The Mathematics of Arbitrage

15.4 A Substitute of Compactness for Bounded Subsets ofH^1349

Vt= lims↘t

,s∈Q+

V̂s,t∈R+

is an a.s. well-defined cadlag super-martingale.
LetWdenote the family of all cadlag super-martingalesW=(Wt)t∈R+
such thatW−V is increasing and such that there is a sequence (Ln)n≥ 1 of
w-admissible integrands such that

Wt= lims↘t

s∈Q+

lim

n→∞

(Ln·M)s, fort∈R+

is a.s. well-defined.
Introducing — similarly as in [K 96a] — the orderW^1 ≥W^2 onW,if
W^1 −W^2 is increasing, we may find a maximal elementW ∈W,withan
associated sequence (Ln)n≥ 1 ofw-admissible integrands.
Indeed, let (Wα)α∈Ibe a maximal chain inWwith associated sequences of
integrands (Lα,n)n≥ 1 ;then(W∞α)α∈Iis an increasing and bounded family of
elements ofL^1 (Ω,F,P) and therefore there is an increasing sequence (αj)j≥ 1
such that (W∞αj)j≥ 1 increases to the essential supremum of (W∞α)α∈I.The
cadlag super-martingaleW = limj→∞Wαjis well-defined and we may find
a sequence (Lαj,nj)j≥ 1 , which we reliable by (Ln)n≥ 1 ,sothat

Wt= lims↘t

s∈Q+

lim

n→∞

(Ln·M)s.

ClearlyWsatisfies the required maximality condition.

Lemma 15.4.13.Under the assumptions of the preceding Lemma 15.4.12 we
have that forT∈R+, the maximal functions

((Ln·M)−(Lm·M))∗T=sup

t≤T

|(Ln·M)t−(Lm·M)t|

tend to zero in measure asn, m→∞.

Proof.The proof of the lemma will use — just as in (9.4.6) and [K 96a] — the
buy low - sell highargument motivated by the economic interpretation ofLn
as trading strategies (see Remark 9.4.7).
Assuming that the assertion of the lemma is wrong there isT∈R+,α> 0
and sequences (nk,mk)k≥ 1 tending to∞such that

P

[

sup

t≤T

((Lnk−Lmk)·M)t>α

]

≥α.

Defining the stopping times

Tk=inf{t≤T|((Lnk−Lmk)·M)t≥α}

we haveP[Tk≤T]≥α.