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320 9. Mesure de la dépendance extrême entre deux actifs financiers
Proof :
Pr{X > FX −1 (u), Y > FY −1 (u)} =
E 1 {X>FX −1 (u)} · 1 {Y >FY −1
(u)}
= E E 1 {X>FX −1 (u)} · 1 {Y >FY −1 (u)} |Y
= E 1 {Y >FY −1
(u)} · E 1 {X>FX −1 (u)} |Y
= E 1 {Y >FY −1
(u)} · E 1 {ε>FX −1
=
(u)−βY }
E 1 {Y >FY −1 (u)} · ¯ Fε(FX −1
(u) − βY )
Assuming that the variable Y admits a density PY with respect to the Lebesgue measure, this
yields
Pr{X > FX −1 (u), Y > FY −1 ∞
(u)} =
F −1
Y (u)
(34)
(35)
(36)
(37)
(38)
dy PY (y) · ¯ Fε[FX −1 (u) − βy] . (39)
Performing the change of variable y = FY −1 (u) · x, in the equation above, we obtain
Pr{X > FX −1 (u), Y > FY −1 (u)} = F −1
Y (u)
∞
1
dx PY (F −1
and, dividing by ¯
FY FY −1 (u) = 1 − u, this concludes the proof.
Let us now define the function
fu(x) =
We can state the following result
−1
FY (u)
1 − u PY (F −1
Y (u) x) · ¯ Fε[FX −1 (u) − βF −1
Y
Lemma 2
Under assumption H1 and H3, for all x ∈ [1, ∞),
almost everywhere, as u goes to 1.
fu(x) −→ 1x> l
Proof: Let us apply the assumption H1. We have
lim
u→1
F −1
Y (u)
Applying now the assumption H3, we have
lim
u→1 FX −1 (u) − βF −1
Y
Y (u) x) · ¯ Fε[FX −1 (u) − βF −1
Y
(u) x] ,
(40)
(u) x] . (41)
β · f(x), (42)
1 − u PY (F −1
Y (u) x) =
t PY (t x)
lim
t→∞ FY
¯
,
(t)
(43)
= f(x). (44)
(u) x = lim
=
21
u→1
βF −1
Y (u)
−∞ if x > l
β ,
∞ if x < l
β ,
FX −1 (u)
βF −1 − x
Y (u)
(45)
(46)
(47)