statistique, théorie et gestion de portefeuille - Docs at ISFA

410 13. Gestion de portefeuille sous contraintes de capital économique
and
e −x ∗ x∗ i
i +h i
dtt
x ∗ i
du |f ′′ (u)−f ′′ (x ∗ i )| ≤ e −x ∗
x ∗ i
i +h i
since whatever the sign of hi, the quantity x ∗ i +hi
x ∗ i
dtt
x ∗ i
du [f ′′ (u)−f ′′ (x∗ i )] ≤ ex ∗ i +hi x∗ i
dt t
x∗ du |f
i
′′ (u) − f ′′ (x∗ i
dtt
x∗ du |f
i
′′ (u)−f ′′ (x∗ i )| ,
(117)
)| remains always positive.
But, |u − x∗ i | ≤ |hi| ≤ C
f ′′ (x∗ i ), which leads, by the mean value theorem and assumption 1, to
which yields
Thus
|f ′′ (u) − f ′′ (x ∗ i )| ≤ sup
ξ∈(x∗ i ,x∗ i +hi)
|f (3) (ξ)| · |u − x ∗ i |, (118)
0 ≤
1
− 2i
e sup |f (3) (ξ)|
i
≤ sup
ξ∈(x∗ i ,x∗ i +hi)
|f (3) (ξ)| C
f ′′ (x∗ i ),
≤ sup |f
ξ∈Gi
(3) (ξ)| C
f ′′ (x∗ i ), where Gi =
x ∗
i +hi
x ∗ i
t
dt
x∗ du |f
i
′′ (u) − f ′′ (x ∗ i )| ≤ 1
2
i
C
f ′′ (x∗ i ) h2 i
≤ e −x ∗ i +hi x∗ i
dtt
x ∗ i
(119)
x ∗ i − C
f ′′ (x∗ i ), x∗i + C
f ′′ (x∗ i )
, (120)
sup
ξ∈Gi
|f (3) (ξ)|
du [f ′′ (u)−f ′′ (x∗ i )] 1
2i
≤ e
sup |f (3) (ξ)|
C
f ′′ (x ∗ i )h2 i . (121)
C
f ′′ (x ∗ i ) h2 i , (122)
where sup ξ∈Gi |f (3) (ξ)|, have been denoted by sup |f (3) (ξ)| in the previous expression, in order not to
cumber the notations.
By proposition 4, we know that for all h ∈ AC and all ɛi > 0
sup
|f
(3) (ξ)
f ′′ (x∗ i )
≤ ɛi, for x ∗ i large enough, (123)
so that
∀ɛ ′ C·ɛ′
i −
> 0 and ∀h ∈ AC, e 2 h2 i ≤ e −x ∗ x∗ i
for |x| large enough.
i +h i
Moreover, from proposition 3, we have for all ɛi > 0 and x∗ i large enough:
so, for all ɛ ′′ > 0
dtt
x∗ du [f
i
′′ (u)−f ′′ (x∗ i )] ≤ e C·ɛ′
i 2 h2 i , (124)
∀h ∈ AC, (1 − ɛi) ν ≤ g(x∗i + hi)
g(x∗ i ) ≤ (1 + ɛi) ν , (125)
∀h ∈ AC, (1 − ɛ ′′ ) Nν ≤
Then for all ɛ > 0 and |x| large enough, this yields :
i
g(x ∗ i
+ hi)
g(x∗ i ) ≤ (1 + ɛ′′ ) Nν . (126)
(1 − ɛ) Nν 1
−
e 2i (f ′′ (x∗ i )+C·ɛ)·h2 i ≤ g(xi)
g(x∗ i )e−f(xi) Nν −
≤ (1 + ɛ) e 1
2i (f ′′ (x∗ i )−C·ɛ)·h2 i , (127)
i
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