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Calibration of Lévy Processes with American Options 273
where V X is defined in (35). Then (σ ∗ ,α ∗ ,ψ ∗ ),u ∗ is a solution of (41), where
we agree to use the notation u ∗ for the function E X (u ∗ ). We have that
(i) u ∗ ε n
converges to u ∗ uniformly in [0,T] × [0,X], andinL 2 (0,T; V X );
(ii) 1 {x>S} rxV εn (u ∗ ε n
) converges to µ ∗ strongly in L 2 ((0,T) × (0,X));
(iii) for all smooth function χ with compact support contained in [0,X), χu ∗ ε n
converges to χu ∗ strongly in L 2 (0,T; V 2 ) and in L ∞ (0,T; V ).
4.4 The Optimality Conditions
We fix X as above. Let a subsequence (σε ∗ n
,αε ∗ n
,ψε ∗ n
,u ∗ ε n
) of solutions of (42)
converge to (σ ∗ ,α ∗ ,ψ ∗ ,u ∗ ) as in Proposition 8, then (σ ∗ ,α ∗ ,ψ ∗ ,u ∗ ) is a solution
of (41).
The optimality conditions will involve an adjoint problem. Since the cost
functional involves point-wise values of u, the adjoint problem will have a
singular data. In that context, the notion of very weak solution of boundary
value problems will be relevant: for that, we introduce the spaces ˜Z and Z,
˜Z =
{
v ∈ L 2 (0,T; V X );
Z = {v ∈ ˜Z; v(t =0)=0},
}
∂v
∂t + A Xv ∈ L 2 ((0,T) × (0,X))
,
(43)
where A X is the operator given by (36), (29) and (13), with the parameters
(σ ∗ ,α ∗ ,ψ ∗ ). These spaces endowed with the graph norm are Banach spaces.
We also need to introduce some functionals before stating the optimality
conditions. We assume that u ∗ (T i ,x i ) >u ◦ (x i ), for all i ∈ I. It is clear from
the continuity of u ∗ and from the uniform convergence of u ∗ ε n
that there exists
a positive real number a and an integer N such that for n>N, u ∗ ε n
(t, x) >
u ◦ (x) +ε n for all (t, x) such that |t − T i |