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

tel-00703797, version 2 - 7 Jun 2012
Chapitre 2. Theorie des jeux et cycles de marché
For the two considered price function, we have
f ′ xj
j2(xj, xl) = −αj
x2 l
and ˜ f ′ j2(xj, xl) = −˜αj,
which are negative. So, the function φl will also be a decreasing function.
Therefore the portfolio size xj ↦→ ˆ Nj(x) function has the following derivative
∂ ˆ ⎛
Nj(x)
= −nj ⎝
∂xj
f ′ j1(xj, xl) lg l ⎞
j(x) ⎠ lg j
j (x) + nlf ′ j2(xl, xj) lg j
l (x)(1 − lgj l (x)).
l=j
Hence, it is decreasing from the total market size
l nl to 0. So the function xj ↦→ ˆ Nj is both
a quasiconcave and a quasiconvex function.
Therefore, using the C 2 characterization of quasiconcave and quasiconvex functions, we
have that
∂ ˆ Nj(x)
∂xj
l=j
= 0 ⇒ ∂2 ˆ Nj(x)
∂x 2 j
Note that the function ˆ Nj(x) is horizontal (i.e. has gradient of 0) when xj → 0 and xj → +∞
for fixed x−j.
Finally, we also need
∂ 2 lg j
j (x)
∂x 2 j
= −
l=j
f ′ j1(xj, xl) ∂ lgl j
∂xj
as f ′′
j11 is 0 for the two considered functions. Since,
∂ lg l j
= − lg
∂xj
l=j
l j f
n=j
′ j1(xj, xn) lg n j + lg l j f ′ j1(xj, xl) and
then we get
∂2 lg j
j (x)
∂x2 = − lg
j
j
j
Convexity concepts
= 0.
lg j
j −
⎛
⎝
f ′ j1(xj, xl) lg l ⎞
∂ lgj
⎠ j
j ,
∂xj
l=j
∂ lgj
j
∂xj
⎛
= − ⎝
f ′ j1(xj, xl) lg l ⎞
⎠
j lg j
j ,
l=j
′
f j1(xj, xl) ⎛
2 l
lgj +2 ⎝
f ′ j1(xj, xl) lg l ⎞
⎠
j
Numerical applications for refined one-period game
l=j
2.6.2 On the dynamic game
Borel-Cantelli lemma and almost sure convergence
A random variable sequence (Xn)n is said to converge almost surely to X, if P (Xn →
X) = 1. A simple characterization of almost sure convergence is
128
Xn
l=j
p.s.
−→ X ⇔ P (∩n0≥0 ∪n≥0 |Xn − X| ≥ ɛ) = 0, ∀ɛ > 0.
2
lg j
j .