32 2 Models of Financial Markets on Finite Probability Spaces
Proof of Theorem 2.6.1.First assume thatT= 1, i.e., we have a one-period
modelS=(S 0 ,S 1 ). In this case the present theorem is just a reformulation
of Theorem 2.4.2: ifVis a super-martingale under eachQ∈Me(S), then
EQ[V 1 −V 0 ]≤0, for allQ∈Me(S).Hence there is a predictable trading strategyH(i.e., anF 0 -measurableRd-
valued function - in the present caseT= 1) such that (H·S) 1 ≥V 1 −V 0.
LettingC 0 =0andwriting∆C 1 =C 1 =−V 1 +(V 0 +(H·S) 1 )wegetthe
desired decomposition. This completes the construction for the caseT=1.
For generalT>1 we may apply, for each fixedt∈{ 1 ,...,T},thesame
argument as above to the one-period financial market (St− 1 ,St) based on
(Ω,F,P) and adapted to the filtration (Ft− 1 ,Ft). We thus obtain anFt− 1 -
measurable,Rd-valued functionHtand a non-negativeFt-measurable function
∆Ctsuch that
∆Vt=(Ht,∆St)−∆Ct,
where again (., .) denotes the inner product inRd. This will finish the con-
struction of the optional decomposition: define the predictable processHas
(Ht)Tt=1and the adapted increasing processC byCt =
∑t
u=1∆Cu.This
proves the implication (i)⇒(ii).
The implications (ii)⇒(i’)⇒(i) are trivial.