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
r(x)
0.0 0.2 0.4 0.6 0.8 1.0
Total lapse rate func. (Pj) - price ratio
1.5 2.0 2.5 3.0
x_j
P 1
P 2
P 3
x_-j
r(x)
0.0 0.2 0.4 0.6 0.8 1.0
Total lapse rate func. (P1)
1.5 2.0 2.5 3.0
x_j
price ratio
price diff
x_-1
r(x)
Figure 2.5: Total lapse rate functions
0.0 0.2 0.4 0.6 0.8 1.0
Total lapse rate func. (Pj) - price diff
Derivatives of higher order use the same notation principle.
The lg function has the good property to be infinitely differentiable. We have
∂ lg j
j (x)
= −
∂xi
∂
⎛
⎝
∂xi
e
l=j
fj(xj,xl)
⎞
⎠
1
1 +
efj(xj,xl) 2 .
Since we have
∂
e
∂xi
fj(xj,xl)
= δji
we deduce
∂ lg j
j (x)
∂xi
l=j
= −δji
l=j
l=j
f ′ j1 (xj, xl)e fj(xj,xl)
l=j
1.5 2.0 2.5 3.0
f ′ j1(xj, xl)e fj(xj,xl) + (1 − δji)f ′ j2(xj, xl)e fj(xj,xi) ,
1 +
efj(xj,xl) 2 − (1 − δji)f ′ j2(xj, xl)
1 +
e f ′ j1 (xj,xl)
2 .
l=j
x_j
f ′ j1 (xj, xi)
This is equivalent to
∂ lg j
j (x)
⎛
= − ⎝
∂xi
f ′ j1(xj, xl) lg l ⎞
j(x) ⎠ lg j
j (x)δij − f ′ j2(xj, xl) lg i j(x) lg j
j (x)(1 − δij).
Furthermore,
∂ lg j
j (x)
∂xi
and also
lg j ∂
j (x)
∂xi
126
l=j
fj(xj,xk)
δjk + (1 − δjk)e
⎛
= − ⎝
f ′ j1(xj, xl) lg l ⎞
j(x) ⎠ lg k j (x)δij−f ′ j2(xj, xi) lg i j(x) lg k j (x)(1−δij).
l=j
fj(xj,xk)
δjk + (1 − δjk)e = lg j
j (x)(1−δjk)
δikf ′ j2(xj, xk)e fj(xj,xk)
+ δijf ′
fj(xj,xk)
j1(xj, xk)e .
l=j
P 1
P 2
P 3
x_-j