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

3.1. DISCRETIZING THE CALIBRATION PROBLEM 97
Proof. For a function f ∈ C b (R), define
⎧
0, x ≥ 2 √ n,
⎪⎨
f n (x) := 0, x < −2 √ n,
⎪⎩
f(x i ), x ∈ [x i − 1/ √ n, x i + 1/ √ n) with 1 ≤ i ≤ 2n,
Then clearly
∫
∫
(1 ∧ x 2 )f(x)ν n (dx) =
f n (x)µ(dx).
Since f(x) is continuous, f n (x) → f(x) for all x and since f is bounded, the dominated convergence
theorem implies that
lim
n
∫
(1 ∧ x 2 )f(x)ν n (dx) = lim
n
∫
∫
f n (x)µ(dx) =
∫
f(x)µ(dx) =
(1 ∧ x 2 )f(x)ν(dx). (3.3)
With f(x) ≡ h2 (x)
1∧x 2
the above yields:
∫
∫
lim h 2 (x)ν n (dx) =
n
h 2 (x)ν(dx).
On the other hand, for every g ∈ C b (R) such that g(x) ≡ 0 on a neighborhood of 0, f(x) := g(x)
1∧x 2
belongs to C b (R). Therefore, from Equation (3.3), lim
n
∫
g(x)νn (dx) = ∫ g(x)ν(dx), and by
Proposition (1.7), P n ⇒ P .
Theorem 3.2. Let P, {P n } n≥1 ⊂ L NA ∩ L + B such that P n ⇒ P , let α > 0, let C M be a data
set of option prices and for each n let Q n be a solution of the calibration problem (2.27) with
prior P n , regularization parameter α and data C M . Then the sequence {Q n } n≥1 has a weakly
convergent subsequence and the limit of every weakly convergent subsequence of {Q n } n≥1 is a
solution of the calibration problem (2.27) with prior P .
Proof. By Lemma 2.12, there exists C < ∞ such that for every n, one can find ˜Q n ∈ M∩L with
I( ˜Q n |P n ) ≤ C. Since, by Lemma 2.2, ‖C ˜Q n
− C M ‖ 2 w ≤ S 2 0 for every n and Q n is the solution
of the calibration problem, I(Q n |P n ) ≤ S0 2 /α + C < ∞ for every n. Therefore, by Lemma
2.10, {Q n } is tight and, by Prohorov’s theorem and Lemma 2.4, weakly relatively compact in
M ∩ L + B . Choose a subsequence of {Q n}, converging weakly to Q ∈ M ∩ L + B
. To simplify
notation, this subsequence is also denoted by {Q n } n≥1 . It remains to show that Q is indeed a
solution of the calibration problem with prior P .