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

98 CHAPTER 3. NUMERICAL IMPLEMENTATION
Lemma 2.11 entails that
I(Q, P ) ≤ lim inf
n I(Q n, P n ), (3.4)
and since, by Lemma 2.2, the pricing error is weakly continuous, we also have
‖C Q − C M ‖ 2 w + αI(Q, P ) ≤ lim inf
n {‖CQn − C M ‖ 2 w + αI(Q n , P n )}. (3.5)
Let φ ∈ C b (Ω) with φ ≥ 0 and E P [φ] = 1. Without loss of generality we can suppose that for
every n, E Pn [φ] > 0 and therefore Q ′ n, defined by Q ′ n(B) := EPn [φ1 B ]
, is a probability measure
E Pn [φ]
on Ω. Clearly, Q ′ n converges weakly to Q ′ defined by Q ′ (B) := E P [φ1 B ]. Therefore, by Lemma
2.2,
Moreover,
lim
n
I(Q ′ n|P n ) = lim
n
∫
Ω
lim
n
‖C Q′ n
− C M ‖ 2 w = ‖C Q′ − C M ‖ 2 w. (3.6)
φ
E Pn [φ] log
= lim
n
1
E Pn [φ]
φ
E Pn [φ] dP n
∫
φ log φdP n − lim
n
Ω
∫ ∫
log φdP n = φ log φdP. (3.7)
Ω
Ω
For the rest of this proof, for every φ ∈ L 1 (P ) with φ ≥ 0 and E P [φ] = 1 let Q φ denote the
probability measure on Ω, defined by Q φ (B) := E P [φ1 B ] for every B ∈ F. Using (3.5–3.7) and
the optimality of Q n , we obtain that for every φ ∈ C b (Ω) with φ ≥ 0 and E P [φ] = 1,
‖C Q − C M ‖ 2 w + I(Q, P ) ≤ ‖C Q φ
− C M ‖ 2 w + I(Q φ |P ) (3.8)
To complete the proof of the theorem, we must generalize this inequality to all φ ∈ L 1 (P ) with
φ ≥ 0 and E P [φ] = 1.
First, let φ ∈ L 1 (P ) ∩ L ∞ (P ) with φ ≥ 0 and E P [φ] = 1. Then there exists a sequence
{φ n } ⊂ C b (Ω) such that φ n → φ in L 1 (P ), φ n ≥ 0 for all n and φ n are bounded in L ∞ norm
uniformly on n. Moreover, φ ′ n := φ n /E P [φ n ] also belongs to L 1 (P ), is positive and φ ′ n
because by the triangle inequality,
‖φ ′ n − φ‖ L 1 ≤
1 (
‖φn
E P − φ‖
[φ n ]
L 1 + ‖φ − φE P )
[φ n ]‖ L 1 −−−→ 0.
n→∞
In addition, it is easy to see that Q φ ′ n
⇒ Q φ . Therefore,
L 1 (P )
−−−→ φ
lim
n
‖C Q φ ′ n − C M ‖ 2 w = ‖C Q φ
− C M ‖ 2 w