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

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1332 THeinandBHofmann where Sε im is an intermediate function satisfying the inequalities min(S(t), S(t) + ε[J(h)](t)) � S ε im (t) � max(S(t), S(t) + ε[J(h)](t)). This limiting process leads to a composition G = M ◦ J of the convolution operator J with a multiplication operatorM described by a multiplier function m in the form [G(h)](t) = m(t)[J(h)](t) (0�t�T, h ∈ L 2 (0, T )). (33) The multiplier function attains the form m(0) = 0, m(t) = ∂UBS(X, K, r, t, S(t)) > 0 (0 < t � T ) (34) ∂s with S = J(a) and we can prove the following. Theorem 5.4. In the case X �= K,thelinear operator G defined by the formulae (33) and (34) maps continuously in L2 (0, T ) with m ∈ L∞ (0, T ). Then condition (i) of proposition 5.3 is satisfied with aconstant � � L = TC2, where C2 := sup � ∂ � (t,s)∈Mc 2UBS(X, K, r, t, s) ∂s 2 � � � � < ∞ is determined from the set Mc := {(t, s) ∈ R 2 : s � ct, 0 < t � T }. Proof. To prove the continuity of G = M ◦ J in L2 (0, T ) with the continuous convolution operator J,itissufficient to show m ∈ L∞ (0, T ),since then the multiplication operator M is also continuous in L2 (0, T ). Fromformula (6) we obtain for (t, s) ∈ [0, T ] × (0, ∞) in the case X �= K the estimate � � � � ∂UBS(X, K, r, t, s) � � � ∂s � � � � X � XK 1 [ln( K ) + rt]2 √ exp − . 8π s 2s This implies for (t, s) ∈ Mc � � � � ∂UBS(X, K, r, t, s) � � � ∂s � � � � � r � XK K c X 1 [ln( K √s exp − 8π X )]2 � . (35) 2s The right-hand expression in inequality (35) is continuous with respect to s ∈ (0, ∞) and tends ∂UBS(X,K,r,t,s) to zero as s → 0andass →∞.With a finite constant C1 := sup | (t,s)∈Mc ∂s | < ∞ we have m ∈ L∞ (0, T ),where�m�L∞ (0,T ) � C1 comes from the inequality S(t) � ct (0 � t � T ), whichisaconsequence of a ∈ D † (F). Inorder to prove condition (i) of proposition 5.3 we verify the structure of the second derivative ∂ 2 UBS(X,K,r,t,s) ∂s 2 from formula (7). Similar considerations as in the case of the first derivative also show the existence of a constant C2 := sup (t,s)∈Mc | ∂ 2UBS(X,K,r,t,s) ∂s 2 | < ∞. Thenwecan estimate with S = J(a), ˜S = J(ã) and a, ã ∈ D † (F) for all t ∈ (0, T ]: |[F(ã) − F(a) − G(ã − a)](t)| � � = � �UBS(X, K, r, t, ˜S(t)) − UBS(X, K, r, t, S(t)) − ∂UBS(X, � K, r, t, S(t)) � ( ˜S(t) − S(t)) � ∂s � = 1 � � � ∂ 2 � 2UBS(X, K, r, t, Sim(t)) ∂s 2 ( ˜S(t) − S(t)) 2 � �� t �2 C2 � � � (ã(τ) − a(τ)) dτ , 2 0
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
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