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

84 CHAPTER 2. THE CALIBRATION PROBLEM
From (2.23), A Pn
is bounded uniformly on n. Therefore, inequality (2.22) shows that |γ Q | is
bounded uniformly with respect to Q, which proves property 2 of Proposition 1.6.
Once again, for u sufficiently large,
A Q +
∫ ∞
−∞
∫
(x 2 ∧ 1)φ Q ν P Q
(dx) ≤ A Q + u (x 2 ∧ 1)ν P Q
(dx)
∫
φ Q ≤u
∫ ∞
+ φ Q ν P Q
(dx) ≤ A P Q
+ u
φ Q >u
−∞
(x 2 ∧ 1)ν P Q
(dx) +
2r
T ∞ log u
and (2.23) implies that the right hand side is bounded uniformly with respect to Q ∈ L r .
Therefore, property 3 of Proposition 1.6 also holds and the proof is completed.
Lemma 2.11. Let Q and P be two probability measures on (Ω, F). Then
I(Q|P ) =
where C b (Ω) is space of bounded continuous functions on Ω.
{∫ ∫
}
sup fdQ − (e f − 1)dP , (2.24)
f∈C b (Ω) Ω
Ω
Proof. Observe that
⎧
⎨ x log x + 1 − x, x > 0,
φ(x) =
⎩
∞, x ≤ 0
and φ ∗ (y) = e y − 1 are proper convex functions on R, conjugate to each other and apply
Corollary 2 to Theorem 4 in [83].
Corollary 2.1. The relative entropy functional I(Q|P ) is weakly lower semicontinuous with
respect to Q for fixed P .
Lemma 2.12. Let P, {P n } n≥1 ⊂ L NA ∩ L + B for some B > 0 such that P n ⇒ P . There exists a
sequence {Q n } n≥1 ⊂ M ∩ L + B and a constant C < ∞ such that I(Q n|P n ) ≤ C for every n.
Proof. For every n ≥ 1, by Remark 2.1, Theorem 2.7 can be applied to P n . Let Q n be the
minimal entropy martingale measure and β n be the corresponding constant, defined in (2.14).
We must show that the minimal relative entropy, given by Equation (2.15), is bounded uniformly
on n.
First, let us show that the sequence {β n } n≥1 is bounded. Let h be a continuous bounded
truncation function, satisfying h(x) = x in a neighborhood of zero and for any Lévy process Q