A Compactness Principle for Bounded Sequences of Martingales with Applications (1999)

Freddy Delbaen () and Walter Schachermayer ()
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Freddy Delbaen: ETH Zürich
Walter Schachermayer: Technische Universität Wien

Chapter 15 in The Mathematics of Arbitrage, 2006, pp 319-356 from Springer

Abstract: For ℋ1-bounded sequences of martingales, we introduce a technique, related to the Kadeč-Pełczyński decomposition for L1 sequences, that allows us to prove compactness theorems. Roughly speaking, a bounded sequence in ℋ1 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.

Date: 2006
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DOI: 10.1007/978-3-540-31299-4_15

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