The Mathematics of Arbitrage

14.3 One-period Processes 285

below for a general version of this result; we refer to [S 94] for an account on
the history of this result, in particular on the work of J.A. Yan [Y 80] and
D. Kreps [K 81]):

Lemma 14.3.1.IfSsatisfies (NA) the convex coneCis weak-star-closed
inL∞(Ω,F,P),andC∩L∞+(Ω,F,P)={ 0 }. Therefore the Hahn-Banach
theorem implies that there is a probability measureQ 1 onF,Q 1 ∼Psuch
that
EQ 1 [f]≤ 0 , forf∈C.

In the case, whenS 1 is uniformly bounded, the measureQ 1 is already the
desired equivalent martingale measure. Indeed, in this case the cone Adm of
admissible elements is the entire spaceRdand therefore

EQ 1 [(x, S 1 )]≤ 0 , forx∈Rd,

which implies that

EQ 1 [(x, S 1 )] = 0, forx∈Rd,

whenceEQ 1 [S]=0.
But if Adm is only a sub-cone ofRd(possibly reduced to{ 0 }), we can only
say much less: first of all,S 1 need not beQ 1 -integrable. But even assuming
thatEQ 1 [S 1 ] exists we cannot assert that this value equals zero; we can only
assert that

(x,EQ 1 [S 1 ]) =EQ 1 [(x, S 1 )]≤ 0 , forx∈Adm,

which means thatEQ 1 [S 1 ] lies in the cone Adm◦polar to Adm, i.e.,

EQ 1 [S 1 ]∈Adm◦={y∈Rd|(x, y)≤ 1 ,forx∈Adm}

={y∈Rd|(x, y)≤ 0 ,forx∈Adm}.

The next lemma will imply that, by passing fromQ 1 to an equivalent prob-
ability measureQwith distance‖Q−Q 1 ‖in total variation norm less than
ε>0, we may remedy both possible defects ofQ 1 :underQthe expectation
ofS 1 is well-defined and it equals zero.
The idea for the proof of this lemma goes back in the special cased=1
and Adm ={ 0 }to the work of D. McBeth [MB 91].

Lemma 14.3.2.LetQ 1 be a probability measure as in Lemma 14.3.1 and
ε> 0 .DenotebyBthe set of barycenters

B={EQ[S 1 ]|Qprobability onF,Q∼P,‖Q−Q 1 ‖<ε,

andS 1 isQ-integrable}.

ThenBis a convex subset ofRdcontaining 0 in its relative interior. In
particular, there isQ∼Q 1 ,‖Q−Q 1 ‖<ε, such thatEQ[S 1 ]=0.