The Mathematics of Arbitrage

330 15 A Compactness Principle

infinity. If (fk)k=1is a dense sequence in the unit ball ofC 0 , then for bounded
sequences (μn)n≥ 1 inM, the weak-star convergence of the sequenceμnis
equivalent to the convergence, for eachk,of

∫

fkdμn. The mappingΦ(μ)=
(2−k

∫

fkdμ)k≥ 1 maps the space of measures into the space^2. The image of
a bounded weak-star-closedconvex set is closed in^2. Moreover on bounded
subsets ofM, the weak-star topology coincides with the norm topology of its
image in^2.
For eachnthe cadlag processNnof finite variation can now be seen as
a function of Ω intoM, mapping the pointωonto the measuredNtn(ω). Using
Theorem 15.1.3, we may find convex combinationsPn∈conv{Nn,Nn+1,...},
Pn=

∑

k≥nα

k

nN

ksuch that the sequence∑

k≥nα

k

nvar(N

k) converges a.s..

This implies that a.s. the sequencePn(ω) takes its values in a bounded set
ofM. Using Theorem 15.1.4 on the sequence (Φ(Pn))n≥ 1 we find convex

combinationsRn=

∑

k≥nβ

k

nP

kof(Pk)

k≥nsuch that the sequenceΦ(dR

n)=

Φ(

∑

k≥nβ

k

ndP

k

t) converges a.s.. Since a.s. the sequence of measuresdR

n(ω)

takes its values in a bounded set ofM, the sequencedRnt(ω) converges a.s.
weak-startoameasuredZt(ω). The last statement is an obvious consequence
of the weak-star convergence. It is also clear thatZ is optional and that
E[var(Z)]≤1.

Remark 15.2.8.If we want to obtain the processZas a limit of a sequence
of processes then we can proceed as follows. Using once more convex com-
binations together with a diagonalisation argument, we may suppose that
Rns converges a.s. for each rationals.Inthiscasewecanwritethata.s.
Zt= lims↘t;s∈Qlimn→∞Rns. We will use such descriptions in Sections 15.4
and 15.5.

Remark 15.2.9.Even if the sequenceNnconsists of predictable processes, the
processZneed not be predictable. Take e.g.Ta totally inaccessible stopping
time and letNndescribe the point mass atT+^1 n. Clearly this sequence

tends, in the sense described above, to the process (^1) [[T,∞[[, i.e. the point mass
concentrated at timeT, a process which fails to be predictable. Also in general,
there is no reason that the processZshould start at 0.
Remark 15.2.10.It might be useful to observe that ifT is a stopping time
such thatZis continuous atT, i.e. ∆ZT=0,thena.s.ZT= limRnT.
We next recall well-known properties on weak compactness inH^1 .The
results are due to Dellacherie, Meyer and Yor (see [DMY 78]).
Theorem 15.2.11.For a family (Mi)i∈Iof elements ofH^1 the following
assertions are equivalent:
(1)the family is relatively weakly compact inH^1 ,
(2)the family of square functions([Mi,Mi]
(^12)
∞)i∈Iis uniformly integrable,
(3)the family of maximal functions((Mi)∗∞)i∈Iis uniformly integrable.