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
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2.5
c(u,v)
Continuous copula density - Gamma G(2,4)
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(a) Continuous case
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c+(u,v)
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Templar copula density - Gamma G(2,4)
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u
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(b) Discrete case
Figure 4.3: Comparison of copula cY1,Y2 and interpolated c W1,W2
Ge(q, ρ) where the mixing is done over the parameter ρ. We want again to study the associated
dependence structure.
Continuous case
Let Yi, i = 1, 2 be conditionnaly independent exponential random variable Yi|Θ = θ ∼ E(θ)
and Ii be independent Bernoulli variable B(1 − q). We define Zi = IiYi. By conditioning on
the variable Ii, we get
P (Zi ≤ x) = 1 − (1 − q)LΘ(x),
and
P (Z1 ≤ x, Z2 ≤ y) = 1 − (1 − q)(LΘ(x) + LΘ(y)) + (1 − q) 2 LΘ(x + y),
for x, y ≥ 0. Let DZ1,Z2 be the distribution function of the pair (FZ1 (Z1), FZ2 (Z2)). We have
0 if u < q or v < q,
DZ1,Z2 (u, v) =
P (Z1 ≤ lu, Z2 ≤ lv) otherwise.
where for p ≥ q, lp = L −1
Θ ((1 − p)/(1 − q)). Hence, we get for u, v ≥ q,
DZ1,Z2 (u, v) = u + v − 1 + (1 − q)2 LΘ(lu + lv).
As q tends 0, we get back to the Archimedean continuous copula CY1,Y2 of the previous
subsection. Note that there is a jump when u or v equal q. Indeed, we have DZ1,Z2 (q, v) = qv
and DZ1,Z2 (u, v) ≥ q2 for u, v ≥ q.
Discrete case
If Y follows an exponential distribution E(θ) and I follows a Bernoulli distribution Ber(1−
q), then W = I⌈Y ⌉ follows a 0-modified geometric distribution G(q, 1 − e −θ ), where ⌈x⌉ is the
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