The Mathematics of Arbitrage

15.2 Notations and Preliminaries 331

This theorem immediately implies the following:

Theorem 15.2.12.If(Nn)n≥ 1 is a relatively weakly compact sequence in
H^1 ,if(Hn)n≥ 1 is a uniformly bounded sequence of predictable processes with
Hn→ 0 pointwise onR+×Ω,thenHn·Nntends weakly to zero inH^1.

Proof.We may and do suppose that|Hn|≤1and‖Nn‖H 1 ≤1foreachn.For
eachnand eachε>0, we defineEnas the predictable setEn={|Hn|>ε}.

We split the stochastic integralsHn·Nnas ( (^1) EnHn)·Nn+

(

(^1) (En)cHn

)

·Nn.
We will show that the first terms form a sequence that converges to 0 weakly.
Because obviously‖

(

(^1) (En)cHn

)

·Nn‖H 1 ≤ε, the theorem follows.

From the previous theorem it follows that the sequence (Hn (^1) En·Nn)n≥ 1
is already weakly relatively compact inH^1. Clearly (^1) En→0pointwise.It
follows thatFn=

⋃

k≥nE

ndecreases to zero asntends to∞.LetN be

a weak limit point of the sequence

((

Hk (^1) Ek

)

·Nk

)

k≥ 1. We have to show that

N=0.Foreachk≥nwe have that (^1) Fn·

((

Hk (^1) Ek

)

·Nk

)

=

(

Hk (^1) Ek

)

·Nk.

From there it follows that (^1) Fn·N=Nand hence by taking limits asn→∞,
we also haveN= (^1) ∅·N=0.

Related to the Davis’ inequality, is the following lemma, due to Garsia and
Chou, (see [G 73, pp. 34–41] and [N 75, p. 198] for the discrete time case); the
continuous time case follows easily from the discrete case by an application of
Fatou’s lemma. The reader can also consult [M 76, p 351, (31.6)] for a proof in
the continuous time case.
Lemma 15.2.13.There is a constantcsuch that, for eachH^1 -martingaleX,
we have
E

[

[X, X]∞

X∞∗

]

≤c‖X‖H 1.

This inequality together with an interpolation technique yields:

Theorem 15.2.14.There is a constantCsuch that for eachH^1 -martingale
Xand for each 0 <p< 1 we have:
∥
∥
∥[X, X]

(^12)
∞

∥

∥

∥

p

≤C‖X‖

21

H^1 ‖X

∗

∞‖

(^12)
p
2 −p

.

Proof.The following series of inequalities is an obvious application of the
preceding lemma and H ̈older’s inequality for the exponents^2 pand 2 −^2 p.The
constantcis the same as in the preceding lemma.

E

[

[X, X]

p

∞^2

]

=E

[

(X∞∗)

p

2

(

[X, X]∞

X∞∗

)p 2 ]

≤

(

E

[

[X, X]∞

X∗∞

])p 2 (

E

[

(X∗∞)

p

2 −p

])^2 − 2 p

≤c

p

(^2) ‖X‖
p 2
H^1 ‖X
∗
∞‖
p 2
p
2 −p

.