The Mathematics of Arbitrage

338 15 A Compactness Principle

or which is the same:

Nn=(Mn)Tn∧σn−(∆ (Mn))Tn∧σn (^1) [[Tn∧σn,∞[[+(Cn)σn−.
The maximal functions satisfy
(Nn)∗≤(Mn)∗∧βn+(varCn)∧ηn
and hence form a uniformly integrable sequence. It follows that the sequence
Nnis a relatively weakly compact sequence inH^1. Using the appropriate
convex combinations will then yield a limitM^0 inH^1.
The problem is that the difference betweenMnandNndoes not tend
to zero in any reasonable sense as shown by Example 15.3.1 above. Let us
therefore analyse this difference:
Mn−Nn
= Mn−(Mn)Tn∧σn+(∆Mn)Tn∧σn (^1) [[Tn∧σn,∞[[−(Cn)σn−.
The maximal function of the first part
(
Mn−

(

(Mn)Tn∧σn

))∗

,

tends to zero a.s. because ofP[Tn<∞]andP[σn<∞] both tending to zero.
The same argument yields that the maximal function of the second part
(

(∆Mn)Tn∧σn (^1) [[Tn∧σn,∞[[

)∗

also tends to zero. The remaining part is (−Cn)σn−. Applying Theorem 15.1.4
then yields convex combinations that converge in the sense of Theorem 15.1.4
to a cadlag process of finite variationZ.
Summing up, we can find convex coefficients

(

αkn

)

k≥nsuch that the mar-
tingales

∑

k≥nα

k

nN

nwill converge inH (^1) -norm to a martingaleM (^0) and such
that, at the same time,

∑

k≥nα

k

nC

nconverge to a process of finite variation

Z, in the sense described in Lemma 15.2.7.
In the case where the jumps ∆Mnare bounded below by an integrable
functionw, or more generally when the set
{
∆(Mn)−ζ

∣

∣

∣n≥1;ζstopping time

}

is uniformly integrable, we do not have to compensate the negative part of
these jumps. So we replace (∆Mn)Tnby the more appropriate ((∆Mn)Tn)

• .
  In this case their compensatorsCnare increasing and therefore the process
  Zis decreasing.
  The case where the jumps form a uniformly integrable family is treated in
  the remark after the proof of Theorem 15.A. The proof of Theorem 15.C is
  therefore completed.