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

176 CHAPTER 5. APPLICATIONS OF LEVY COPULAS
Lemma 5.5. Let F be a Lévy copula on (−∞, ∞] d , satisfying (5.5), and F ξ be the corresponding
conditional distribution function, defined by (5.8). Then, there exists N ⊂ R with λ(N) = 0
such that for every fixed ξ ∈ R \ N, F ξ (∗) is a probability distribution function, satisfying
F ξ (x 2 , . . . , x d ) = sgn(ξ) ∂ ∂ξ V F ((ξ ∧ 0, ξ ∨ 0] × (−∞, x 2 ] × · · · × (−∞, x d ]) (5.9)
in every point (x 2 , . . . , x d ), where F ξ is continuous.
Remark 5.2. Since the law of a random variable is completely determined by the values of its
distribution function at the continuity points of the latter, being able to compute F ξ at all
points where it is continuous is sufficient for all practical purposes.
Proof. Since it has already been observed that K(ξ, R d−1 ) = 1 λ-almost everywhere, we only
need to prove the second part of the lemma. Let
G(x 1 , . . . , x d ) := sgn x 1 V F ((x 1 ∧ 0, x 1 ∨ 0] × (−∞, x 2 ] × · · · × (−∞, x d ])
By Theorem 2.28 in [1], for each f ∈ L 1 (R d , µ),
∫
∫ ∞
f(x 1 , . . . , x d )µ(dx 1 · · · dx d ) = dx 1 f(x 1 , . . . , x d )K(x 1 , dx 2 · · · dx d ), (5.10)
R d −∞
∫R d
which implies that
∫
G(x 1 , . . . , x d ) = sgn x 1
(x 1 ∧0,x 1 ∨0]
dξF ξ (x 2 , . . . , x d ),
Therefore, for fixed (x 2 , . . . , x d ), (5.9) holds ξ-almost everywhere. Since a union of countably
many sets of zero measure is again a set of zero measure, there exists a set N ⊂ R with λ(N) = 0
such that for every ξ ∈ R \ N, (5.9) holds for all (x 2 , . . . , x d ) ∈ Q d , where Q denotes the set of
rational numbers.
Fix ξ ∈ R \ N and let x ∈ R d−1 and {x + n } and {x − n } be two sequences of d − 1-dimensional
vectors with coordinates in Q, converging to x from above and from below (componentwise).
Since F ξ is increasing in each coordinate (as a probability distribution function), the limits
lim n F ξ (x + n ) and lim n F ξ (x − n ) exist. Suppose that
lim F ξ (x +
n
n ) = lim F ξ (x −
n
n ) = F ∗ (5.11)
and observe that for every δ ≠ 0,
G(ξ + δ, x − n ) − G(ξ, x − n )
δ
≤
G(ξ + δ, x) − G(ξ, x)
δ
≤ G(ξ + δ, x+ n ) − G(ξ, x + n )
.
δ