The Mathematics of Arbitrage

6.10 The DMW TheoremforT≥1 via Closedness ofC 107

Since (Hn·S)−T is a.s. bounded we have that ((ψ 1 ,∆S 1 )+f)≥0andby
the(NA)condition this implies (ψ 1 ,∆S 1 )+f= 0. This meansψ 1 =0by
(i). Since|ψ 1 |=1onAwe must haveP[A]=0provingthat(H 1 n)∞n=1is a
bounded sequence. The sums from 2 toT

∑T

t=2

Htn∆St=(Hn·S)T−(H 1 n,∆S 1 )

satisfy (
∑T

t=2

Htn∆St

)

−

≤(Hn·S)−T+|(H 1 n,∆S 1 )|

and hence are a.s. bounded. The inductive hypothesis shows that ((Htn)Tt=2)∞n=1
is then a.s. bounded too.

We are now ready to prove the main result of this section.

Theorem 6.9.2.If(St)Tt=0isRd-valued and adapted with respect to the fil-
tration(Ft)Tt=0,ifSsatisfies the (NA) condition, then the cone

C=K−L^0 +(Ω,FT,P)={(H·S)T−h|Hpredictable,h≥ 0 }

is closed inL^0 (Ω,FT,P).

Proof.Letfn=(Hn·S)T andhn≥0 be such thatgn=fn−hn→g.
Clearly (Hn·S)−T =fn−≤gn−forms a bounded sequence. By Proposition
6.9.1 (ii) and (iii) we have that (Hn)∞n=1itself is already bounded and we also
have the existence of aFT-measurably parameterised subsequenceσnso that
fσn →f∈K. Then necessarilyhσn =fσn−gσntends a.s. tof−g=h
and we have thereforeh≥0. Moreoverg=f−h∈Ca.s. which proves the
theorem.

6.10 Proof of the Dalang-Morton-Willinger Theorem forT≥

forT≥1 using the Closedness ofC

Proof of Theorem 6.1.1.The proof is the same as in Sect. 6.5 for Theorem
6.5.1, except for some obvious variations. Let us indicate the changes. First
we take an equivalent probability measureP 1 so thatddPP^1 is bounded and
St∈L^1 (P 1 ) for all 0≤t≤T. For example we may takeddPP^1 =cexp(−|S 0 |−
...−|ST|), wherecis the normalisation constant given byc−^1 =E[exp(−|S 0 |−
...−|ST|)].
The next step consists in considering the set

C 1 =C∩L^1 (Ω,FT,P 1 ).