9.5 The Set of Representing Measures 183Remark 9.5.4.The corollary was known long before Theorem 9.5.2 was known.
The earliest versions of it are due to Yor [Y 78a]. Contrary to intuition, the
boundedness condition needed in (iii) of Theorem 9.5.2 cannot be relaxed to
fbeing a member ofL^1 +(R)foreachRinMe(P). A counter-example can be
found in [S 93].
The next theorem is a new criterion for the completeness of the market.Theorem 9.5.5.IfSis locally bounded andPis a local martingale measure
forS,then
(i)M(P)is a closed convex bounded set ofL^1 (Ω,F,P)
(ii)M(P)=Me(P)implies thatM(P)={P}.
Proof.(i): We only have to show thatM(P)isclosed.TakeQna sequence
inM(P) and suppose thatQnconverges toQ.TakeTa stopping time such
thatSTis bounded. Ift<sandA∈Ftthen we can see that:EQ[STt (^1) A]
= limEQn[StT (^1) A] = limEQn[SsT (^1) A]=EQn[SsT (^1) A]. This provesQ∈M(P).
(ii): IfM(P)=Me(P)thenMe(P) is a closed, bounded, convex set.
The Bishop-Phelps theorem, see [D 75], states that the setGof elementsfof
L∞(Ω,F,P) that attains their supremum onMe(P), is a norm dense set in
L∞(Ω,F,P). The preceding theorem, part (ii), states thatGis a subset of
W.SinceWis weak-star-closed it is certainly norm closed. SinceWis closed
andGis dense for the norm topology we obtainW=L∞(Ω,F,P).Bythe
Hahn-Banach theorem, two distinct elements ofL^1 can be separated by an
element ofL∞i.e. by an element ofW. However, elements ofWare constant
onM(P). This implies thatM(P)={P}.
As we remarked in the introduction our results remain true forRd-valued
processes. The same holds for Theorem 9.5.5. As the example of [AH 95] shows,
Theorem 9.5.5 is no longer true for an infinite number of assets. The example
uses the set{ 0 , 1 }as time set, but as easily seen and stated in [AH 95] it is
easy to transform the example into a setting with continuous time.
In [D 92] the following identity was proved for a continuous processS.For
everyf∈L∞:
sup
Q∈M(P)
EQ[f]=inf{x|there ish∈W^0 withx+h≥f}.
In the general case this equality becomes false as the following example in
discrete time shows. The left hand side of the equality is always dominated by
the right hand side. The example shows that a “gap” is possible. Some further
properties displayed by Example 9.5.6 are:W^0 is weak-star-closed but the set
W^0 −L∞+is not even norm closed. We will also see that the norm closure and
the weak-star-closure ofW^0 −L∞+are different.
Example 9.5.6.The set Ω is the setN={ 1 , 2 , 3 ,...}of natural numbers. The
σ-algebraFnis theσ-algebra generated by the atoms{k}fork≤ 3 nand the