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We give an elementary proof of the celebrated Bichteler-Dellacherie Theoremwhich states that the class of stochastic processes $S$ allowing for a usefulintegration theory consists precisely of those processes which can be writtenin the form $S=M+A$, where $M$ is a local martingale and $A$ is a finitevariation process. In other words, $S$ is a good integrator if and only if itis a semi-martingale. We obtain this decomposition rather directly from anelementary discrete-time Doob-Meyer decomposition. By passing to convexcombinations we obtain a direct construction of the continuous timedecomposition, which then yields the desired decomposition. As a by-product ofour proof we obtain a characterization of semi-martingales in terms of avariant of \emph{no free lunch}, thus extending a result from [DeSc94].

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