TITLE:
Semimartingale Property and Its Connections to Arbitrage
AUTHORS:
Sallieu Kabay Samura, Junjun Mao, Dengbao Yao
KEYWORDS:
Bichteler-Dellaccherie Theorem; Doob-Meyer Decomposition; Semi-Martingales; Arbitrage; Komlos Lemma
JOURNAL NAME:
Journal of Mathematical Finance,
Vol.3 No.2,
May
24,
2013
ABSTRACT: 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.