The Mathematics of Arbitrage

15.4 A Substitute of Compactness for Bounded Subsets ofH^1337

is uniformly integrable. In this case the set

{Mσn|n≥ 1 ,σa stopping time}

is, as easily seen, also uniformly integrable. The maximal function of the
stopped martingale (Mn)
Tn
is bounded by
(
(Mn)Tn

)∗

≤max

(

(Mn)∗∧βn,

∣

∣MTn

n

∣

∣

)

.

It is then clear that they form a uniformly integrable sequence. It is this
situation that arises in the proof of M. Yor’s theorem.

15.4.2Proof of Theorem 15.C.The case of anH^1 -bounded sequenceMn
of cadl ag martingales.

We again turn to the general situation. In this case we cannot conclude that
the stopped martingales (Mn)Tnform a relatively weakly compact set inH^1.
Indeed the size of the jumps at timesTnmight be too big. In order to remedy
this situation we will compensate these jumps in order to obtain martingales
that havesmallerjumps at these stopping timesTn.Foreachnwe denote

byCnthe dual predictable projection of the process (∆Mn)Tn (^1) [[Tn,∞[[.The
processCnis predictable and has integrable variation
E[varCn]≤E[|(∆Mn)Tn|]≤ 2 c.
The Kadeˇc-Pelczy ́nski decomposition 2.1 above yields the existence of a se-
quenceηntending to∞,

∑

n≥ 1

1

ηn <∞and such that (varC

n)∧ηnforms

a uniformly integrable sequence (again we replaced the original sequence by
a subsequence). For eachnwe now define thepredictablestopping timeσnas

σn=inf{t|varCtn≥ηn}.

Because the processCnstops at timeTnwe necessarily have thatσn≤Tn
on the set{σn<∞}.
We remark that whenX is a martingale and whenνis a predictable
stopping time, then the process stopped atν−and defined byXtν−=Xtfor
t<νandXtν−=Xν−fort≥ν, is still a martingale.
Let us now turn our attention to the sequence of martingales

Nn=

(

(Mn)Tn−

(

(∆Mn)Tn (^1) [[Tn,∞[[−Cn
))σn−
.
The processesNncan be rewritten as
Nn=

(

(Mn)Tn

)σn

−(∆ (Mn))σn (^1) [[σn,∞[[
−(∆Mn)Tn (^1) {σn=∞} (^1) [[Tn,∞[[+(Cn)σn−,