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

On the nature of ill-posedness of an inverse problem arising in option pricing 1333
where Sim with min( ˜S(t), S(t)) � Sim(t) � max( ˜S(t), S(t)) for 0 < t � T is an intermediate
function such that the pairs of real numbers (t, ˜S(t)), (t, S(t)) and (t, Sim(t)) all belong to the
set Mc. Byapplying Schwarz’s inequality this provides
�F(ã) − F(a) − G(ã − a)�L 2 (0,T ) � TC2
�ã − a�2 L 2 2 (0,T )
and hence the required condition (i), which proves the theorem. �
For X �= K the nature of local ill-posedness of (31) at a point a ∈ D † (F) arises from two
components, namely from the global decay rate of singular values θi(J) ∼ 1/i of the linear
integral operator J forming the compact part in G and from the local decay rate of m(t) → 0as
t tends to zero of the multiplication operator M as the noncompact part in G. Both components
will occur again in the following if we consider the source condition (ii) and the closeness
condition (iii) of proposition 5.3.
In ordertointerpret the conditions (ii) and (iii) in the case X �= K ,wewrite (ii) as
(a − a ∗ )(t) =
� T
t
m(τ)w(τ) dτ (0 � t � T, w∈ L 2 (0, T )) (36)
using the equations G∗ = J ∗ ◦ M∗ = J ∗ ◦ M and [J ∗ (h)](t) = � T
t h(τ) dτ (0� t � T ).
Formula (36) directly implies
(a − a ∗ )(T ) = 0 and
(a − a∗ ) ′
∈ L
m
2 (0, T ) (37)
with a difference a−a ∗ ∈ H 1 (0, T ),forwhichthe generalized derivative belongs to a weighted
L2-space with a weight 1
m �∈ L∞ (0, T ). The closeness condition (iii) then attains the form
�
�
�
(a − a
�
∗ ) ′ �
�
�
m � <
L2 (0,T )
1
. (38)
L
The right-hand condition in (37) and condition (38) express the character of ill-posedness
of (31) at the point a as smoothness and smallness requirements on the difference a − a∗ .
Following the concept of ill-posedness rates developed in [24, section 4] for IPs including
multiplication operators it should be noted that we have an exponential growth rate of
1
m(t) →∞as t → 0. Based on formula (6) we derive for X �= K
1
m(t) = K � S(t) exp(ψ(t)) (0 < t � T )
with a constant K > 0and
ψ(t) = ν2
2S(t) + r 2t 2
� �
νrt ν rt S(t)
X
+ + + + , ν := ln �= 0.
2S(t) S(t) 2 2 8 K
For S ∈ I (D † (F)) we have ct � S(t) � ¯c √ t (0 � t � T ) with ¯c := �a�L2 (0,T ). Thisimplies
for positive constants K and ¯K the estimates
K √ � 2 ν
t exp
2 ¯c √ �
�
t
1
m(t) � ¯K 4√ � 2 �
ν
t exp
(0 < t � T ) (39)
2ct
below and above. Since, for fixed ν �= 0, the function 1
m(t) exponentially grows to infinity as
t → 0, the condition (38) on the difference a − a∗ is very rigorous with respect to small t.
Formula (39) also shows that for X − K → 0implying ν → 0thenorm �m�L ∞ (0,T ) tends to
infinity.
Here we also see that at-the-money options with X = K represent a singular situation in
our purely time-dependent model, since we derive from (6) and (7) for ν = 0theformulae