Furthermore, C is unique on the Cartesian product of the ranges of the marginal distributions.
As a consequence, the copula C is not unique outside the support of the random variables
X and Y . When X, Y are discrete variables in N, C is only unique on N 2 but not on R 2 \ N 2 .
The non-identifiability is a major source of issues. An example of discrete copulas is the
empirical copula for observation sample (Xi, Yi)1≤i≤n.
Some problems in the discrete case arise from the discontinuity of the distribution function
of discrete random variables. Let B be a Bernoulli variable B(p). The distribution function
FB, given by FB(x) = (1 − p) 1x≥0 + p 1x≥1, is discontinuous. Thus, the random variable
FB(B) is not a uniform variable U(0, 1).
Let us introduce Genest and Neslehova (2007)’s notation. Let A be the class of functions
verifying Equation (4.9) for all x, y ∈ R for a given FX and FY . Let us define the function
B as for all u, v ∈ [0, 1], B(u, v) = (F −1 −1
X (u), FY (v)). We also denote by D the distribution
function of the pair (FX(X), FY (Y )).
With a simple bivariate Bernoulli vector, Example 1 of Genest and Neslehova (2007) shows
that (i) functions B and D are different, (ii) B is not a distribution function despite both B
and D belong to the class A. Even in that simple support {0, 1} 2 , the identifiability issue of
the copula C cannot be discarded. Proposition 1 of Genest and Neslehova (2007) extends to
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