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

142 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES
Definition 4.5 (F-volume). Let F : S ⊂ ¯R d → ¯R. For a, b ∈ S, with a ≤ b, the F -volume of
|a, b| is defined by
V F (|a, b|) := ∑ (−1) N(c) F (c), (4.4)
where the sum is taken over all vertices c of |a, b|, and N(c) := #{k : c k = a k }.
The notion of F-volume should only be seen as a convenient notation for the sum in the
right-hand side of (4.4). It does not in general correspond to any measure because the measure
of |a, b| will depend on whether the boundary is included or not.
In particular, in two dimensions (d = 2) the F -volume of a rectangle B = |x 1 , x 2 | × |y 1 , y 2 |
satisfies
V F (B) = F (x 2 , y 2 ) − F (x 2 , y 1 ) − F (x 1 , y 2 ) + F (x 1 , y 1 ).
If F (u) = ∏ d
i=1 u i, the F -volume of any interval is equal to its Lebesgue measure.
Definition 4.6 (d-increasing function). A function F : S ⊂ ¯R d → ¯R is called d-increasing
if for all a, b ∈ S with a ≤ b, V F (|a, b|) ≥ 0.
Definition 4.7 (Grounded function). For each k, let S k ⊂ ¯R be such that inf S k ∈ S k . A
function F : S 1 × · · · × S d → ¯R is grounded if F (x 1 , . . . , x d ) = 0 whenever x k = inf S k for at
least one k.
Definition 4.8 (margins of a d-increasing grounded function). For each k, let S k ⊂ ¯R
be such that inf S k ∈ S k and sup S k ∈ S k and let F : S 1 × · · · × S d → ¯R be d-increasing and
grounded. Then for I ⊂ {1, . . . , d} nonempty, the I-margin of F is a function F I : ∏ i∈I S i → ¯R
defined by
F I ((x i ) i∈I ) := F (x 1 , . . . , x d )| xi =sup S i , i∈Ic. (4.5)
When I = {k}, to simplify notation, the (one-dimensional) I-margin of F is denoted by F k .
Example 4.3. The distribution function F of a random vector X ∈ R d is usually defined by
F (x 1 , . . . , x d ) := P [X 1 ≤ x 1 , . . . , X d ≤ x d ]