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

4.4. LEVY COPULAS: SPECTRALLY POSITIVE CASE 149
The following properties of Ũ (−1)
k
follow directly from its definition:
Ũ (−1)
k
(t) is nonincreasing in t,
Ũ (−1)
k
(∞) = 0, (4.17)
Ũ k (Ũ (−1)
k
(t)) = t ∀t ∈ Ran Ũk. (4.18)
For every (x 1 , . . . , x d ) ∈ [0, ∞] d , we define
⎧
⎪⎨
0, x k = ∞ for some k
Ũ(x 1 , . . . , x d ) =
⎪⎩ U(x 1 , . . . , x d ), otherwise.
Finally, introduce a measure ˜ν on [0, ∞] d \{0} by ˜ν(B) := ν(B ∩R d +) for all B ∈ B([0, ∞] d \{0}).
Clearly, Equation (4.14) still holds for all x, y ∈ [0, ∞] d \{0} with x ≤ y, if U and ν are replaced
by Ũ and ˜ν.
First part. To prove the existence of a Lévy copula, we construct the required Lévy copula
in two stages.
1. First consider the function ˜F : D := Ran Ũ1 × · · · × Ran Ũd → [0, ∞] defined by
˜F (x 1 , . . . , x d ) =
Ũ(Ũ
(−1)
1 (x 1 ), . . . , Ũ (−1)
d
(x d )).
Suppose that x k = 0 for some k. Without loss of generality we can take k = 1. Then
˜F (0, x 2 , . . . , x d ) =
Ũ(z, Ũ
(−1)
2 (x 2 ), . . . , Ũ (−1)
d
(x d )),
where z is such that Ũ1(z) = 0. Since Ũ is nonnegative and nonincreasing in each argument,
0 ≤
Ũ(z, Ũ
(−1)
2 (x 2 ), . . . , Ũ (−1)
d
(x d )) ≤ Ũ(z, 0, . . . , 0) = Ũ1(z) = 0.
Therefore, ˜F is grounded. Let a, b ∈ D with ak ≤ b k , k = 1, . . . , d and denote
B := (a 1 , b 1 ] × · · · × (a d , b d ]
and
Since Ũ (−1)
k
˜B := [Ũ (−1)
1 (b 1 ), Ũ (−1)
1 (a 1 )) × · · · × [Ũ (−1)
(b k ) ≤ Ũ (−1)
k
(a k ) for every k, formula (4.14) entails that
d
(b d ), Ũ (−1)
d
(a d )).
V F (B) = (−1) d VŨ( ˜B) = ˜ν( ˜B) ≥ 0,