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
asymptotically
P (X > k) ∼
k→+∞ (1 − q)λα Γ(λ + 1) (log k)
Γ(α)
α−1
kλ+1 ,
which decreases slightly slower than a Zipf distribution due to the logarithm in the numerator.
When Θ follows a Lévy distribution Le(α), again we use an asymptotic of alternating
binomial sums with the function f2(z) = e −α√ z . Thus, the tail distribution is asymptotically
⎛
P (X > k) ∼ (1 − q)α ⎝
1
γe
√ −
k→+∞ π log(k)
2 π log 3 ⎞
⎠ ,
(k)
which decreases extremely slowly. Such a tail behaviour is heavier than for a Pareto distribution.
With continuous distributions, a similar behaviour is obtained for the log-Cauchy
distribution, for example.
Numerical illustrations
On Figure 4.1, we plot the tails of the distributions derived above. The exponentialgeometric
distribution has a very tractable survival function, since incomplete beta function is
available most softwares, e.g., in R via the pbeta function. Therefore, we can benchmark the
asymptotic with the true value. However, for the two other distributions, we have to compute
two binomial alternating sums. These sums are particularly unstable because the central term
C n/2
n reaches very quickly infinity, which drives the binomial alternating sum between +∞ or
−∞.
In modern computers, a real number is stored in eight bytes (i.e. 64 bits), but only
53 bits are reserved for the precision (see, e.g., http://en.wikipedia.org/wiki/Double-precision_
floating-point_format). In our numerical experiment, the alternating binomial sum Sn(φ)
becomes unstable for n ≥ 48 with the standard double precision. To compute the alternating
sum Sn(φ) for large n, we have no other option than to use high precision floating-point
arithmetic libraries such as the GMP library of Grandlund Torbjoern & the GMP Devel.
Team (2011) and the MPFR library of Fousse et al. (2011). Using GMP and MPFR libraries
allow us to a high number of bits, say 500 or 1000. Reliability of those libraries has been
established in Fousse et al. (2007). Those libraries are interfaced in R via the Rmpfr package
of Maechler (2012).
Figures 4.1a, 4.1b and 4.1c correspond to a mixing distribution when Θ is exponential,
gamma and Lévy-stable distributed, respectively. Note that all plots have log scale for x and
y axes. On Figures 4.1a, 4.1b, the distribution tail show a Pareto-type behavior, as we observe
a straight line. The Lévy stable mixing on Figure 4.1c shows clearly a heavier tail.
4.4 Focus on the dependence structure
This section studies the dependence structure of the dependent risk models described in
Subsections 4.2.1 and 4.2.2, respectively for discrete-time and continuous-time settings. Let
us start by recalling Property 2.1 of Albrecher et al. (2011) for the continuous-time model.
Proposition. When claim sizes fullfill for each n ≥ 1,
P (X1 > x1, . . . , Xn > xn|Θ = θ) =
n
e −θxi ,
i=1
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