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

108 CHAPTER 3. NUMERICAL IMPLEMENTATION
The limit relations for J δ (α) follow from the relations for ε δ (α).
We will now present an alternative a posteriori parameter choice rule, which reduces to the
discrepancy principle when inequality (3.13) has a solution but also works when this is not
the case. However, if the parameter is chosen according to the alternative rule, the sequence
of regularized solutions does not necessarily converge to a minimum entropy solution as in
Proposition 3.3 but to a solution with bounded entropy (see Proposition 3.7). Our treatment
partly follows [50] where this parameter choice rule is applied to Tikhonov regularization.
Alternative principle
For a given noise level δ, if there exists α > 0 that satisfies
c 1 δ 2 ≤ ε δ (α) ≤ c 2 δ 2 , (3.18)
choose one such α; otherwise, choose an α > 0 that satisfies
ε δ (α) ≤ c 1 δ 2 , J δ (α) ≥ c 2 δ 2 . (3.19)
Proposition 3.6. Suppose that the hypotheses 1–3 of page 103 are satisfied and let c 1 and c 2
be as in (3.12). Then there exists α > 0 satisfying either (3.18) or (3.19).
Proof. Suppose that (3.18) does not admit a solution. We need to prove that there exists α > 0
satisfying (3.19). Let
B := {α > 0 : ε δ (α) ≤ c 1 δ 2 } and U := {α > 0 : ε δ (α) > c 2 δ 2 }.
The limit relations of Lemma 3.5 imply that both sets are nonempty. Moreover, since we have
assumed that (3.18) does not admit a solution, necessarily sup B = inf U. Let α ∗ := sup B ≡
inf U. Now we need to show that
By continuity of J δ (α),
J δ (α ∗ ) > c 2 δ 2 . (3.20)
J δ (α ∗ ) ≥ c 2 δ 2 + lim
α↓α ∗ γ δ(α).
If lim α↓α ∗ γ δ (α) > 0 then (3.20) holds. Otherwise from (3.16), P is the minimal entropy martingale
measure and (3.17) implies that Q δ α ⇒ P as α ↓ α ∗ . Therefore, J δ (α ∗ ) = lim α↓α ∗ ε δ (α) =
‖C Q∗ − C δ M ‖2 w > c 2 δ 2 and (3.20) is also satisfied.