15
A Compactness Principle
for Bounded Sequences of Martingales
with Applications (1999)
Abstract. ForH^1 -bounded sequences of martingales, we introduce a technique,
related to the Kadeˇc-Pelczy ́nski decomposition forL^1 sequences, that allows us to
prove compactness theorems. Roughly speaking, a bounded sequence inH^1 can
be split into two sequences, one of which is weakly compact, the other forms the
singular part. If the martingales are continuous then the singular part tends to zero
in the semi-martingale topology. In the general case the singular parts give rise to
a process of bounded variation. The technique allows to give a new proof of the
optional decomposition theorem in Mathematical Finance.
15.1 Introduction
Without any doubt, one of the most fundamental results in analysis is the
theorem of Heine-Borel:
Theorem 15.1.1.From a bounded sequence (xn)n≥ 1 ∈Rdwe can extract
a convergent subsequence(xnk)k≥ 1.
If we pass fromRdto infinite dimensional Banach spacesXthis result does
not hold true any longer. But there are some substitutes which often are useful.
The following theorem can be easily derived from the Hahn-Banach theorem
and was well-known to S. Banach and his contemporaries (see [DRS 93] for
related theorems).
Theorem 15.1.2.Given a bounded sequence(xn)n≥ 1 in a reflexive Banach
spaceX(or, more generally, a relatively weakly compact sequence in a Banach
spaceX) we may find a sequence(yn)n≥ 1 of convex combinations of(xn)n≥ 1 ,
yn∈conv{xn,xn+1,...},which converges with respect to the norm ofX.
[DS 99] A Compactness Principle for Bounded Sequences of Martingales with Ap-
plications. Proceedings of the Seminar on Stochastic Analysis, Random Fields and
Applications,Progress in Probability, vol. 45, pp. 137–173, Birkh ̈auser, Basel (1999).