The Mathematics of Arbitrage

340 15 A Compactness Principle

show that these supports form a sequence of sets that tends to the empty set.
This requires some extra arguments.

Since|∆(Hn·M)Tn|≤[Hn·M, Hn·M]

(^12)
∞we obtain that the sequence
|∆(Hn·M)Tn|∧βn

(

(trace([M, M]∞))

(^12)
+1

)

is uniformly integrable. As in the proof of the Kadeˇc-Pelczy ́nski theorem we
then find a sequenceγn≥βnsuch thatγβnn→∞and such that the sequence

|∆(Hn·M)Tn|∧γn

(

trace([M, M]∞)

(^12)
+1

)

is still uniformly integrable. As a consequence also the sequences

|∆(Hn·M)Tn|∧βn

(

trace([M, M]Tn)

(^12)
+1

)

and
|∆(Hn·M)Tn|∧γn

(

trace([M, M]Tn)

(^12)
+1

)

are uniformly integrable.
By passing to a subsequence we may suppose that

(1) the sequencesβn,γnare increasing,
(2)

∑

n≥ 1

1

βn<∞and hence

∑

P[Tn<∞]<∞,
(3) γβnn→∞,

(4) for eachnwe have

κnβn+1(d+1)^2

γn+1

≤

1

(d+1)^2

,

which can be achieved by choosing inductively a subsequence, sinceγβnn

becomes arbitrarily large.

We now turn the sequence of stopping timesTn into a sequence of stop-
ping times having mutually disjoint graphs. This is done exactly as in Sub-
sect. 15.4.1 above. SinceP[Tn<∞] tends to zero, we may, taking a subse-
quence if necessary, suppose that

lim

n→∞

E

[

sup

j≤n

[Hj·M, Hj·M]

(^12)
∞^1 ⋃k>n{Tk<∞}

]

=0.

We now replace each stopping timeTnby the stopping timeτndefined by

τn=

{

Tn ifTn<Tkfor allk>n,

∞ otherwise.