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270 9. Mesure de la dépendance extrême entre deux actifs financiers
and applying the result given in eqation (B.60), we finally obtain
E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A] = (1 − ρ 2 )
which concludes the proof.
B.3 Conditioning on Y larger than v
The conditioning set is A = [v, +∞), thus
Pr{Y ∈ A | ν} = ¯ Tν(v) = ν ν−1
2
y∈A
Cν
ν
ν − 2 ·
Pr
+ O
vν
ν
Pr Y ∈ A | ν − p =
ν − p ¯
ν − p
Tν−p v =
ν
ν ν−p
2
ν
dy y · ty(y) =
ν − 2 tν−2
ν − 2
v =
ν
ν ν
ν
ν−2 Y ∈ A | ν − 2
Pr{Y ∈ A | ν}
, (B.65)
v −(ν+2)
, (B.66)
(ν − p) 1
2
Cν−p
+ O v
vν−p −(ν−p+2)
, (B.67)
2
√ ν − 2
Cν−2
+ O
vν−1 v −(ν−3)
(B.68) ,
where tν(·) and ¯ Tν(·) denote respectively the density and the Student’s survival distribution with ν degrees
of freedom and Cν is defined in (B.47).
Using equation (B.42), one can thus give the exact expression of ρ + v . Since it is very cumbersomme, we will
not write it explicitely. We will only give the asymptotic expression of ρ + v . In this respect, we can show that
Thus, for large v,
Var(Y | Y ∈ A) =
E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A] =
B.4 Conditioning on |Y | larger than v
ν
(ν − 2)(ν − 1) 2 v2 + O(1) (B.69)
ν 1 − ρ
ν − 2
2
ν − 1 v2 + O(1) . (B.70)
ρ + ρ
v −→
ρ2
ν−2
+ (ν − 1) ν (1 − ρ2 . (B.71)
)
The conditioning set is now A = (−∞, −v]∪[v, +∞), with v ∈ R+. Thus, the right hand sides of equations
(B.66) and (B.67) have to be multiplied by two while
dy y · ty(y) = 0, (B.72)
y∈A
for symmetry reasons. So the equation (B.70) still holds while
Thus, for large v,
Var(Y | Y ∈ A) =
ρ s v −→
ν
(ν − 2) v2 + O(1) . (B.73)
ρ
ρ2 + 1
ν−2
(ν−1) ν (1 − ρ2 )
32
. (B.74)