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

144 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES
for x 1 , . . . x d ∈ ¯R. This function satisfies Equation (4.6) and can therefore play the role of a
distribution function. However, the margins of ˜F (e.g. the distribution functions computed
using Equation (4.7) for the components of X) are no longer given by (4.5). It can be shown
that
˜F I ((x i ) i∈I ) := P [X i ∈ (x i ∧ 0, x i ∨ 0], i ∈ I] ∏ i∈I
sgn x i
=
∑
(x i ) i∈I c ∈{−∞,∞} |Ic |
˜F (x 1 , . . . , x d ) ∏ i∈I c sgn x i .
The above example motivates the following definitions.
notion of “increasing and grounded” function.
First, we need to generalize the
Definition 4.9 (Volume function). Let S k ⊂ ¯R for k = 1, . . . , d. A function F : S 1 × · · ·×S d
is called volume function if it is d-increasing and there exists x ∗ ∈ S 1 × · · · × S d such that
F (x 1 , . . . , x d ) = 0 whenever x k = x ∗ k
for some k.
The term “volume function” is due to the fact that an increasing function is a volume
function if and only if there exists x ∗ ∈ S 1 × · · · × S d such that
d∏
F (x 1 , . . . , x d ) = V F (|x 1 ∧ x ∗ 1, x 1 ∨ x ∗ 1| × · · · × |x d ∧ x ∗ d , x d ∨ x ∗ d |) sgn(x i − x ∗ i )
for all x ∈ S 1 ×· · ·×S d . Every increasing grounded function is a volume function. The function
˜F , defined in Example 4.4 is a volume function but is not grounded.
Definition 4.10 (Margins of a volume function). Let S k ⊂ ¯R for k = 1, . . . , d and let
F : S 1 × · · · × S d → ¯R be a volume function. 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 ) := sgn(x i − x ∗ i )
i∈I
∑
× sup
(−1) N((x i) i∈I c ) F (x 1 , . . . , x d ), (4.8)
a i ,b i ∈S i :i∈I c
(x i ) i∈I c ∈ ∏
j∈I c{a j,b j }
where N((x i ) i∈I c) = #{i ∈ I c : x i = a i }.
In particular, for d = 2 we have F 1 (x) = sgn(x − x ∗ ) sup y1 ,y 2 ∈S 2
{F (x, y 2 ) − F (x, y 1 )}.
Equation (4.8) looks so complicated because it applies without modification to all cases that
i=1