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

2.5. REGULARIZING THE CALIBRATION PROBLEM 93
Proof. By Lemma 2.6, there exists at least one MELSS with data C M and prior P , that has
finite relative entropy with respect to the prior. Let Q + be any such MELSS. Since Q δ k
solution of the regularized problem, for every k,
is the
‖C Qδk − C δ k
M ‖2 + α(δ k )I(Q δ k
|P ) ≤ ‖C Q+ − C δ k
M ‖2 + α(δ k )I(Q + |P ).
Using the fact that for every r > 0 and for every x, y ∈ R,
(1 − r)x 2 + (1 − 1/r)y 2 ≤ (x + y) 2 ≤ (1 + r)x 2 + (1 + 1/r)y 2 ,
we obtain that
(1 − r)‖C Qδk − C M ‖ 2 + α(δ k )I(Q δ k
|P )
≤ (1 + r)‖C Q+ − C M ‖ 2 + 2δ2 k
r + α(δ k)I(Q + |P ), (2.34)
and since Q + is a least squares solution with data C M , this implies for all r ∈ (0, 1) that
α(δ k )I(Q δ k
|P ) ≤ 2r‖C Q+ − C M ‖ 2 + 2δ2 k
r + α(δ k)I(Q + |P ). (2.35)
If the constraints are reproduced exactly, then ‖C Q+ −C M ‖ = 0 and with the choice r = 1/2,
the above expression yields:
I(Q δ k
|P ) ≤
4δ2 k
α(δ k ) + I(Q+ |P ).
Since, by the theorem’s statement, in the case of exact constraints
δk
2
α(δ k )
→ 0, this implies that
lim sup{I(Q δ k
|P )} ≤ I(Q + |P ). (2.36)
k
If ‖C Q+ − C M ‖ > 0 (misspecified model) then the right-hand side of (2.35) achieves its
maximum when r = δ k ‖C Q+ − C M ‖ −1 , in which case we obtain
I(Q δ k
|P ) ≤
Since in the case of approximate constraints,
4δ k
α(δ k ) ‖CQ+ − C M ‖ + I(Q + |P ).
δ k
α(δ k )
→ 0, we obtain (2.36) once again.
Inequality (2.36) implies in particular that I(Q δ k|P ) is uniformly bounded, which proves,
by Lemmas 2.10 and 2.4, that {Q δ k} is relatively weakly compact in M ∩ L + B .