In this paper, we prove the celebrated Bichteler-Dellaccherie Theorem which states that the class of stochastic processes X allowing for a useful integration theory consists precisely of those processes which can be written in the form X = X0 + M + A, where M0 = A0 = 0, M is a local martingale, and A is of finite variation process.We obtain this decomposition rather direct form an elementary discrete-time Doob-Meyer decomposition.By moving to convex combination we obtain a direct continuous time decomposition, which then yield the desired decomposition. We also obtain a characterization of semi-martingales in terms of a variant no free lunch with vanishing risk.
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