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A Direct Proof of the Bichteler--Dellacherie Theorem and Connections to Arbitrage

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  • Mathias Beiglb\"ock
  • Walter Schachermayer
  • Bezirgen Veliyev

Abstract

We give an elementary proof of the celebrated Bichteler-Dellacherie Theorem 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 $S=M+A$, where $M$ 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 \emph{no free lunch}, thus extending a result from [DeSc94].

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File URL: http://arxiv.org/pdf/1004.5559
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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1004.5559.

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Date of creation: Apr 2010
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Handle: RePEc:arx:papers:1004.5559

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Web page: http://arxiv.org/

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  1. Kardaras, Constantinos & Platen, Eckhard, 2011. "On the semimartingale property of discounted asset-price processes," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2678-2691, November.
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Cited by:
  1. Constantinos Kardaras, 2011. "On the closure in the Emery topology of semimartingale wealth-process sets," Papers 1108.0945, arXiv.org, revised Jul 2013.
  2. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
  3. Christoph K\"uhn & Bj\"orn Ulbricht, 2013. "Modeling capital gains taxes for trading strategies of infinite variation," Papers 1309.7368, arXiv.org.
  4. Constantinos Kardaras, 2013. "On the closure in the Emery topology of semimartingale wealth-process sets," LSE Research Online Documents on Economics 44996, London School of Economics and Political Science, LSE Library.

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