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

In: The Mathematics of Arbitrage

Author

Listed:
  • Freddy Delbaen

    (ETH Zürich)

  • Walter Schachermayer

    (Technische Universität Wien)

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.

Suggested Citation

  • Freddy Delbaen & Walter Schachermayer, 2006. "A Compactness Principle for Bounded Sequences of Martingales with Applications (1999)," Springer Finance, in: The Mathematics of Arbitrage, chapter 15, pages 319-356, Springer.
    • Handle: RePEc:spr:sprfcp:978-3-540-31299-4_15
    DOI: 10.1007/978-3-540-31299-4_15
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