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

92 CHAPTER 2. THE CALIBRATION PROBLEM
and 2.10 and Prohorov’s theorem, {Q n } n≥1 is weakly relatively compact, which proves the first
part of the theorem.
Choose any subsequence of {Q n } n≥1 , converging weakly to a process Q ∗ ∈ M ∩ L + B . , To
simplify notation, this subsequence is denoted again by {Q n } n≥1 . The triangle inequality and
Lemma 2.2 imply that
‖C Qn − C n M‖ 2 −−−→
n→∞ ‖CQ∗ − C M ‖ 2 (2.33)
Since, by Lemma 2.11, the relative entropy functional is weakly lower semicontinuous in Q,
for every Q ∈ M ∩ L + B ,
‖C Q∗ − C M ‖ + αI(Q|P ) ≤ lim inf{‖C Qn − C n
n
M‖ 2 + αI(Q n |P )}
≤ lim inf{‖C Q − C n
n
M‖ 2 + αI(Q|P )}
= lim
n
‖C Q − C n M‖ 2 + αI(Q|P )
= ‖C Q − C M ‖ 2 + αI(Q|P ),
where the second inequality follows from the fact that Q m is the solution of the calibration
problem with data C m M
and the last line follows from the triangle inequality.
2.5.2 Convergence of regularized solutions
In this section we study the convergence of solutions of the regularized calibration problem
(2.27) to the solutions of the minimum entropy least squares calibration problem (2.11) when
the noise level in the data tends to zero.
Theorem 2.17. Let {C δ M } be a family of data sets of option prices such that ‖C M − C δ M ‖ ≤ δ,
let P ∈ L NA ∩ L + B and suppose that there exist a solution Q of problem (2.4) with data C M (a
least squares solution) such that I(Q|P ) < ∞.
and
If ‖C Q − C M ‖ = 0 (the constraints are reproduced exactly), let α(δ) be such that α(δ) → 0
δ2
δ
α(δ)
→ 0 as δ → 0. Otherwise, let α(δ) be such that α(δ) → 0 and
Then every sequence {Q δ k}, where δ k → 0 and Q δ k
α(δ)
→ 0 as δ → 0.
is a solution of problem (2.27) with data
C δ k
M , prior P and regularization parameter α(δ k), has a weakly convergent subsequence. The
limit of every convergent subsequence is a solution of problem (2.11) (MELSS) with data C M
and prior P . If such a MELSS Q + is unique then lim δ→0 Q δ = Q + .