The Mathematics of Arbitrage

6.4 The Closedness of the ConeC 93

As regards (i) note that the assumption of the a.s. boundedness of
((Hn,∆S))∞n=1implies that (L̂k,∆S) tends to zero a.s.; indeed, on the set

Bthe ratio
̂Lk
Lk tends to zero, while outside ofBthe integrand
L̂kvanishes.
Hence (
L,̂∆S

)

= lim

k→∞

(

L̂k,∆S

)

= 0, a.s.,

which by Lemma 6.2.6 implies thatL̂= 0, the required contradiction.
As regards (ii) note that
(
L,̂∆S

)

−

= lim

k→∞

(

L̂k,∆S

)

−

= 0, a.s..

From the no-arbitrage assumption(NA)we now may conclude that

(

L,̂∆S

)

=

0 a.s., which again implies thatL̂= 0 and yields the desired contradiction.
(i’): Suppose now that (Hn)∞n=1 does not converge a.s., assume that
((Hn,∆S))∞n=1does so and let us work towards a contradiction. By (i) above
we may assume, that (Hn)∞n=1is a.s. bounded. Applying Proposition 6.3.3
toK=Rd∪{∞}, we may find a measurably parameterised subsequence
(Hτk)∞k=1converging a.s. to someH^0. Applying Proposition 6.3.4 (ii) tof 0 =
H^0 we may find another measurably parameterised subsequence (Hσk)∞k=1

converging a.s. to someĤ^0 ∈Hfor which we haveP

[

Ĥ^0 =H^0

]

>0. Note

thatH^0 as well asĤ^0 are in canonical form.
As

(

H^0 −Ĥ^0 ,∆S

)

= limn→∞(Hn,∆S)−limn→∞(Hn,∆S) = 0, we ob-

tain again a contradiction to Lemma 6.2.6.
(ii’): Suppose now thatSsatisfies(NA),that(Hn)∞n=1does not converge to
zero a.s., while ((Hn,∆S)−)∞n=1does. Again we may use Proposition 6.3.4 (ii),
this time applied tof 0 = 0, to find a measurably parameterised subsequence
of (Hn)∞n=1converging a.s. to someH^0 withP[H 0 =0]>0. We have

(H^0 ,∆S)−= lim

n→∞

(Hn,∆S)−= 0, a.s..

From(NA)we conclude that (H^0 ,∆S) = 0 a.s. so that we get again a con-
tradiction to Lemma 6.2.6.

We can now formulate a theorem on the closedness of the space (resp.
cone) of the stochastic integrals (resp. of functions dominated by stochastic
integrals). Assertion (i) is due to Ch. Stricker [Str 90] and (ii) is due to the
second named author [S 92].

Theorem 6.4.2.Let theRd-valued one-step processS=(S 0 ,S 1 )be adapted
to(Ω,(Ft)^1 t=0,P)

(i) The vector space

K={(H,∆S)|H∈L^0 (Ω,F 0 ,P;Rd)}

is closed inL^0 (Ω,F 1 ,P).