The Mathematics of Arbitrage

336 15 A Compactness Principle

tends to zero in probability), we easily see thatMTnn tends to zero inL^1.

Doob’s maximum inequality then implies that

(

(Mn)Tn

)∗

tends to zero in

probability. It is then obvious that also (Mn)∗tends to zero in probability.

Because

(

(Mn)Tn

)∗

tends to zero in probability and because this sequence

is uniformly integrable, we deduce that the sequence (Mn)Tntends to zero in
H^1. The sequenceMntherefore tends to zero in the semi-martingale topol-
ogy.

Remark 15.4.2.The above corollary, together with Theorem 15.2.14, show
thatMntends to zero inHp(i.e., (Mn)∗tends to zero inLp)andinhp

(i.e., [Mn,Mn]

(^12)
∞tends to zero inLp)foreachp<1. For continuous local
martingales, however,Hpandhpare the same.
Remark 15.4.3.That we actually need that the sequenceMnis bounded in
H^1 , and not just inL^1 , is illustrated in the followingnegativeresult.
Lemma 15.4.4.Suppose that (Mn)n≥ 1 is a sequence of continuous, non-
negative, uniformly integrable martingales such thatM 0 n=1and such that
M∞n→ 0 in probability. Then‖Mn‖H 1 →∞.
Proof.Forβ>1wedefineσn=inf{t|Mtn>β}.Since

1=E

[

Mσnn

]

=βP[σ<∞]+

∫

{(Mn)∗≤β}

M∞n,

we easily see that limn→∞P[σn<∞]=^1 β. It follows from the Davis’ inequal-

ity that limn→∞‖Mn‖H^1 ≥climn→∞

∫∞

0 P[σn>β]dβ=∞. 

Remark 15.4.5.There are two cases where Theorem 15.A can easily be gener-
alised to the setting ofH^1 -martingales with jumps. Let us describe these two
cases separately. The first case is when the set

{∆Mσn|n≥ 1 ,σa stopping time}

is uniformly integrable. Indeed, using the same definition of the stopping times
Tnwe arrive at the estimate

(Mn)∗Tn≤(Mn)∗∧βn+

∣

∣∆MTn

n

∣

∣.

Because of the hypothesis on the uniform integrability of the jumps and by the

selection of the sequenceβnwe may conclude that the sequence

(

(Mn)Tn

)

n≥ 1
is relatively weakly compact inH^1. The corollary generalises in the same way.
The other generalisation is when the set

{M∞n|n≥ 1 }