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

110 CHAPTER 3. NUMERICAL IMPLEMENTATION
3.3.2 A posteriori parameter choice rule for non-attainable calibration problems
In this subsection we suppose that hypothesis 1 of page 103 is satisfied and that hypotheses 2
and 3 are replaced with the following ones:
2a. There exists a solution Q + of problem (2.11) with data C M such that
I(Q + |P ) < ∞
and
‖C Q+ − C M ‖ w > 0
(the data is not attainable by an exp-Lévy model).
3a. There exists δ 0 such that
Given two constants c 1 and c 2 satisfying
ε max := inf
δ≤δ 0
{‖C Q∗ − C δ M‖ 2 w − ‖C Q+ − C δ M‖ 2 w} > 0.
the discrepancy principle can be stated as follows:
0 < c 1 ≤ c 2 < εmax
δ0
2 , (3.22)
Discrepancy principle for non-attainable data For a given noise level δ, choose α > 0
that satisfies
c 1 δ 2 ≤ ε δ (α) − ε δ (0) ≤ c 2 δ 2 , (3.23)
Proofs of the following two results are similar to the proofs of Propositions 3.4 and 3.3 of
the preceding subsection and are therefore omitted.
Proposition 3.8. Suppose that hypothesis 1 of page 103 and hypotheses 2a and 3a of page 110
are satisfied and let c 1 and c 2 satisfy (3.22). If ε δ (α) is a single-valued function then there exists
an α satisfying (3.23).
Proposition 3.9. Suppose that hypothesis 1 of page 103 and hypotheses 2a and 3a of page 110
are satisfied and let c 1 and c 2 satisfy (3.22) for all δ. 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.