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

104 CHAPTER 3. NUMERICAL IMPLEMENTATION
completely blurred and the data does not allow to construct a better approximation of the true
solution than Q ∗ .
Let Q δ α denote a solution of the regularized problem (2.27) with data C δ M
parameter α. The function
and regularization
ε δ (α) := ‖C Qδ α
− C δ M‖ 2 w
is called the discrepancy function of the calibration problem (2.27). Note that since this problem
can have many solutions, ε δ (α) is a priori a multivalued function. Given two constants c 1 and
c 2 satisfying
the discrepancy principle can be stated as follows:
1 < c 1 ≤ c 2 < εmax
δ0
2 , (3.12)
Discrepancy principle
For a given noise level δ, choose α > 0 that satisfies
δ 2 < c 1 δ 2 ≤ ε δ (α) ≤ c 2 δ 2 , (3.13)
If, for a given α, the discrepancy function has several possible values, the above inequalities
must be satisfied by each one of them.
The intuition behind this principle is as follows. We would like to find a solution Q of
the equation C Q = C M . Since the error level in the data is of order δ, it is the best possible
precision that we can ask for in this context, so it does not make sense to calibrate the noisy
data CM δ with a precision higher than δ. Therefore, we try to solve ‖C Qδ α − C
δ
M
‖ 2 w ≤ δ 2 . In
order to gain stability we must sacrifice some precision compared to δ, therefore, we choose a
constant c with 1 c, for example, c = 1.1 and look for Q δ α in the level set
‖C Qδ α
− CM‖ δ 2 w ≤ cδ 2 . (3.14)
Since, on the other hand, by increasing precision, we decrease the stability, the highest stability
is obtained when the inequality in (3.14) is replaced by equality and we obtain
ε δ (α) ≡ ‖C Qδ α
− CM‖ δ 2 w = cδ 2 .
To make the numerical solution of the equation easier, we do not impose a strict value of the
discrepancy function but allow it to lie between two bounds, obtaining (3.13).