On the martingale property of certain local martingales

• 成果类型: Article
• 署名作者: Mijatovic, Aleksandar; Urusov, Mikhail
• 署名单位: Ulm University; University of Warwick
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0314-7
• 发表日期: 2012
• 页码: 1-30
• 关键词: Bubbles; options

摘要

The stochastic exponential Z(t) = exp{M-t - M-0 - (1/2) < M, M >(t)} of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where M-t = integral(t)(0)b(Yu) dW(u) and Y is a one-dimensional diffusion driven by a Brownian motion W. Furthermore, we provide a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.