The Mathematics of Arbitrage

15.2 Notations and Preliminaries 329

As a first application of the Kadeˇc-Pelczy ́nski decomposition we prove the
vector-valued Komlos-type theorem stated in the introduction:

Theorem 15.2.6.IfEis a reflexive Banach space and(fn)n≥ 1 abounded
sequence inL^1 (Ω,F,P;E)we may find convex combinations

gn∈conv{fn,fn+1,...}

andg 0 ∈L^1 (Ω,F,P;E)such that(gn)n≥ 1 converges tog 0 almost surely, i.e.,

lim

n→∞

‖gn(ω)−g 0 (ω)‖E=0, for a.e.ω∈Ω.

Proof.By the remark made above there is a subsequence, still denoted by
(fn)n≥ 1 as well as a sequence (An)n≥ 1 of mutually disjoint sets such that

the sequence‖fn‖ (^1) Acnis uniformly integrable. By a well-known theorem on
L^1 (Ω,F,P;E)ofareflexive space E, [DU 77], see also [DRS 93], the se-
quence (fn (^1) Acn)n≥ 1 is therefore relatively weakly compact inL^1 (Ω,F,P;E).
Therefore (see Theorem 15.1.2 above) there is a sequence of convex combi-
nationshn∈conv{fn (^1) Acn,fn+1 (^1) Acn+1,...},hn=

∑

k≥nα
k
nfk^1 Acksuch that
hnconverges to a functiong 0 with respect to the norm ofL^1 (Ω,F,P;E).

Since the sequencefn (^1) An converges to zero a.s. we have that the sequence
gn=

∑

k≥nα

k
nfkconverges tog^0 in probability. If needed one can take a fur-
ther subsequence that converges a.s., i.e.,‖gn(ω)−g 0 (ω)‖Etends to zero for
almost eachω.

The preceding theorem allows us to give an alternative proof of [K 96a,
Lemma 4.2].

Lemma 15.2.7.Let(Nn)n≥ 1 be a sequence of adapted cadl ag stochastic pro-
cesses,N 0 n=0, such that

E[varNn]≤ 1 ,n∈N,

wherevarNndenotes the total variation of the processNn.
Then there is a sequenceRn∈conv{Nn,Nn+1...}and an adapted cadl ag
stochastic processZ=(Zt)t∈R+such that

E[varZ]≤ 1

and such that almost surely the measuredZt, defined on the Borel sets ofR+,
is the weak-star limit of the sequencedRnt. In particular we have that

Zt= lim

s↘t

lim sup

n→∞

Rsn= lim

s↘t

lim inf

n→∞

Rsn.

Proof.We start the proof with some generalities of functional analysis that
will allow us to reduce the statement to the setting of Theorem 15.1.4.
ThespaceoffinitemeasuresMon the Borel sets ofR+is the dual of
the spaceC 0 of continuous functions onR+ =[0,∞[, tending to zero at