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Note that p ∗ satisfies
Calibration of Lévy Processes with American Options 275
∂p ∗
∂t − AT Xp ∗ = −2 ∑ ω i (u ∗ (T i ,x i ) − ū i )δ t=Ti ⊗ δ x=xi (53)
i∈I
in the sense of distributions in the open set {x, t : u ∗ (t, x) >u ◦ (x)} and that
(48) implies that p ∗ vanishes in the coincidence set.
5 The Variational Inequality when σ =0
We focus on the case when σ =0andwhen(α, ψ) ∈F 2 with
[ ] {
1
F 2 =
2 + α, 1 − α × ψ ∈B:
∣ ‖ψ‖ B ≤ ¯ψ; ψ ≥ 0,
ψ ≥ ψ a.e. in [−¯z, ¯z]
for three constants α, ψ ¯ψ, 0ψ> 0.
}
. (54)
Remark 6. In the case when σ =0andα 0andλ ≥ 0suchthat
〈Av, v〉 ≥c|v| 2 V α − λ‖v‖2 L 2 (R +) , ∀v ∈ V α (55)
and
〈Av, v + 〉≥c|v + | 2 V α − λ‖v +‖ 2 L 2 (R +) , ∀v ∈ V α ; (56)
• the operator A + λI is one to one and continuous from V 2α onto L 2 (R + ).
The goal is to obtain the existence of a weak solution to (15), (17), (18)
by a singular perturbation argument: we fix (α, ψ) ∈ F 2 and for η > 0,
we call u η the solution to (15), (17), (18) corresponding to σ = η, given
by Theorem 1. It can be proven that ‖u η ‖ L∞ (0,T ;V α ) and ‖u η ‖ L2 (0,T ;V 2α ) are
bounded independently of η, and that the free boundary associated to u η stays
in [0,T] × [0, ˇX], where ˇX does not depend on η. By the results contained in
[Lio73, in particular, Théorème 4.1, p. 286], one may pass to the limit as η
tends to zero, and prove the following result:
Theorem 3. We choose σ =0and (α, ψ) ∈F 2 and we define
K = {v ∈ V α , v(x) ≥ u ◦ (x) in R + }.
There exists a unique weak solution of (15), (17) and (18) in (0,T) × R + ,
i.e. a function u which belongs to C 0 ([0,T]; K) and to L 2 (0,T; V 2α ), and with