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

tel-00703797, version 2 - 7 Jun 2012
4.4. Focus on the dependence structure
any bivariate pair (X, Y ): B is not a distribution, whereas D is a distribution function but
not a copula.
Let us now consider two exponential random variables that are conditionally independent
given a factor Θ. We choose to focus here only on the bivariate case. Suppose that these
variables are discretized by taking their integer parts. We will see that the two resulting
random variables have geometric distributions; these are, of course, correlated. Our purpose
in Subsections 4.4.1 - 4.4.4 below is precisely to study the dependence involved.
4.4.1 Dependence induced by the mixing approach
We start by comparing the joint distributions of two mixed exponential random variables
and of geometric approximations obtained by discretization.
Continous case
Now, consider Yi, i = 1, 2 to be conditionnaly independent exponential random variables,
i.e. Yi|Θ = θ i.i.d.
∼ E(θ). We have
FY1 (x) = P (Y1 ≤ x) =
∞
P (Y1 ≤ x|Θ = θ)dFΘ(θ) = 1 − LΘ(x),
0
where LΘ stands for the Laplace transform of the random variable Θ, assuming LΘ exists.
Using this formulation, we can check that the random variable FYi (Yi) is uniformly distributed.
Indeed we have
P (FY1 (Y1) ≤ u) = P (Y1 ≤ L −1
Θ
−1
(1 − u)) = 1 − LΘ(LΘ (1 − u)) = u.
Furthermore, the (unique) copula of the couple (Y1, Y2) can be derived by
CY1,Y2 (u, v) = P (FY1 (Y1) ≤ u, FY2 (Y2) ≤ v) = P (Y1 ≤ L −1
=
∞
0
(1 − e −θL−1
Θ (1−u) )(1 − e −θL−1
Θ (1−v) )dFΘ(θ).
Hence the copula function is given by
CY1,Y2 (u, v) = u + v − 1 + LΘ
Θ (1 − u), Y2 ≤ L −1
Θ
(1 − v))
−1
LΘ (1 − u) + L−1
Θ (1 − v) . (4.10)
We retrieve the fact that the couple (Y1, Y2) has an Archimedean survival copula with generator
L −1
Θ , see, e.g., Albrecher et al. (2011). In other words, the joint distribution of the tail is given
by
P ( ¯ FY1 (Y1) > u, ¯ FY2 (Y2)
−1
> v) = LΘ LΘ (u) + L−1
Θ (v) .
From Theorem 4.6.2 of Nelsen (2006), the above expression can be extended to any dimension.
A n-dimension function C(u1, . . . , un) defined by a generator φ and C(u1, . . . , un) =
φ−1 (φ(u1) + · · · + φ(un)) is a n-copula for all n ≥ 2 if and only if the generator φ−1 is
completely monotone, i.e. for all k ∈ N,
(−1) k dk φ −1
dt k (t) ≥ 0.
As the generator is φ−1 (t) = LΘ(t) in our model, then the Laplace transform is completely
monotone. In particular, LΘ is a decreasing convex function. As φ is the generator of an
Archimedean copula, φ is a convex decreasing function. Since φ(t) = L −1
Θ (t), L−1
Θ is also a
convex decreasing function.
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