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

3.3. CHOICE OF THE REGULARIZATION PARAMETER 109
By continuity of J δ (α), there exists ∆ > 0 such that J δ (α ∗ − ∆) > c 2 δ 2 . However, since
α ∗ = sup B and ε δ (α) is nondecreasing, necessarily ε δ (α ∗ − ∆) ≤ c 1 δ 2 . Therefore, α − ∆ is a
solution of (3.19).
Remark 3.2. If c 1 < c 2 , one can show, along the lines of the above proof, that there exists not
a single α that satisfies either (3.18) or (3.19) but an interval of nonzero length (α 1 , α 2 ), such
that each point inside this interval satisfies one of the two conditions. From the computational
viewpoint this means that a feasible α can be found by bisection with a finite number of
iterations.
Proposition 3.7. Suppose that the hypotheses 1–3 of page 103 are satisfied and that c 1 and c 2
are chosen according to (3.12). Let {C δ k
M } k≥1 be a sequence of data sets such that ‖C M −C δ k
M ‖ w <
δ k and δ k → 0.
Then the sequence {Q δ k
αk
} k≥1 where Q δ k
αk
is a solution of problem (2.27) with data C δ k
M , prior
P and regularization parameter α k chosen according to the alternative principle, has a weakly
convergent subsequence. The limit Q ′ of every such subsequence of {Q δ k
αk
} k≥1 satisfies
‖C Q′ − C M ‖ w = 0
I(Q ′ |P ) ≤ c 2
c 2 − 1 I(Q+ |P ).
Proof. Using the optimality of Q δk
α k
, we can write:
ε δk (α k ) + α k I(Q δk
α k
|P ) ≤ ‖C Q+ − C δ k
M ‖2 w + α k I(Q + |P ) ≤ δ 2 + α k I(Q + |P ). (3.21)
If α k satisfies (3.18), the above implies that
otherwise (3.19) entails that
and together with (3.21) this gives
I(Q δk
α k
|P ) ≤ I(Q + |P ),
ε δk (α k ) + α k I(Q δk
α k
|P ) ≥ c 2 δ 2 ,
I(Q δk
α k
|P ) ≤ c 2
c 2 − 1 I(Q+ |P ).
In both cases I(Q δk
α k
|P ) is uniformly bounded, which means, by Lemma 2.10, that the sequence
{Q δ k
αk
} k≥1 is tight and therefore, by Prohorov’s theorem, relatively weakly compact. The rest
of the proof follows the lines of the proof of Proposition 3.3.