The Mathematics of Arbitrage

350 15 A Compactness Principle

DefineL̂kas

L̂k=Lnk (^1) [[ 0,T
k]]+L
mk 1
]]Tk,∞[[
so thatL̂kis aw-admissible predictable integrand.
Denote bydkthe function indicating the difference betweenLnk·Mand
Lmk·Mat timeTk,ifTk<∞, i.e.,
dk=((Lnk−Lmk)·M)Tk (^1) {Tk<∞}.
Note that, fort∈R+,
(L̂k·M)t=(Lnk·M)t (^1) {t≤Tk}+((Lmk·M)t+dk) (^1) {t>Tk}.
By passing to convex combinations

∑∞

j=kαjL̂

jofL̂kwe therefore get that,

for eacht∈Q+,

(∑∞

j=k

αjL̂j·M

)

t

=

(∑∞

j=k

αjLnj·M

)

t

(^1) {t≤Tk}+

(∑∞

j=k

αjLmj·M

)

t

(^1) {t>Tk}+Dkt
where (Dkt)k≥ 1 =

(∑∞

j=kαjdj^1 {t>Tk}

)

k≥ 1 is a sequence of random variables
which converges almost surely to a random variableDtso that (Dt)t∈Q+is
an increasing adapted process which satisfiesP[DT>0]>0 by Lemma 9.8.1.
Hence (L̂k)k≥ 1 is a sequence ofw-admissible integrands such that, for all

t∈Q+,(L̂k·M)tconverges almost surely tôWt=Wt+Dt,andP[DT>
0]>0, a contradiction to the maximality ofWfinishing the proof.

Lemma 15.4.14.Under the conditions of Theorem 15.D and Lemma 15.4.12
there is a subsequence of the sequence(Ln)n≥ 1 , still denoted by(Ln)n≥ 1 ,and
an increasing sequence(Tj)j≥ 1 of stopping times,Tj≤j andP[Tj=j]≥
1 − 2 −j, such that, for eachj, the sequence of processes((Ln·M)(Tj)−)n≥ 1 is
uniformly bounded and the sequence((Ln·M)Tj)n≥ 1 is a bounded sequence of
martingales inH^1 (P).

Proof.First note that, fixingj∈N,C>0, and defining the stopping times

Un=inf{t||(Ln·M)t|≥C}∧j,

the sequence ((Ln·M)Un)n≥ 1 is bounded inH^1 (P). Indeed, this is a sequence
of super-martingales by [AS 94], hence

E[|(Ln·M)Un|]≤ 2 E

[

((Ln·M)Un)−

]

≤2(C+E[w]),

whence
E[|∆(Ln·M)Un|]≤2(C+E[w]) +C.
As the maximal function (Ln·M)∗Unis bounded byC+|∆(Ln·M)Un|
we obtain a uniform bound on theL^1 -norms of the maximal functions ((Ln·