The Mathematics of Arbitrage

332 15 A Compactness Principle

Hence ∥
∥
∥[X, X]

(^12)
∞

∥

∥

∥

p

≤c

(^12)
‖X‖
(^12)
H^1 ‖X
∗
∞‖
(^12)
p
2 −p

.

Corollary 15.2.15.IfXnis a sequence ofH^1 -martingales such that‖Xn‖H 1
is bounded and such that(Xn)∗∞tendstozeroinprobability,then[Xn,Xn]∞
tends to zero in probability.

In fact, for eachp< 1 ,(Xn)∗∞as well as[Xn,Xn]

(^12)
∞tend to zero in the
quasi-norm ofLp(Ω,F,P).
Proof.Fix 0<p<1. Obviously we have by the uniform integrability of the
sequence

(

(Xn)∗∞

) 2 −pp

,that‖(Xn)∗∞‖ 2 −ppconverges to zero. It then follows

from the theorem that also [Xn,Xn]∞→0 in probability.

Remark 15.2.16.It is well-known that, for 0≤p<1, there is no connection
between the convergence of the maximal function and the convergence of the
bracket, [MZ 38], [BG 70], [M 94]. But as the theorem shows, forboundedsets
inH^1 the situation is different. The convergence of the maximal function
implies the convergence of the bracket. The result also follows from the result
on convergence in law as stated in [JS 87, Corollary 6.7]. This was kindly
pointed out to us by A. Shiryaev. The converse of our Corollary 15.2.15 is
not true as the example in the next section shows. In particular the relation
between the maximal function and the bracket is not entirely symmetric in
the present context.

Remark 15.2.17.In the case ofcontinuousmartingales there is also an inverse
inequality of the type

E

[

(X∗∞)^2

[X, X]

(^12)
∞

]

≤c‖X‖H 1.

The reader can consult [RY 91, Example 4.17 and 4.18].

15.3 An Example

Example 15.3.1.There is a uniformly bounded martingaleM =(Mt)t∈[0,1]
and a sequence (Hn)n≥ 1 of M-integrands satisfying

‖Hn·M‖H 1 ≤ 1 , forn∈N,

and such that

(1) for eacht∈[0,1] we have

lim

n→∞

(Hn·M)t=−

t

2

a.s.

(2) [Hn·M, Hn·M]∞→0 in probability.