The Mathematics of Arbitrage

15.4 A Substitute of Compactness for Bounded Subsets ofH^1335

inequality this implies the existence of a constantc<∞such that for alln:
E[(Mn)∗]≤c. From the Kadeˇc-Pelczy ́nski decomposition theorem we deduce
the existence of a sequence (βn)n≥ 1 , tending to∞and such that (Mn)∗∧βn
is uniformly integrable. The reader should note that we replaced the original
sequence by a subsequence. Passing to a subsequence once more also allows
to suppose that

∑∞

n=1

1

βn<∞.Foreachnwe now define

τn=inf{t||Mtn|>βn}.

ClearlyP[τn<∞]≤βcn for some constantc.IfweletTn=infk≥nτkwe
obtain an increasing sequence of stopping times (Tn)n≥ 1 such thatP[Tn<
∞]≤

∑

k≥n

c
βkand hence tends to zero. Let us now start with the case of
continuous martingales.

15.4.1Proof of Theorem 15.A.The case when the martingalesMnare
continuous.

Because of the definition of the stopping timesTn, we obtain that ((Mn)Tn)∗≤
(Mn)∗∧βnand hence the sequence ((Mn)Tn)n≥ 1 forms a relatively weakly
compact sequence inH^1. Also the maximal functions of the remaining parts
Mn−(Mn)Tn tend to zero a.s.. As a consequence we obtain the existence
of convex combinationsNn=

∑

k≥nα

k

n(M

k)Tkthat converge inH (^1) -norm to
a continuous martingaleM^0 .WealsohavethatRn=

∑

k≥nα

k

nM

kconverge

toM^0 in the semi-martingale topology and that (M^0 −Rn)∗∞tends to zero
in probability. From Corollary 15.2.15 in Sect. 15.2 we now easily derive that
[M^0 −Rn,M^0 −Rn]∞as well as (M^0 −Rn)∗∞tend to zero inLp,foreach
p<1.
If all the martingalesMnare of the formHn·Mfor a fixed continuousRd-
valued local martingaleM, then of course the elementM^0 is of the same form.
This follows from Yor’s Theorem 15.1.6, stating that the space of stochastic
integrals with respect toM, is a closed subspace ofH^1. This concludes the
proof of Theorem 15.A.

Corollary 15.4.1.If(Mn)n≥ 1 is a sequence of continuousH^1 -martingales
such that

sup

n

‖Mn‖H 1 <∞ and M∞n→ 0 in probability,

thenMntends to zero in the semi-martingale topology. As a consequence we
have that(Mn)∗→ 0 in probability.

Proof.Of course we may take subsequences in order to prove the statement.
So let us take a subsequence as well as stopping times as described in Theo-
rem 15.A. The sequence (Mn)Tnis weakly relatively compact inH^1 and since
MTnntends to zero in probability (becauseP[Tn<∞] tends to zero andM∞n