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

tel-00703797, version 2 - 7 Jun 2012
Strictly convex
Convex
Strictly
pseudoconvex
Under
differentiability
Pseudoconvex
Under
differentiability
Strongly
quasiconvex
Strictly
quasiconvex
Under lower
semicontinuity
2.6. Appendix
Quasiconvex
Figure 2.6: Convexity and generalized convexity, from Bazaraa et al. (2006)
x ⋆ 1 x ⋆ 2 x ⋆ 3 ||x ⋆ − a|| 2 ||x ⋆ − m|| 2 ∆ ˆ N1 ∆ ˆ N2 ∆ ˆ N3 Nb
1.4048 1.0537 1.0642 0.1413 0.0802 -2583 1357 1226 11
1.507 1.1725 1.5968 0.2059 0.3061 -1609 2695 -1086 128
1.5527 1.6031 1.2504 0.2267 0.3525 -1359 -1217 2576 106
1 1.4896 1.5299 0.1449 0.2675 3127 -1790 -1338 56
2.4177 2.4107 2.3994 4.0024 4.6571 -83 -21 104 542
3 2.8239 2.8087 7.9441 8.838 -1191 509 683 38
1.5495 1.2359 3 3.0797 3.5277 -1182 3479 -2297 2
1.6611 3 1.3993 3.172 3.6768 -616 -3197 3813 6
3 3 2.9022 8.812 9.7701 -305 -238 543 2
3 2.9326 3 8.912 9.877 -219 321 -102 6
2.8877 2.874 3 8.3306 9.273 175 264 -439 11
2.8559 3 2.8279 8.0789 9.0088 259 -759 500 20
1 1.0214 3 3.065 3.4201 1277 1023 -2299 1
3 1.7906 1.5134 3.5479 3.8892 -4498 -357 4856 9
3 3 1.7839 6.436 7.1316 -4487 -3167 7654 1
Table 2.12: Premium equilibria - PLN with price difference function
Lemma (Borel–Cantelli). Let Bn
be a sequence of events on a probability space. If the serie
n P (Bn) is finite
n P (Bn) < +∞, then
P (∩n0≥0 ∪n≥0 Bn) = 0.
An application of the Borel-Cantelli lemma to almost sure convergence is
∀ɛ > 0,
p.s.
P (|Xn − X| ≥ ɛ) < +∞ ⇒ Xn −→ X.
Notation and definition of classic stochastic orders
n
Using the notation of Shaked and Shanthikumar (2007), we denote by ≤st the stochastic
order, which is characterised as X ≤st Y if ∀x ∈ R, P (X > x) ≤ P (Y > x). They are
129