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

4.5. LEVY COPULAS: GENERAL CASE 157
If I = {i}, then the definition of m implies that this equals
(
νi ⊗ λ| (0,1)
) (
{(x, y) ∈ R × [0, 1] : Ũ i (x) + y∆U i (x) ∈ (0, u i ]} ) + (u i − ν i ((0, ∞)))1 {ui >ν i ((0,∞))}.
Introducing x ∗ := inf{x ≥ 0 : Ũi(x) + ∆U i (x) ≤ u i }, this can be expressed as
ν i ((x ∗ , ∞)) + (u i − Ũi(x ∗ ) − ∆U i (x ∗ ))1 {x∗≠0} + (u i − ν i ((0, ∞)))1 {x ∗ =0} = u i ,
i.e. property 4 in Definition 4.16 is met.
Now, let (x i ) i∈I ∈ (R \ {0}) I . Again, we consider only the case where all the x i are nonnegative.
Then
F I ((U i (x i )) i∈I ) = m( {
(˜x 1 , . . . , ˜x d , y 1 , . . . , y d , z) ∈ (−∞, ∞] d × [0, 1] d × R :
= ν
} )
Ũ i (˜x i ) + y i ∆U i (˜x i ) + z ∈ (0, U i (x i )] for i ∈ I
( ∏
i∈I[x i , ∞) × R Ic )
= ν I ( ∏
i∈I[x i , ∞)
= U I ((x i ) i∈I )
as claimed. The uniqueness statement follows from (4.24) and Lemma 4.4.
The converse statement.
)
Since F is d-increasing and continuous (by Lemma 4.4), there
exists a unique measure µ on (−∞, ∞] d \ {∞, . . . , ∞} such that V F (|a, b|) = µ((a, b]) for any
a, b ∈ (−∞, ∞] d \ {∞, . . . , ∞} with a ≤ b. (see [60], Section 4.5). For a one-dimensional tail
integral U(x), we define
⎧
⎨
U (−1) sup{x > 0 : U(x) ≥ u} ∨ 0, u ≥ 0
(u) :=
⎩
sup{x < 0 : U(x) ≥ u}, u < 0.
Let ˜ν := f(µ) be the image of µ under
f : (u 1 , . . . , u d ) ↦→ (U (−1)
1 (u 1 ), . . . , U (−1)
d
(u d ))
(4.25)
and let ν be the restriction of ˜ν to R d \ {0}. We need to prove that ν is a Lévy measure and
that its marginal tail integrals U I ν satisfy
U I ν ((x i ) i∈I ) = F I ((U i (x i )) i∈I )