Department of Economics and Business Economics

A direct proof of the Bichteler-Dellacherie Theorem and connections to arbitrage

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  • Mathias Beiglböck, Fakultät Für Mathematik, Denmark
  • Walter Schachermayer, Fakultät Für Mathematik, Denmark
  • Bezirgen Veliyev

We give an elementary proof of the celebrated Bichteler-Dellacherie the-orem which states that the class of stochastic processes S allowing for a useful integration theory consists precisely of those processes which can be written in the form S = M + A,whereM is a local martingale and A is a finite variation process. In other words, S is a good integrator if and only if it is a semi-martingale. We obtain this decomposition rather directly from an elementary discrete- time Doob-Meyer decomposition. By passing to convex combinations, we obtain a direct construction of the continuous time decomposition, which then yields the desired decomposition. As a by-product of our proof, we obtain a characterization of semi- martingales in terms of a variant of no free lunch, thus extending a result from [Math. Ann. 300 (1994) 463-520].

Original languageEnglish
JournalAnnals of Probability
Volume39
Issue6
Pages (from-to)2424-2440
Number of pages17
ISSN0091-1798
DOIs
Publication statusPublished - 1 Nov 2011

    Research areas

  • Arbitrage, Bichteler-dellacherie theorem, Doob-meyer decomposition, Komlós' lemma

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