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November 2011 A direct proof of the Bichteler–Dellacherie theorem and connections to arbitrage
Mathias Beiglböck, Walter Schachermayer, Bezirgen Veliyev
Ann. Probab. 39(6): 2424-2440 (November 2011). DOI: 10.1214/10-AOP602

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 no free lunch, thus extending a result from [Math. Ann. 300 (1994) 463–520].

Citation

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Mathias Beiglböck. Walter Schachermayer. Bezirgen Veliyev. "A direct proof of the Bichteler–Dellacherie theorem and connections to arbitrage." Ann. Probab. 39 (6) 2424 - 2440, November 2011. https://doi.org/10.1214/10-AOP602

Information

Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1232.60028
MathSciNet: MR2932672
Digital Object Identifier: 10.1214/10-AOP602

Subjects:
Primary: 60G05 , 60H05
Secondary: 91G99

Keywords: Arbitrage , Bichteler–Dellacherie theorem , Doob–Meyer decomposition , Komlós’ lemma

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 6 • November 2011
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