statistique, théorie et gestion de portefeuille - Docs at ISFA

9.1. Les différentes mesures de dépendances extrêmes 273
D Conditional Spearman’s rho
The conditional Spearman’s rho has been defined by
We have
ρs(˜v) =
Cov(U, V | V ≥ ˜v)
Var(U | V ≥ ˜v)Var(V | V ≥ ˜v) , (D.85)
1 1
˜v 0 · dC(u, v)
1 1 1
E[· | V ≥ ˜v] =
=
dC(u, v) 1 − ˜v ˜v 0
1
˜v
1
0
· dC(u, v) , (D.86)
thus, performing a simple integration by parts, we obtain
E[U | V ≥ ˜v] = 1 + 1
1
du C(u, ˜v) −
1 − ˜v 0
1
,
2
(D.87)
E[V | V ≥ ˜v]
E[U
=
1 + ˜v
,
2
(D.88)
2 | V ≥ ˜v] = 1 + 2
1
du u C(u, ˜v) −
1 − ˜v
1
,
3
(D.89)
which yields
so that
0
E[V 2 | V ≥ ˜v] = ˜v2 + ˜v + 1
, (D.90)
3
1 1
1
1 + ˜v 1
E[U · V | V ≥ ˜v] = + dv du C(u, v) + ˜v du C(u, ˜v) −
2 1 − ˜v
1
, (D.91)
2
Cov(U, V | V ≥ ˜v) =
ρs(˜v) =
Var(U | V ≥ ˜v) =
Var(V | V ≥ ˜v) =
1 − 4˜v + 24 (1 − ˜v) 1
˜v
0
1 1
1
dv du C(u, v) −
1 − ˜v ˜v 0
1
1
du C(u, ˜v) −
2 0
1
, (D.92)
4
1
1 − 4˜v 2
2˜v − 1
+ du u C(u, ˜v) +
12 (1 − ˜v) 2 1 − ˜v 0
(1 − ˜v) 2
1
du C(u, ˜v)
0
1
−
(1 − ˜v) 2
1
2
du C(u, ˜v) , (D.93)
(1 − ˜v)2
12
0
, (D.94)
12 1
1−˜v ˜v dv 1
0 du C(u, v) − 6 1
0
0 du u C(u, ˜v) + 12 (2˜v − 1) 1
0
0
du C(u, ˜v) − 3
du C(u, ˜v) − 12
E Tail dependence generated by the Student’s factor model
We consider two random variables X and Y , related by the relation
1
2
0 du C(u, ˜v)
(D.95)
X = αY + ɛ, (E.96)
35