The Mathematics of Arbitrage

6.8 The Closedness ofKin the CaseT≥ 1 103

∫

A

S 0 f 1 dQ 1 =

∫

A

S 1 f 1 dQ 1.

Let us finally defineQonFTby the rule

Q[A]=

∫

A

f 1 dQ^1 for allA∈FT.

Of course this means thatddQP=f 1 dQ

1
dP and hence this is a bounded random
variable. FurthermoreddQP>0 almost surely and henceQandPare equiva-
lent. Now let us check the integrability properties as well as the martingale
properties. Fort=1,...,Twe have
∫

Ω

|St|dQ=

∫

Ω

|St|f 1 dQ^1 <∞,

by construction ofQ^1 and the boundedness off 1.
The martingale property of (St)Tt=0 with respect toQis also an easy
calculation. Indeed, for allA∈F 0 we have
∫

A

S 0 dQ=

∫

A

S 0 f 1 dQ^1 =

∫

A

S 1 f 1 dQ^1 =

∫

A

S 1 dQ

by construction off 1 .Fort≥1weremarkthatf 1 wasF 1 -measurable and
bounded, which means that the sequence of the following equalities is easily
justified. IfA∈Ft,t≥1wehave
∫

A

StdQ=

∫

A

Stf 1 dQ^1 =

∫

A

St+1f 1 dQ^1 =

∫

A

St+1dQ.

This ends the proof of the induction step.

6.8 Proof of the Closedness ofKin the CaseT≥1 .............

In this section we extend Stricker’s lemma (Theorem 6.4.2 (i)) to the case
T≥1.

Proposition 6.8.1.Let the processS=(St)Tt=0beRd-valued and(Ft)Tt=0-
adapted. The space

K=

{ T

∑

t=1

(Ht,∆St)

∣

∣

∣

∣

∣

(Ht)Tt=1Rd-valued and predictable

}

is a closed subspace ofL^0 (Ω,FT,P).