A DIRECT PROOF OF THE BICHTELER-DELLACHERIE THEOREM AND CONNECTIONS TO ARBITRAGE

• 成果类型: Article
• 署名作者: Beiglboeck, Mathias; Schachermayer, Walter; Veliyev, Bezirgen
• 署名单位: University of Vienna
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP602
• 发表日期: 2011
• 页码: 2424-2440
• 关键词: martingales; meyer

摘要

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].