The Mathematics of Arbitrage

342 15 A Compactness Principle

Let us defineLn=

∑

kα

k
nH
k. Clearly‖Ln·M‖
H^1 ≤1foreachn.From
Theorem 15.1.3, it follows that there are convex combinations (ηnk)k≥n, dis-

jointly supported, such that

∑

kη

k

n[L

k·M, Lk·M]∞ (^21) converges a.s.. Hence
we have that supn

∑

kη

k

n[L

k·M, Lk·M]∞^12 <∞a.s.. We also may suppose

that maxkηkn→0asn→∞. From Minkowski’s inequality for the bracket it

follows that also supn

[(∑

kη

k

nL

k)·M,(∑

kη

k

nL

k)·M]^12 <∞a.s.. Because

the convex combinations were disjointly supported we also obtain a.s. and for
Rn=

∑

kη

k

n

∑

jα

j

kV

j:

sup

n

[Rn·M, Rn·M]

(^12)
∞≤supn

[(

∑

k

ηknLk

)

·M,

(

∑

k

ηnkLk

)

·M

] (^12)
∞

<∞.

From the fact that the convex combinations were disjointly supported and
from

∑

n^1 En≤d, we conclude that for each point (t, ω)∈R+×Ω, onlyd
vectorsRn(t, ω) can be nonzero. Let us putPn=

∑s=2n+1

s=2n+1^2

−nRs. It follows

that a.s.

∫

Pnd[M, M]Pn≤d

∫

⎛

⎝

s=2∑n+1

s=2n+1

2 −^2 nRsd[M, M]Rs

⎞

⎠

≤d 2 −n

s=2∑n+1

s=2n+1

2 −n[Rs·M, Rs·M]∞

≤d 2 −nsup

s

[Rs·M, Rs·M]∞

→ 0.

If we now putUn=

∑k=2n+1

k=2n+1^2

−n∑

kη

k

l

∑

lα

l

kH

l, we arrive at convex

combinationsUn=

∑

λlnHlsuch that

(1) the convex combinationsλknare disjointly supported,
(2)

(∑

kλ

k

nK

k)·M→H (^0) ·MinH (^1) ,
(3)

[(∑

kλ

k

nV

k)·M,(∑

kλ

k
nV
k)·M]
∞→0 in probability,
(4)

[(∑

kλ

k

nW

k)·M,(∑

kλ

k

nW

k)·M]
∞→0 in probability, and even
(5)

((∑

kλ

k

nW

k)·M)∗→0 in probability.

As a consequence we obtain that

[(

Un−H^0

)

·M,

(

Un−H^0

)

·M

]

∞→0in
probability, and hence inLp(Ω,F,P)foreachp<1.
We remark that these properties will remain valid if we take once more
convex combinations of the predictable processes(∑ Un. The stochastic integrals

kλ

k

nV

k)·M need not converge in the semi-martingale topology as the

example in Sect. 15.3 shows. But exactly as in the Subsect. 15.4.2 we will show
that after taking once more convex combinations, they converge in a pointwise
sense, to a process of finite variation.