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

3.3. CHOICE OF THE REGULARIZATION PARAMETER 107
From [30, Theorem 2.2],
γ δ (α) ≥ I(Q δ α|Q ∗ ) + I(Q ∗ |P ). (3.16)
Therefore
Using the inequality
I(Q δ α|Q ∗ ) ≤ ‖CQ∗ − C δ M ‖2 w
α
−−−→
α→∞ 0.
|P − Q| ≤ √ 2I(P |Q), (3.17)
where |P − Q| denotes the total variation distance (see [30, Equation (2.3)]), this implies that
Q δ α converges to Q ∗ in total variation distance (and therefore also weakly) as α goes to infinity.
The limit relation now follows from Lemma 2.2.
To prove the continuity of ε δ (α), let {α n } be a sequence of positive numbers, converging to
α > 0. By the optimality of Q δ α n
, I(Q δ α n
|P ) is bounded and one can choose a subsequence of
{Q δ α n
}, converging weakly toward some measure Q ′ , and denoted, to simplify notation, again
by {Q δ α n
} n≥1 . We now need to prove that Q ′ is the solution of the calibration problem with
regularization parameter α. By weak continuity of the pricing error (Lemma 2.2) and weak
lower semicontinuity of the relative entropy (Lemma 2.11), we have for any other measure Q:
‖C Q′ − C δ M‖ 2 w + αI(Q ′ |P ) ≤ lim inf
n {‖CQδ αn − C
δ
M ‖ 2 w + αI(Q δ α n
|P )}
= lim inf
n {‖CQδ αn − C
δ
M ‖ 2 w + α n I(Q δ α n
|P ) + (α − α n )I(Q δ α n
|P )}
≤ lim inf
n {‖CQ − C δ M‖ 2 w + α n I(Q|P ) + (α − α n )I(Q δ α n
|P )}
= ‖C Q − C δ M‖ 2 w + αI(Q|P ).
Therefore, the sequence {ε δ (α n )} converges to one of the possible values of ε δ (α). If this function
is single-valued in α, this means that every subsequence of the original sequence {ε δ (α n )} has
a further subsequence that converges toward ε δ (α), and therefore, the original sequence also
converges toward ε δ (α).
The fact that J δ (α) is nondecreasing as a function of α is trivial. To show the continuity,
observe that for 0 < α 1 < α 2 ,
J δ (α 2 ) − J δ (α 1 ) ≤ ε δ (α 1 ) + α 2 γ δ (α 1 ) − ε δ (α 1 ) − α 1 γ δ (α 1 )
= (α 2 − α 1 )γ δ (α 1 ) ≤ (α 2 − α 1 )I(Q + |P ).