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
Proof. Since for u, v ≤ q = FW1 (0), both functions DW1,W2 and DZ1,Z2 are zero, we have
max
i,j≥0 ∆i,j = max
i,j≥1 ∆i,j.
Using Lemma 4.4.5, the function x ↦→ LΘ(x) − (1 − q)LΘ(x + c) is convex decreasing. Thus,
x ↦→ LΘ(x) − (1 − q)LΘ(x + c) is concave increasing. We get back to the proof of Proposition
4.4.4. So, it follows that the maximal difference is the stairstep ∆∞,1.
Graphical comparison
As already said, functions DZ1,Z2 and DW1,W2 are not copula but only distribution functions.
We plot on Figure 4.4 these two functions when Θ is gamma distributed Ga(2, 4) with
parameter q = 0.3. The appropriate way to deal with non-identifiability issues is again to use
the interpolated copula C .
1.0
0.8
C(u,v)
0.6
Continuous case - Gamma G(2,4), q=0.3
0.4
0.2
0.0
0.0
0.2
0.4
u
0.6
4.5 Conclusion
0.8
1.00.0
(a) Continuous case
0.2
0.4
0.6
v
1.0
0.8
0.0
0.0
0.2
0.4
Figure 4.4: Comparison of distribution functions DZ1,Z2
1.0
0.8
H(u,v)
0.6
0.4
Discrete case - Gamma G(2,4), q=0.3
0.2
u
0.6
0.8
1.00.0
(b) Discrete case
0.2
0.4
0.6
v
and DW1,W2
This paper uses a new class of dependent risk models, where the dependence is based on a
mixing approach. Emphasis has been put on infinite time ruin probability asymptotics. This
paper validates the A + B/u rule suggested in Example 2.3 of Albrecher et al. (2011). The
asymptotic rule applies both when the claim amounts or the waiting times are correlated. In
the ruin literature, even when some dependence is added in the claim arrival process, e.g.,
a Markovian setting or a specific dependence for the claim severity and waiting time, the
decreasing shape of the ruin probability remains unchanged compared to the corresponding
independent case, either exponential e −γu or polynomial u −α . Hence, our particular mixing
approach, leading to A + B/u asymptotics, significantly worsens the situation for the insurer.
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1.0
0.8