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

2.4. RELATIVE ENTROPY OF LEVY PROCESSES 83
uniformly with respect to Q ∈ L r . Let ε > 0 and choose u such that ∫ {φ Q >u} νQ (dx) ≤ ε/2 for
every Q ∈ L r . By Proposition 1.7,
∫ ∞
−∞
f(x)ν Pn (dx) →
∫ ∞
−∞
f(x)ν P (dx)
for every continuous bounded function f that is identically zero on a neighborhood of zero.
Since the measures ν P and ν Pn
for all n ≥ 1 are finite outside a neighborhood of zero, we can
choose a compact K such that ν Pn (R \ K) ≤ ε/2u for every n. Then
∫
ν Q (R \ K) =
(R\K)∩{φ Q ≤u}
which proves property 1 of Proposition 1.6.
∫
φ Q ν P Q
(dx) +
ν Q (dx) ≤ ε,
(R\K)∩{φ Q >u}
It is easy to check by computing derivatives that for every u > 0, on the set {x : φ Q (x) ≤ u},
(φ Q − 1) 2 ≤ 2u(φ Q log φ Q + 1 − φ Q ).
Therefore, for u sufficiently large and for all Q ∈ L r ,
∫
∣
|x|≤1
x(φ Q − 1)ν P Q (dx) ∣
∫
∣ ∣∣∣∣ ≤
x(φ Q − 1)ν P Q (dx)
∣ |x|≤1, φ Q ≤u
∣
∫|x|≤1, + x(φ Q − 1)ν P Q (dx)
φ Q >u
∣
∫
∫
∫
≤ x 2 ν P Q
(dx) +
(φ Q − 1) 2 ν P Q
(dx) + 2 φ Q ν P Q
(dx)
∫
≤
∫
≤
|x|≤1
|x|≤1
|x|≤1
x 2 ν P Q
(dx) + 2u
|x|≤1, φ Q ≤u
∫ ∞
−∞
φ Q >u
(φ Q log φ Q + 1 − φ Q )ν P Q
(dx) +
4r
T ∞ log u
x 2 ν P Q
(dx) + 3ru
T ∞
. (2.22)
By Proposition 1.6, applied to the sequence {P n } n≥1 ,
∫
A Pn + x 2 ν Pn (dx) (2.23)
|x|≤1
is bounded uniformly on n, which implies that the right hand side of (2.22) is bounded uniformly
with respect to Q ∈ L r . From Proposition 1.5, A Q = A P Q
for all Q ∈ L r because for the relative
entropy to be finite, necessarily Q ≪ P Q . From Theorem 2.9 and Proposition 1.5 it follows that
{ ∫ 1
} 2
γ Q − γ P − x(ν Q − ν P )(dx) ≤ 2AP Qr
.
−1
T ∞