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

2010/04/30 by Beiglböck, Mathias, Schachermayer, Walter, Veliyev, Bezirgen · 1 citation
#60G05 #60H05 #91G99 #Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1004.5559

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 [DeSc94].

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