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

## Author Info

• 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

## Bibliographic Info

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

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Handle: RePEc:arx:papers:1004.5559

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Web page: http://arxiv.org/

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## References

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1. Constantinos Kardaras & Eckhard Platen, 2008. "On the semimartingale property of discounted asset-price processes," Papers 0803.1890, arXiv.org, revised Nov 2009.
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## Citations

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Cited by:
1. Christoph K\"uhn & Bj\"orn Ulbricht, 2013. "Modeling capital gains taxes for trading strategies of infinite variation," Papers 1309.7368, arXiv.org.
2. 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.
3. Constantinos Kardaras, 2011. "On the closure in the Emery topology of semimartingale wealth-process sets," Papers 1108.0945, arXiv.org, revised Jul 2013.
4. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.

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