statistique, théorie et gestion de portefeuille - Docs at ISFA

322 9. Mesure de la dépendance extrême entre deux actifs financiers
where the only change in (59) compared to (33) is to replace ¯ Fε(·) by ¯ F −1
ε|Y =FY (u) x(·). Let us
now assume that the function ¯ Fε|Y =y(x) admits a uniform limit when x and y tend to ±∞. Then,
equation (48) still holds and lemma 2 remains true.
As an example, let F denote any one-dimensional distribution fonction. Then, one can easily check
that, for any conditional distribution whose form is
¯F ε|Y =y(x) = ¯
y2 F x , (60)
+ y2
the uniform limit condition is satisfied and theorem 1 and lemma 2 still hold. In contrast, conditional
distributions of the form
¯F ε|Y =y(x) = ¯ F (x − ρy) (61)
do not fulfill the uniform limit condition, so that the result given by theorem 1 does not hold.
The full understanding of the impact of more general dependences between the factor and the
idiosyncratic noise on the coefficient of tail dependence requires a full-fledge investigation that we
defer to a future work. Our goal here has been to show that one can reasonably expect our results
to survice in the presence of weak dependence.
A.2 Tail dependence between two assets
A.2.1 Statement
We consider three random variables X1, X2 and Y , related by the relations
y 2 0
X1 = β1 · Y + ε1 (62)
X2 = β2 · Y + ε2, (63)
where ε1 and ε2 are two random variables independent of Y and β1, β2 two non-random positive
coefficients.
Let PY and FY denote respectively the density with respect to the Lebesgue measure and the
distribution function of the variable Y . Let F1, (resp. F2) denotes the distribution function of
X1 (resp. X2) and Fε1 (resp. Fε2 ) the marginal distribution function of ε1 (resp. ε2). Let Fε1,ε2
denotes the joined distribution of (ε1, ε2). We state the following theorem:
Theorem 2
Assuming that
H0: The variables Y , ε1 and ε2 have distribution functions with infinite support,
H1: For all x ∈ [1, ∞),
lim
t→∞
t PY (tx)
¯
FY (t)
= f(x), (64)
H2: There are real numbers t0 > 0, δ > 0 and A > 0, such that for all t ≥ t0 and all x ≥ 1
¯FY (tx)
¯FY (t)
23
A
≤ , (65)
xδ