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

3.2. CHOICE OF THE PRIOR 99
Since φ ′ n are bounded in L ∞ norm uniformly on n, φ ′ n log φ ′ n is also bounded and the dominated
convergence theorem implies that lim n I(Q φ ′ n
|P ) = I(Q φ |P ). Passing to the limit in (3.8), we
obtain that this inequality holds for every φ ∈ L 1 (P ) ∩ L ∞ (P ) with φ ≥ 0 and E P [φ] = 1.
Let us now choose a nonnegative φ ∈ L 1 (P ) with E P [φ] = 1. If I(Q φ |P ) = ∞ then surely
(3.8) holds, therefore we can suppose I(Q φ |P ) < ∞. Let φ n = φ ∧ n. Then φ n → φ in L 1 (P )
because
∫
‖φ n − φ‖ L 1 ≤
φ≥n
∫
φdP =
φ≥n
Denoting φ ′ n := φ n /E P [φ n ] as above, we obtain that
φ log φ
log φ dP ≤ I(Q φ|P )
log n → 0.
lim
n
‖C Q φ ′ n − C M ‖ 2 w = ‖C Q φ
− C M ‖ 2 w
Since, for a sufficiently large n, |φ n (x) log φ n (x)| ≤ |φ(x) log φ(x)|, we can once again apply the
dominated convergence theorem:
∫
lim φ ′
n
n log φ ′ 1
ndP =
lim n E P [φ n ] lim n
∫
∫
φ n log φ n dP − lim log E P [φ n ] =
n
φ log φdP
Therefore, by passage to the limit, (3.8) holds for all φ ∈ L 1 (P ) with φ ≥ 0 and E P [φ] = 1,
which completes the proof of the theorem.
To approximate numerically the solution of the calibration problem (2.27) with a given prior
P , we need to construct, using Lemma 3.1, an approximating sequence {P n } of Lévy processes
with atomic measures such that P n ⇒ P . The sequence {Q n } of solutions corresponding to this
sequence of priors will converge (in the sense of Theorem 3.2) to a solution of the calibration
problem with prior P .
3.2 Choice of the prior Lévy process
The prior Lévy process must, generally speaking, reflect the user’s view of the model. It is
one of the most important ingredients of the calibration method and cannot be determined
completely automatically because the choice of the prior has a strong influence on the outcome
of the calibration. The user should therefore specify a characteristic triplet (A P , ν P , γ P ) of
the prior P . A natural solution, justified by the economic considerations of Section 2.3 is
to take the historical probability, resulting from statistical estimation of an exponential Lévy