On the nature of ill-posedness of an inverse problem arising in option

1330 THeinandBHofmann
for arbitrarily small values β>0. For any given ε>0thereisavalueβ0 > 0suchthat
S(β0) < ε
4 ,since limβ→0 S(β) = 0. For sufficiently large n we moreover have with (30)
�Sδn − S�L ∞ (β0,γ ) < ε
2 and hence �Sδn − S�L ∞ (0,γ ) 0}
t∈(0,T )
with agiven uniform positive lower bound c. Since J : L2 (0, T ) → L2 (0, T ) defined by
formula (15) is a compact linear operator (see, e.g., [28, p 235]) and N : D+ ∩ L2 (0, T ) ⊂
L2 (0, T ) → L2 (0, T ) defined by formula (11) is a continuous nonlinear operator as a
consequence of lemma 2.3, the composite nonlinear operator F = N ◦ J : D † (F) ⊂
L2 (0, T ) → L2 (0, T ) is also compact and continuous. Then based on results of section 2
we have the following lemma.
Lemma 5.1. The nonlinear operator F : D † (F) ⊂ L 2 (0, T ) → L 2 (0, T ) possessing a convex
and weakly closed domain D † (F) is injective, compact, continuous, weakly continuous and
consequently weakly closed, and the inverse operator F −1 defined on F(D † (F)) exists.
Then proposition A.3 of [15] applies and we can formulate the following as a corollary of
lemma 5.1.
Corollary 5.2. Foragiven right-hand side u ∈ F(D † (F)) the operator equation (31) has a
uniquely determined solution a ∈ D † (F). Forany ball Br(a) with centre a and radius r > 0
there existsasequence {an} ∞ n=1 ⊂ Br(a) ∩ D † (F) with
an ⇀ a butan �→ a and F(an) → u in L 2 (0, T ) as n →∞.
Thus, equation (31) is locally ill posed in the sense of [25, definition 2] and F −1 is not
continuous in u.