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

TL;DR

This paper provides an elementary proof of the Bichteler-Dellacherie Theorem, showing that semi-martingales are exactly the processes suitable for stochastic integration, and connects this to arbitrage concepts.

Contribution

It offers a direct, elementary proof of the semi-martingale decomposition theorem using discrete-time methods and extends no free lunch characterizations.

Findings

Elementary proof of Bichteler-Dellacherie Theorem

Characterization of semi-martingales via no free lunch

Construction of continuous-time decomposition from discrete-time results

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].