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

4.3. INCREASING FUNCTIONS 145
are of interest to us: distribution functions, copulas, Lévy copulas with closed or open domains.
In particular cases we will obtain simplifications that are easier to work with. Using the F-
volume notation, Equation (4.8) can be rewritten as follows:
⎛ ⎧
⎞
( )
∏
F I ((x i ) i∈I ) = sgn(x i − x ∗ ⎜
d∏
⎪⎨ |x i ∧ x ∗ i , x i ∨ x ∗ i |, i ∈ I ⎟
i ) sup V F ⎝
⎠ .
i∈I
a i ,b i ∈S i :a i ≤b i ,i∈I c i=1
⎪⎩ |a i , b i |, i ∈ I c
(4.9)
When F satisfies the conditions of Definition 4.8, since F is d-increasing, the sup is achieved
when for every i ∈ I c , a i = inf S i and b i = sup S i . Therefore, in this case the above formula
yields F I ((x i ) i∈I ) = F (x 1 , . . . , x d )| xi =sup S i , i∈I c
and the two definitions coincide.
The following important property of increasing functions will be useful in the sequel.
Lemma 4.4. Let S k ⊂ ¯R for k = 1, . . . , d and let F : S 1 × · · · × S d → ¯R be a volume function.
Let (x 1 , . . . , x d ) and (y 1 , . . . , y d ) be any points in Dom F . Then
|F (x 1 , . . . , x d ) − F (y 1 , . . . , y d )| ≤
Proof. From the triangle inequality,
d∑
|F k (x k ) − F k (y k )|. (4.10)
k=1
|F (x 1 , . . . , x d ) − F (y 1 , . . . , y d )| ≤ |F (x 1 , . . . , x d ) − F (y 1 , x 2 . . . , x d )| + . . .
+ |F (y 1 , . . . , y d−1 , x d ) − F (y 1 , . . . , y d )|,
hence it suffices to prove that
|F (x 1 , . . . , x d ) − F (y 1 , x 2 . . . , x d )| ≤ |F 1 (x 1 ) − F 1 (y 1 )|. (4.11)
Without loss of generality suppose that x 1 ≥ y 1 ≥ x ∗ 1 and that x k ≥ x ∗ k
B := |x ∗ 2 , x 2| × · · · × |x ∗ d , x d|. Then
for k = 2, . . . , d and let
F (x 1 , . . . , x d ) − F (y 1 , x 2 . . . , x d ) = V F (|y 1 , x 1 | × B). (4.12)
Moreover, for all x ≥ x ∗ 1 ,
F 1 (x) =
sup V F (|x ∗ 1, x| × |a 2 , b 2 | × · · · × |a d , b d |).
a i ,b i ∈S i :a i ≤b i ,i=2,...,d