The Mathematics of Arbitrage

344 15 A Compactness Principle

tend to a process of finite variation. The third term has a maximal function
that tends to zero since

∑

n

P

⎡

⎣

⋃

k≥n

({Tk<∞}∪{Sn<∞})

⎤

⎦<∞.

Modulo the proof of the claim above, the proof of Theorem 15.B is com-
plete. So let us now prove the claim.
It is sufficient to show that for an arbitrary selection ofd+ 1 indices
n 1 <···<nd+1we necessarily have thatE=

⋂

k...d+1E

nk=∅,dλ-a.s.. For

eachkwe look at the compensator of the processes

(

∆(Hnk·M)Tnk

)+

(^1) [[T ̃n
k,∞[[
resp.

(

∆(Hnk·M)Tnk

)−

(^1) [[T ̃n
k,∞[[

.

Let+Enk (resp.−Enk) be the supports of the compensators of these pro-
cesses. For each of the 2d+1sign combinationsεk=+/−we look at the set
⋂d+1
k=1
εkEnk.IfthesetEis non-empty, then at least one of these 2d+1sets

would be non-empty and without loss of generality we may and do suppose
that this is the case forεk=+foreachk.
For eachkwe now introduce the compensatorC ̃kof the process
((
trace([M, M]Tnk)

)^12

+1

)

(^1) {∆(Hnk·M)Tnk> 0 } (^1) [[T ̃n
k,∞[[

.

The processesHnkared-dimensional processes and hence for each (t, ω)we
find that the vectorsHntk(ω) are linearly dependent. Using the theory of linear
systems and more precisely the construction of solutions with determinants
we obtain (d+ 1)-predictable processes (αk)dk+1=1such that

(1) for each (t, ω) at least one of the numbersαk(t, ω) is nonzero
(2)

∑

kα

kHnk=0

(3) the processesαkare all bounded by 1.

We emphasize that these coefficients are obtained in a constructible way and
that we do not need a measurable selection theorem!
We now look at the compensator of the processes

∆(Hnk·M)Tnl (^1) {∆(Hnk·M)Tnk> 0 } (^1) [[T ̃n
k,∞[[

.

This compensator is of the formgl,kdC ̃lfor a predictable processgl,k.Be-
cause of the construction of the coefficients, we obtain that for eachl≤d+1:

∑

gl,kαkn=0.

The next step is to show on the set

⋂d+1

k=1

+Enk,thematrix(gl,k)
l,k≤d+1
is non-singular. This will then give the desired contradiction, because the