The Mathematics of Arbitrage

15.4 A Substitute of Compactness for Bounded Subsets ofH^1339

15.4.3Proof of Theorem 15.B.The case where all martingales are of the
formMn=Hn·M.

This situation requires, as we will see, some extra work. We start the construc-
tion as in the previous case but this time we work with the square functions,
i.e., the brackets instead of the maximal functions.
Without loss of generality we may suppose thatMis anH^1 -martingale.
Indeed let (μn)n≥ 1 be a sequence of stopping times that localises the lo-
cal martingale M in such a way that the stopped martingalesMμn are
all in∑ H^1. Take now a sequence of strictly positive numbersansuch that

nan‖M

μn‖H 1 <∞, putμ 0 = 0 and replaceMby theH (^1) -martingale:
∑
n≥ 1
an(Mμn−Mμn−^1 ).
The integrands have then to be replaced by the integrands
∑
k≥ 1

1

ak

Hn (^1) ]]μk− 1 ,μk]].
In conclusion, we may assume w.l.g., thatMis inH^1.
Also without loss of generality we may suppose that the predictable inte-
grands are bounded. Indeed for eachnwe can takeκnbig enough so that
‖

(

Hn (^1) {|Hn|≥κn}

)

·M‖H 1 < 2 −n.

It is now clear that it is sufficient to prove the theorem for the sequence of

integrandsH (^1) {‖Hn‖≤κn}. So we suppose that for eachnwe have|Hn|≤κn.
We apply the Kadeˇc-Pelczy ́nski construction of Theorem 15.2.1 with the
functiong=(trace([M, M]∞))
(^12)

. Without changing the notation we pass to
a subsequence and we obtain a sequence of numbersβn, tending to∞,such
that the sequence

[Hn·M, Hn·M]

(^12)
∞∧βn

(

(trace([M, M]∞))

(^12)
+1

)

is uniformly integrable.
The sequence of stopping timesTnis now defined as:

Tn=inf

{

t

∣

∣

∣[Hn·M, Hn·M]

(^12)
t ≥βn

(

(trace([M, M])t)

1

(^2) +1

)}

.

In the general case the sequence of jumps ∆ (Hn·Mn)Tnis not uniformly
integrable and so we have to eliminate the big parts of these jumps. But this
time we want to stay in the framework of stochastic integrals with respect to
M. The idea is, roughly speaking, to cut out of the stochastic interval [[0,Tn]] ,
the predictable support of the stopping timeTn.Ofcoursewethenhaveto