The Mathematics of Arbitrage

15.4 A Substitute of Compactness for Bounded Subsets ofH^1343

We consider the martingales

(∑

kλ

k

nV

k)·M.ForeachnletDnbe the

compensator of (

∑

kλ

k
n∆(H
k·M)T
k^1 [[T ̃k,∞[[. This is a predictable process
of integrable variation. MoreoverE[varDn] ≤

∑

kλ
k
nE[|∆(H
k·M)T
k|]≤
2

∑

kλ

k

n‖H

k·M‖H 1 ≤2. We now apply the Kadeˇc-Pelczy ́nski decompo-

sition technique to the sequence varDn and we obtain, if necessary by
passing to a subsequence, a sequence of numbers

∑

n

1
ξn < ∞such that
varDn∧ξnis uniformly integrable. Again we define predictable stopping times
Sn =inf{t|var(Dn)t≥ξn}. We stop the processes at time (Sn−)since
this will not destroy the martingale properties. More precisely we decompose∑

kλ

k

nV

k·Mas follows:

∑

k

λknVk·M

=

(

∑

k

λknVk·M−

(

∑

k

λkn∆(Hk·M)Tk (^1) [[T ̃k,∞[[−D ̃n
))Sn−
first term

+

(

∑

k

λkn∆(Hk·M)Tk (^1) [[T ̃k,∞[[−D ̃n
)Sn−
second term

+

⎛

⎝

(

∑

k

λknVk·M

)

−

(

∑

k

λknVk·M

)Sn−⎞

⎠ third term.

Since

(

[Dn,Dn]Sn−

) (^12)
≤2(varDn)Sn−≤(varDn)∧ξn, we obtain that the
first term defines a relatively weakly compact sequence inH^1. Indeed, for each
nwe have [[T ̃n]]⊂En⊂[[ 0,Tn]] and hence:
[first term,first term]
(^12)
∞
≤

∑

λkn[Vk·M, Vk·M]

(^12)
T ̃n−+[D
n,Dn]^12
Sn−
≤[Hn·M, Hn·M]
(^12)
∧βn([M, M]∞+1)
+[Hn·M, Hn·M]
1
(^21) {T
n=T ̃n}+[D
n,Dn]
∞∧ξ
n.
It follows that the first term defines a relatively weakly compact sequence
inH^1. But the first term is supported by the set

⋃

k≥nE

k, which tends to

the empty set ifn→∞. From Theorem 15.2.12, it then follows that the
sequence defined by the first term tends to zero weakly. The appropriate
convex combinations will therefore tend to 0 in the norm ofH^1.
The second term splits in
∑

k

λkn∆(Hk·M)Tk (^1) [[T ̃k,∞[[,
whose maximal functions tend to zero a.s. and the processes (Dn)Sn−.Onthe
latter we can apply Theorem 15.1.4, which results in convex combinations that