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

4.4. LEVY COPULAS: SPECTRALLY POSITIVE CASE 151
Ran Ũ1, then a straightforward computation using Equation (4.19) shows that
Suppose that x 1
V F (B) = y 1 − x 1
¯x 1 − x 1
V ˜F
(|x 1 , ¯x 1 | × ˜B) ≥ 0.
/∈ Ran Ũ1 but y 1 ∈ Ran Ũ1. Let z = inf{ζ ≥ x 1 , ζ ∈ Ran Ũ1}. Because
Ran Ũ1 is closed, z ∈ Ran Ũ1. The F -volume of B can be decomposed as follows:
V F (B) = V F (|x 1 , z| × ˜B) + V F (|z, y 1 | × ˜B).
The second term is positive because all vertices of the corresponding d-box are in the domain
of ˜F . The first term can again be computed using formula (4.19):
V F (|x 1 , z| × ˜B) = z − x 1
z − x 1
V ˜F
(|x 1 , z| × ˜B) ≥ 0.
The case where x 1 ∈ Ran Ũ1 and y 1 /∈ Ran Ũ1 can be treated in this same way and if x 1 /∈ Ran Ũ1,
y 1 /∈ Ran Ũ1 and between x 1 and y 1 there are points that belong to Ran Ũ1, the interval |x 1 , y 1 |
can be split onto two intervals of types that we have already discussed.
We have thus extended ˜F to a function defined on ([0, λ 1 ]∪{∞})×Ran Ũ2 ×· · ·×Ran Ũd. If
λ 1 = ∞ then we are done; otherwise we extend ˜F to [0, ∞] × Ran Ũ2 × · · · × Ran Ũd by defining
F (x 1 , . . . , x d ) := ˜F (g λ1 (x 1 ), x 2 , . . . , x d ) + (x 1 − λ 1 ) + 1 x2 =∞ . . . 1 xd =∞, (4.20)
where
⎧
x, x ≤ λ 1
⎪⎨
g λ1 (x) := λ 1 , λ 1 < x < ∞
⎪⎩
∞, x = ∞.
The function F , defined by (4.20) is an increasing function because it is a sum of two increasing
functions. The groundedness and marginal properties can be verified by direct substitution.
This completes the construction of the Lévy copula.
To prove the uniqueness, assume that there exist two functions with required properties,
i.e., we have
F 1 (U 1 (x 1 ), . . . , U d (x d )) = F 2 (U 1 (x 1 ), . . . , U d (x d )), ∀x 1 , . . . , x d .
For each vector (t 1 , . . . , t d ) ∈ Ran U 1 × · · · × Ran U d there exists a vector (x 1 , . . . , x d ) ∈ [0, ∞] d
such that U 1 (x 1 ) = t 1 , . . . , U d (x d ) = t d . This means that for every (t 1 , . . . , t d ) ∈ Ran U 1 × · · · ×