The Mathematics of Arbitrage

6.3 The Selection Principle 89

HX=

{

f:Ω→Rd

∣

∣fisF 0 -measurable andPf=f} (6.4)

By construction we have forg∈EXandf∈HXthat (f, g)Rd=0almost
surely. Therefore we could say thatfis “orthogonal” toEXa.s.. The space
P(Rd) is a subspace ofRdwhich is dependent onω∈ΩinanF 0 -measurable
way. In some sense (see below for a precise statement)P(Rd)isthebest
F 0 -measurable prediction of the space spanned byX.
More precisely:

Lemma 6.2.4.LetX∈L^0 (Ω,F 1 ,P;Rd)and letHXbe defined as above. If
h∈L^0 (Ω,F 1 ,P;R)isF 0 -measurable andhX∈L^1 (Ω,F 1 ,P;Rd)thenE[hX|
F 0 ]∈HX.

Proof.Forα∈EXwe have

(α, hX)Rd=h(α, X)Rd= 0, a.s..

Therefore we have for boundedα∈EXthatE[(α, hX)Rd|F 0 ]=(α,E[hX|
F 0 ])Rd= 0 almost surely. In other wordsE[hX|F 0 ]∈HX.

We shall apply the above results to the situation whereX=∆S 1 =S 1 −S 0
for a one-step financial market (St) adapted to the filtration (Ft)^1 t=0.For
H∈L^0 (Ω,F 0 ,P;Rd) the random variable (H, X)Rdthen equals the stochas-
tic integral (H·S) 1. In general the integration mapI:L^0 (Ω,F 0 ,P;Rd)→
L^0 (Ω,F 1 ,P) mappingH to (H·S) 1 is not injective; this was precisely the
theme of the above considerations. We have to restrict the integration mapI
to the spaceHXin order to make it injective. This is done in the subsequent
Definition 6.2.5 and in Lemma 6.2.6.

Definition 6.2.5.We say thatH∈L^0 (Ω,F 0 ,P;Rd)is in canonical form for
(S 0 ,S 1 )ifH∈HXwhereHXis defined in (6.3) and (6.4) withX=S 1 −S 0.

Lemma 6.2.6.The kernel of the mappingI:L^0 (Ω,F 0 ,P,Rd)→L^0 (Ω,F 1 ,P)
equalsEX. The restriction ofItoHXis injective, linear and has full range.

Proof.LetHandH′be inL^0 (Ω,F 0 ,P;Rd) such thatI(H)=I(H′). Then
(H−H′,X) = 0 a.s. and hence (H−H′)∈EX.SinceH−H′∈HXwe
necessarily haveH−H′∈HX∩EX={ 0 }.

6.3 The SelectionPrinciple...................................

In this section we consider a probability space (Ω,F,P) and a compact met-
ric space (K,d). In the applications below (K,d) will typically be the one-
point compactificationRd∪{∞}ofRd. The Bolzano-Weierstrass Theorem
states that, for a sequence (xn)∞n=1inK, we may find a convergent subse-
quence (xnk)∞k=1. We want to generalise this theorem to a sequence (fn)∞n=1