346 15 A Compactness Principle
P
[
sup
t|(Hn·M)t−Nt|≥ 2 −n]
≤ 2 −n.The Borel-Cantelli lemma then implies that
sup
tsup
n|(Hn·M)t|<∞ a.s..For each natural numberkwe then define the stopping timeTkas:
Tk=inf{t| there isnsuch that|(Hn·M)t|≥k}.Because of the uniform boundedness intandnwe obtain that the sequence
TksatisfiesP[Tk<∞]→0. Also the sequenceTkis clearly increasing. For
eachkand eachnwe have that
‖(Hn·M)Tk‖H 1 ≤k+‖(Hn·M)Tk‖L 1.Since the sequence fn =(Hn·M)∞ is uniformly integrable (it is even
norm convergent), we have that also the sequence of conditional expec-
tations, ((Hn·M)Tk)n≥ 1 is uniformly integrable and hence the sequence
(
(Hn·M)Tk
)
n≥ 1 is weakly relatively compact inH(^1). Taking the appropri-
ate linear combinations will give a limit inH^1 of the formKk·MwithKk
supported by [[0,Tk]] and satisfying (Kk·M)=NTk. We now take a sequence
(km)m≥ 1 such that‖NTkm−f 0 ‖≤ 2 −m.Ifwedefine
H^0 =Kk^1 +
∑
m≥ 2Kkm (^1) ]]Tkm− 1 ,Tkm]],
we find thatH^0 ·Mis uniformly integrable and that (H^0 ·M)∞=f 0.
15.4.5Proof of Theorem 15.D
The basic ingredient is Theorem 15.C. Exactly as in M. Yor’s theorem we do
not have — a priori — a sequence that is bounded inH^1. The lower boundw
only permits to obtain a bound for theL^1 -norms and we need again stopping
time arguments. This is possible because of a uniform bound over the time
interval, exactly as in the previous part. The uniformity is obtained as in
Lemma 9.4.6.
Definition 15.4.8.We say that anM-integrable predictable processHisw-
admissible for some non-negative integrable functionwifH·M≥−w, i.e.
the process stays above the level−w.
Remark 15.4.9.The concept ofa-admissible integrands, wherea>0isade-
terministic number, was used in [DS 94] (here reproduced as Chap. 9) where
a short history of this concept is given. The above definition generalises the
admissibility as used in Chap. 9 in the sense that it replaces a constant func-
tion by a fixed non-negative integrable functionw. The concept was also used
by the second named author in [S 94, Proposition 4.5].