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

156 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES
and
⎧
⎨ lim ξ↓x U i (ξ) − U i (x) for x ≠ 0 and x ≠ ∞,
∆U i (x) :=
⎩
0 for x = ∞ or x = 0.
Let m be the measure on ((−∞, ∞] d \ {0}) × [0, 1] d × R defined by
m := ˜ν ⊗ λ| (0,1) d ⊗ δ 0 +
d∑
i=1
δ (0,...,0
,∞,0,...,0) ⊗ δ (0,...,0 ) ⊗ λ| (νi ((0,∞)),∞)∪(−∞,−ν
}{{}
}{{}
i ((−∞,0))),
i−1
d
where ˜ν is the extension of ν to (−∞, ∞] d \ {0}, i.e. ˜ν(B) := ν(B ∩ R d ). Let
g i : (−∞, ∞] × [0, 1] × R → (−∞, ∞],
(x, y, z) ↦→ Ũi(x) + y∆U i (x) + z
and define a measure ˜m on (−∞, ∞] d \ {∞, . . . , ∞} via
˜m(B) := m(˜g −1 (B))
with
˜g(x 1 , . . . , x d , y 1 , . . . , y d , z) := (g 1 (x 1 , y 1 , z), . . . , g d (x d , y d , z)).
Finally, let F be given by
⎧ ( d∏
) d∏
⎪⎨ ˜m (u i ∧ 0, u i ∨ 0] sgn u i , (u 1 , . . . , u d ) ∈ (−∞, ∞] d \ (∞, . . . , ∞)
F (u 1 , . . . , u d ) := i=1
i=1
⎪⎩ ∞, (u 1 , . . . , u d ) = (∞, . . . , ∞).
Properties 1 and 2 in Definition 4.16 are obvious. From the fact that ˜m is a positive measure
it follows immediately that F is d-increasing. Let I ⊂ {1, . . . , d} nonempty and (u i ) i∈I ∈
(−∞, ∞] I . For ease of notation, we consider only the case of non-negative u i . The general case
follows analogously. By definition of F we have
∑
F I ((u i ) i∈I ) = lim
F (u 1 , . . . , u d ) ∏
sgn u j
c→∞
(u j ) j∈I c ∈{−c,∞} Ic j∈I c
( )
∏
u i ] × (−∞, ∞]
i∈I(0, Ic
= ˜m
= m( {
(x 1 , . . . , x d , y 1 , . . . , y d , z) ∈ ((−∞, ∞] d \ {0}) × [0, 1] d × R :
Ũ i (x i ) + y i ∆U i (x i ) + z ∈ (0, u i ] for i ∈ I} ) .