A direct proof of the Bichteler–Dellacherie theorem and connections to arbitrage

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].