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278 9. Mesure de la dépendance extrême entre deux actifs financiers
Putting together equations (E.126) and (E.137) we obtain
∞
1
dx
ɛ
1
1
xν Cν
which concludes the proof.
1 + 1
ν
E.4 Derivation of equation (E.105)
From equation (E.111), we can deduce
¯FY (y) =
Using equations (E.98) and (E.102), we obtain
where
x−x0
ɛ
2 ν+1
2
Pɛ(α ˜ Yu + η − αy) = Pɛ(γ ˜ Yu − αy) ·
− 1
xν
= O(ɛ
0
min{ν,1} ) , (E.138)
ν−1
ν 2 Cν
yν −2
1 + O(y ) . (E.139)
γ = α 1 +
Putting together these results yields for the leading order
∞
dy ¯ FY (y) · Pɛ(α ˜ Yu + η − αy) =
˜Yu
= ν ν−1
1 + O( ˜ Y −2
u ) , (E.140)
σ
ν1/ν . (E.141)
α
∞
dy
˜Yu
2 Cν
ν
α ˜ Yu
ν−1
ν 2 Cν
yν ·
∞
1
σ
dx 1
·
xν where the change of variable x = y
has been performed in the last equation.
˜Yu
Cν
1 + (γ ˜ Yu−αy) 2
ν σ 2
1 + 1
ν
Cν α ˜ Yu
σ
x− γ
α
σ
α ˜ Yu
ν+1
2
2 ν+1
2
, (E.142)
We now apply lemma 2 with x0 = γ
σ
α > 1 and ɛ =
α ˜ which goes to zero as u goes to 1. This gives
Yu
which shows that
thus
which finally yields
∞
dy ¯ FY (y) · Pɛ(α ˜ Yu + η − αy) ∼u→1
˜Yu
Pr[X > F −1
−1
X (u), Y > FY (u)] ∼u→1 F −1
Y ( ˜ Yu)
Pr[X > F −1
X
−1
(u)|Y > F (u)] ∼u→1
λ =
Y
1
1 + σ
α
40
ν ν−1
2 Cν
α ˜ Y ν u
ν α
γ
ν α
= (1 − u)
γ
, (E.143)
, (E.144)
ν α
, (E.145)
γ
ν α
, (E.146)
γ
ν . (E.147)