82 5 The Kreps-Yan Theorem
(iii) For allf∈Cwe haveEQ[f]=
∑∞
n=1βnEQn[f]≤0.
The final assertion is obvious.
We now turn to the question how to characterise theσ(Lp,Lq)-closedness
of a coneC⊆Lp(Ω,F,P) as considered in the preceding Theorem 5.2.3 and
whether this is possible by only considering sequences rather than nets. In the
case 1≤p<∞this is quite obvious: by the Hahn-Banach theorem a convex
setC⊆Lpisσ(Lp,Lq)-closed, where^1 p+^1 q=1,iffCis closed w.r. to the norm
‖.‖pofLp. Hence in this case the closedness ofCcan be characterised by using
sequences. The price to pay for this comfortable situation is that — reading
Theorem 5.2.3 as an “if and only if” result — we only obtain a probability
measureQwith the additional requirementddQP∈Lq(Ω,F,P), i.e.,Qmust
have a finiteq-th moment, for some 1<q≤∞. This additional requirement
onQis not natural in most applications to finance: for example passing from
Pto an equivalent probability measureP 1 (an operation, which we very
often apply) the requirementddQP ∈Lq(P)doesnotimplyddPQ 1 ∈Lq(P 1 ), if
1 <q≤∞. Only for the caseq= 1 this difficulty disappears, as for equivalent
probability measuresQandPwe always have‖ddQP‖L (^1) (P)=1.Thisiswhy,in
general, the casep=∞,q= 1 is of prime importance and there is no “cheap”
way to make the subtleties of the weak-star topology onL∞disappear.
There is, however, a notable exception to these considerations: we shall
see in the next chapter that, for finite discrete timeT={ 0 ,...,T},weobtain
a version of the Fundamental Theorem of Asset Pricing, due to R. Dalang,
A. Morton, W. Willinger, where we get the additional requirement ddQP ∈
L∞(P) for the desired equivalent martingale measureQ“for free”. In this
case we therefore shall use Theorem 5.2.3 for the casep=1andq=∞.
The casep=∞in the above Theorem 5.2.3 is more subtle (and more
interesting). The subsequent result, which is essentially based on the Krein-
Smulian theorem, goes back to A. Grothendieck [G 54]. We only formulate it
for the case of convex cones, but it can be extended to general convex sets in
an obvious way.
Proposition 5.2.4.Let C be a convex cone in L∞(Ω,F,P).Denoteby
σ(L∞,L^1 )(resp.τ(L∞,L^1 )) the weak-star (resp. the Mackey) topology on
L∞and, for 0 <p≤∞,by‖.‖pthe norm topology induced by(Lp,‖.‖p)on
L∞(for 0 <p< 1 ‖.‖pis only a quasi-norm). We denote byball∞the unit
ball ofL∞(Ω,F,P).
The following assertions are equivalent
(i) Cisσ(L∞,L^1 )-closed.
(i’) Cisτ(L∞,L^1 )-closed.
(ii) C∩ball∞isσ(L∞,L^1 )-closed.
(ii’) C∩ball∞isτ(L∞,L^1 )-closed.
(iii) C∩ball∞is‖.‖p-closed, for every 0 <p<∞.
(iii’)C∩ball∞is‖.‖p-closed, for some 0 <p<∞.