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

## Author

Listed:
• Mathias Beiglbock
• 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].

## Suggested Citation

• Mathias Beiglbock & Walter Schachermayer & Bezirgen Veliyev, 2010. "A Direct Proof of the Bichteler--Dellacherie Theorem and Connections to Arbitrage," Papers 1004.5559, arXiv.org.
• Handle: RePEc:arx:papers:1004.5559
as

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

## References listed on IDEAS

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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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## Citations

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Cited by:

1. Christoph Kuhn & Bjorn Ulbricht, 2013. "Modeling capital gains taxes for trading strategies of infinite variation," Papers 1309.7368, arXiv.org, revised Jun 2015.
2. Kardaras, Constantinos, 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. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
4. Dániel Ágoston Bálint & Martin Schweizer, 2018. "Making No-Arbitrage Discounting-Invariant: A New FTAP Beyond NFLVR and NUPBR," Swiss Finance Institute Research Paper Series 18-23, Swiss Finance Institute, revised Mar 2018.
5. Constantinos Kardaras, 2011. "On the closure in the Emery topology of semimartingale wealth-process sets," Papers 1108.0945, arXiv.org, revised Jul 2013.

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