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

tel-00703797, version 2 - 7 Jun 2012
By successive conditioning, we get
P (Card(It) > 1) = P (Card(I0) > 1)
t
P (Card(Is) > 1|Card(Is−1) > 1) <
So, the probability P (Card(It) > 1) decreases geometrically as t increases.
s=1
2.4. Dynamic framework
1 − ˜ t ξ .
Proposition 2.4.2. For the repeated I − player insurance game defined in the previous subsection,
if for all k = j, xj ≤ xk and xj(1 − ej) ≤ xk(1 − ek), then the underwriting result by
policy is ordered UWj ≤icx UWk where UWj is the random variable
UWj = xj(1 − ej) − 1
Nj(x)
Yi.
Nj(x)
Proof. Let us consider a price vector x such that xj < xk for all k = j. Since the change
probability pk→j (for k = j) is a decreasing function (see Appendix 2.6.1), pk→j(x) > pk→l(x)
for l = j given the initial portfolio sizes nj’s are constant.
Below we use the stochastic orders (≤st, ≤cx) and the majorization order (≤m) whose
definitions and main properties are recalled in the Appendices 2.6.2 and 2.6.2 respectively.
Using the convolution property of the stochastic order I times, we can show a stochastic order
of the portfolio size
Nk(x) ≤st Nj(x), ∀k = j.
Let us consider the underwriting result per policy
uwj(x, n) = 1
n
nxj(1 − ej) −
n
i=1
Yi
i=1
= xj(1 − ej) −
n
i=1
1
n Yi,
for insurer j having n policies, where Yi denotes the total claim amount per policy.
Let n < ñ be two policy numbers and añ, an ∈ Rñ be defined as
⎛
⎞
1 1
⎜
añ = , . . . , and an = ⎜
1 1 ⎟
ñ ñ
⎝ , . . . , , 0, . . . , 0 ⎟
n n
⎠
size ñ−n
size n
.
Since añ ≤m an and (Yi)i’s are i.i.d. random variables, we have
i añ,iYi
≤cx i an,iYi i.e.
ñ
i=1
1
ñ Yi ≤cx
n
i=1
1
n Yi.
For all increasing convex functions φ, the function x ↦→ φ(x + a) is still increasing and
convex. Thus for all random variables X, Y such that X ≤icx Y and real numbers a, b, a ≤ b,
we have
E(φ(X + a)) ≤ E(φ(X + b)) ≤ E(φ(Y + b)),
i.e. a + X ≤icx b + Y .
As xj(1 − ej) ≤ xk(1 − ek) and using the fact that X ≤cx Y is equivalent to −X ≤cx −Y ,
we have
uwj(x, ñ) ≤icx uwk(x, n), ∀k = j.
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