Processus de Lévy en Finance - Laboratoire de Probabilités et ...

168 CHAPTER 5. APPLICATIONS OF LEVY COPULAS
stands for u 2 , . . . , u d ):
˜H(∗) =
∫ H(b1 ,∗)
H(a 1 ,∗)
ψ ′ (t)dt
= (H(b 1 , ∗) − H(a 1 , ∗))
∫ 1
= (H(b 1 , ∗) − H(a 1 , ∗))ψ ′ +(0)
+ (H(b 1 , ∗) − H(a 1 , ∗))
0
∫ 1
0
ψ ′ (H(a 1 , ∗) + t(H(b 1 , ∗) − H(a 1 , ∗)))dt
{ψ ′ (H(a 1 , ∗) + t(H(b 1 , ∗) − H(a 1 , ∗))) − ψ ′ +(0)}dt.
Note that ψ ′ +(0), the right derivative of ψ at 0, is well defined because we have supposed that
d ≥ 2 and therefore ψ ′′ exists on (0, 1) and ψ ′ is increasing. In the right-hand side of the above
equation:
• The integrand is d−1-increasing by the induction hypothesis, because H(a 1 , ∗)+t(H(b 1 , ∗)−
H(a 1 , ∗)) is d − 1-increasing and grounded for every t and ˜ψ(t) := ψ ′ (t) − ψ ′ +(0) is an
increasing continuous function with ˜ψ(0) = 0 and nonnegative derivatives of orders up to
d − 1 on (0, 1).
• The second term is d − 1-increasing by Lemma 4.3,
• The first term is d − 1-increasing because H is,
hence ˜H is d − 1-increasing as sum of d − 1-increasing functions and the proof is complete.
One example of an absolutely monotonic (with positive derivatives of all orders) function
that maps [0, 1] into [0, ∞] is φ(x) =
x
1−x .
The following result allows to construct Lévy copulas on (−∞, ∞] d , analogous to the
Archimedean copulas (cf. [76]). It can be used to build parametric families of Lévy copulas
in arbitrary dimension, where the number of parameters does not depend on the dimension.
Theorem 5.2 (Archimedean Lévy copulas). Let φ : [−1, 1] → [−∞, ∞] be a strictly increasing
continuous function with φ(1) = ∞, φ(0) = 0, and φ(−1) = −∞, having derivatives of
orders up to d on (−1, 0) and (0, 1), and satisfying
Let
d d φ(e x )
dx d ≥ 0,
d d φ(−e x )
dx d ≤ 0, x ∈ (−∞, 0). (5.1)
˜φ(u) := 2 d−2 {φ(u) − φ(−u)}