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

3.3. CHOICE OF THE REGULARIZATION PARAMETER 103
A typical example of a parameter choice rule that works in both attainable and unattainable
cases would be
α(δ) = Cδ µ (3.10)
with any C > 0 and any µ ∈ (0, 1).
However, in real calibration problems the error level δ is fixed and finite and rules of type
(3.10) do not tell us how we should choose α in this case. Good performance and error control
for finite values of δ is achieved by using the so called a posteriori parameter choice rules
(α depends not only on δ but also on the data CM δ ), the most widely used of which is the
discrepancy principle, originally developed by Morozov for Tikhonov regularization in Banach
spaces [74, 75], see also [40]. In the rest of this section we apply this principle and its variants
that is better suited for nonlinear problems to our entropic regularization method.
3.3.1 A posteriori parameter choice rules for attainable calibration problems
In this subsection we make the following standing assumptions:
1. The prior Lévy process corresponds to an arbitrage-free model with jumps bounded from
above by B: P ∈ L NA ∩ L + B
measure Q ∗ ).
(this implies the existence of a minimal entropy martingale
2. There exists a solution Q + of problem (2.11) with data C M (minimum entropy least
squares solution) such that
I(Q + |P ) < ∞
and
‖C Q+ − C M ‖ w = 0
(the data is attainable by an exp-Lévy model).
3. There exists δ 0 such that
ε max := inf
δ≤δ 0
‖C Q∗ − C δ M‖ 2 w > δ 2 0. (3.11)
Remark 3.1. In the condition (3.11), δ 0 can be seen as the highest possible noise level of all data
sets that we consider. If, for some δ, ‖C Q∗ − C δ M ‖ w < δ 0 then either Q ∗ is already sufficiently
close to the solution or the noise level is so high that all the useful information in the data is