322 15 A Compactness Principle
Then there is anM-integrable predictable stochastic processH^0 such that
H^0 ·Mis anL^2 -bounded martingale and such that(H^0 ·M)∞=f 0.
It is not hard to extend the above theorem to the case ofLp,for1<p≤∞.
But the extension top= 1 is a much more delicate issue which has been settled
by M. Yor [Y 78a], who proved the analogue of Theorem 15.1.5 for the case of
H^1 andL^1.
Theorem 15.1.6 (Yor). Let(Hn)n≥ 1 be a sequence of M-integrable pre-
dictable stochastic processes such that each(Hn·M)is anH^1 -bounded (resp.
a uniformly integrable) martingale and such that the sequence of random vari-
ables((Hn·M)∞)n≥ 1 converges to a random variablef 0 ∈H^1 (Ω,F,P)(resp.
f 0 ∈L^1 (Ω,F,P)) with respect to theH^1 -norm (resp.L^1 -norm); (or even only
with respect to theσ(H^1 ,BMO) (resp.σ(L^1 ,L∞)) topology).
Then there is anM-integrable predictable stochastic processH^0 such that
H^0 ·Mis anH^1 -bounded (resp. uniformly integrable) martingale and such
that(H^0 ·M)∞=f 0.
We refer to Jacod [J 79, Theor`eme 4.63, p.143] for theH^1 -case. It es-
sentially follows from Davis’ inequality forH^1 -martingales. TheL^1 -case (see
[Y 78a]) is more subtle. Using delicate stopping time arguments M. Yor suc-
ceeded in reducing theL^1 case to theH^1 case. In Sect. 15.4 we take the
opportunity to translate Yor’s proof into the setting of the present paper.
Let us also mention in this context a remarkable result of M ́emin ([M 80,
Theorem V.4]) where the processMis only assumed to be a semi-martingale
and not necessarily a local martingale and which also allows to pass to a limit
H^0 ·Mof a Cauchy sequenceHn·MofM-integrals (w.r. to the semi-martingale
topology).
All these theorems areclosedness resultsin the sense that, if (Hn·M)is
aCauchy-sequencewith respect to some topology, then we may findH^0 such
that (H^0 ·M) equals the limit of (Hn·M).
The aim of our paper is to provecompactness resultsin the sense that,
if (Hn·M)isabounded sequencein the martingale spaceH^1 ,thenwemay
find a subsequence (nk)k≥ 1 as well as decompositionsHnk=rKk+sKkso
that the sequencerKk·Mis relatively weakly compact inH^1 and such that
the singular partssKk·Mhopefully tend to zero in some sense to be made
precise. The regular partsrKk·Mthen allow to take convex combinations
that converge in the norm ofH^1.
It turns out that forcontinuouslocal martingalesMthe situation is nicer
(and easier) than for the general case of local martingales with jumps. We
now state the main result of this paper, in its continuous and in its general
version (Theorem 15.A and 15.B below).
Theorem 15.A.Let(Mn)n≥ 1 be anH^1 -bounded sequence of real-valuedcon-
tinuouslocal martingales.
Then we can select a subsequence, which we still denote by(Mn)n≥ 1 ,as
well as an increasing sequence of stopping times(Tn)n≥ 1 , such thatP[Tn<∞]