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

4.6. EXAMPLES OF LEVY COPULAS 163
Conversely, if F ‖ | [0,∞] d is a Lévy copula of a spectrally positive Lévy process X, then the
Lévy measure of X is supported by an ordered subset of R d +. If, in addition, the tail integrals
U i of X i are continuous and satisfy lim x↓0 U i (x) = ∞, i = 1, . . . , d, then the jumps of X are
completely dependent.
Lévy copulas provide a simple characterization of possible dependence patterns of multidimensional
stable processes.
Theorem 4.13. Let X := (X 1 , . . . , X d ) be a Lévy process on R d and let α ∈ (0, 2). X is
α-stable if and only if its components X 1 , . . . , X d are α-stable and it has a Lévy copula F that
is a homogeneous function of order 1:
∀r > 0, ∀u 1 , . . . , u d , F (ru 1 , . . . , ru d ) = rF (u 1 , . . . , u d ). (4.32)
Proof. The “only if” part. Let X be α-stable. For each i = 1, . . . , d, three situations are
possible: Ran U i = (−∞, 0] (only negative jumps), Ran U i = [0, ∞) (only positive jumps) or
Ran U i = (−∞, 0) ∪ (0, ∞) (jumps of both signs). We exclude the trivial case of a component
having no jumps at all. Let I 1 = {i : Ran U i = (−∞, 0]} and I 2 = {i : Ran U i = [0, ∞)} and
for each i, let ¯X i be a copy of X i , independent from X and from ¯X k for k ≠ i. Define a Lévy
process ˜X on R d by
⎧
⎨ X i , i /∈ I 1 ∪ I 2 ,
˜X i =
⎩
X i − ¯X i , i ∈ I 1 ∪ I 2 .
Let ˜ν be the Lévy measure of ˜X, Ũ be its tail integral and ˜F be its Lévy copula (it exists by
Theorem 4.8). The process ˜X is clearly α-stable and each component of this process has jumps
of both signs (Ran Ũi = R \ {0}). By Theorem 14.3 in [87], for every B ∈ B(R d \ {0}) and for
every r > 0,
˜ν(B) = r α˜ν(rB). (4.33)
Therefore, for every I ⊂ {1, . . . , d} nonempty and for every (x i ) i∈I ∈ (R \ {0}) |I| ,
Ũ I ((x i ) i∈I ) = r α Ũ I ((rx i ) i∈I ). (4.34)
By Theorem 4.8 this implies that for all (u 1 , . . . , u d ) ∈ (R \ {0}) d ,
˜F I ((u i ) i∈I ) = r −1 ˜F I ((ru i ) i∈I ),