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 all x ∈ [x, x] I . Taking supremum and infimum on player j, we get
0 < p l = inf
j
lg j
l (xj− ) and sup lg
j
j
l (xj−
) = pl < 1.
Using the definition of portfolio size Nj,t(x) given in Subsection 2.2.1 as a sum of binomial
random variables Blj,t(x), we have
P (Nj,t(x) = mj|Nj,t−1 > 0, Card(It−1) > 1)
⎛
= P ⎝
Blj,t(x) = mj
l∈It−1
Nj,t−1
⎞
> 0, Card(It−1) > 1⎠
=
P (Blj,t(x) = ˜ml)
Therefore,
=
˜m1,..., ˜mI t−1 ≥0
s.t.
l ˜m l =m j
˜m1,..., ˜mI ≥0 t−1
s.t.
l ˜m l =mj
nl,t−1
l∈It−1
>
P (Card(It) = 0|Card(It−1) > 1) =
⎛
=
≥
Thus, we have
118
˜ml
l∈It−1
˜m1,..., ˜mI t−1 ≥0
s.t.
l ˜m l =m j
lg j
˜ml
l (x) 1 − lg j
l (x)
nl,t−1− ˜mj
l∈It−1
nl,t−1
P ⎝∀j ∈ It−1, Nj,t(x) ≥ 0, Kj,t−1 + Nj,t(x)x ⋆ j,t(1 − ej) <
⎛
≥ P ⎝∀j ∈ It−1, Nj,t(x) > 0, Kj,t−1 + Nj,t(x)x ⋆ j,t(1 − ej) <
m1,...,mI t−1 ≥0
s.t.
l m l =n
m1,...,mI t−1 >0
s.t.
l m l =n
l∈It−1
l∈It−1
> ξ
P (Card(It) > 1|Card(It−1) > 1) =
Pt (Nj,t(x) = mj) P
Pt (Nj,t(x) = mj) P
m1,...,mI t−1 >0
s.t.
l m l =n
l∈It−1
P
˜ml
p ˜ml
l (1 − p l) nl,t−1− ˜mj = ξ > 0
Nj,t(x)
i=1
Yi
Nj,t(x)
i=1
⎞
⎠
Yi
⎞
⎠
Kj,t−1 + mjx ⋆ mj
j,t(1 − ej) < Yi
i=1
Kj,t−1 + mjx ⋆ mj
j,t(1 − ej) < Yi
i=1
Kj,t−1 + mjx ⋆ mj
j,t(1 − ej) < Yi
i=1
1 − P (Card(It) = 0|Card(It−1) > 1) − P (Card(It)1|Card(It−1) > 1)
> 0
≤ 1 − P (Card(It) = 0|Card(It−1) > 1) < 1 − ˜ ξ < 1.