The Mathematics of Arbitrage

15.4 A Substitute of Compactness for Bounded Subsets ofH^1341

For eachnletT ̃nbe defined as

T ̃n=

{

τn if|∆(Hn·M)τn|>γn

(

(trace([M, M]))

(^12)
τn+1

)

,

∞ otherwise.

For eachnletF ̃nbe the compensator of the process (^1) [[T ̃n,∞[[.
∑We now analyse the supports of the measuresdF ̃n. The measuredλ=
n≥ 1
1
2 nd
F ̃nwill serve as a control measure. The measureλsatisfiesE[λ∞]<
∞by the conditions above. Letφnbe a predictable Radon-Nikod ́ym derivative
φn=d
F ̃n
dλ. It is clear that for eachnwe haveE
n={φn=0}⊂[[ 0,Tn]]. T h e
idea is to show the following assertion:
Claim 15.4.6.

∑

n≥ 11 En≤d,dλ-a.s.. Hence there are predictable sets, still
denoted byEn, such that

∑

n≥ 11 En ≤deverywhere and such thatE

n=

{φn=0},dλ-a.s..

We will give the proof at the end of this section.

For eachnwe decompose the integrandsHn=Kn+Vn+Wnwhere:

Kn= (^1) [[ 0,T ̃n]] (^1) (En)cHn
Vn= (^1) EnHn
Wn= (^1) ]]T ̃n,∞[[Hn.
SinceP[T ̃n < ∞] tends to zero, we have that the maximal functions
(Wn·M)∗∞tend to zero in probability.
We now show that the sequenceKn·Mis relatively weakly compact in
H^1. The brackets satisfy
[Kn·M, Kn·M]
(^12)
∞≤[H
n·M, Hn·M]^12
∞∧γn

(

[M, M]

21

∞+1

)

+[Hn·M, Hn·M]

(^12)
∞^1 {T ̃n=Tn}.
The first term defines a uniformly integrable sequence, the second term
defines a sequence tending to zero inL^1. It follows that the sequence [Kn·M,
Kn·M]
(^12)
∞is uniformly integrable and hence the sequenceKn·Mis relatively
weakly compact inH^1.
There are convex combinations (αkn)k≥nsuch that

(∑

kα

k

nK

k)·Mcon-

verges inH^1 to a martingale which is necessarily of the formH^0 ·M.Wemay
of course suppose that these convex combinations are disjointly supported, i.e.
there are indices 0 =n 0 <n 1 <n 2 < ...such thatαkj is 0 fork≤nj− 1 and

k>nj. We remark that if we take convex combinations of

(∑

kα

k

nK

k)·M,

then these combinations still tend toH^0 ·MinH^1 .Wewillusethisremark
in order to improve the convergence of the remaining parts ofHn.