FUNCTIONAL ITO CALCULUS AND STOCHASTIC INTEGRAL REPRESENTATION OF MARTINGALES

• 成果类型: Article
• 署名作者: Cont, Rama; Fournie, David-Antoine
• 署名单位: Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Columbia University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP721
• 发表日期: 2013
• 页码: 109-133
• 关键词: differential-equations; malliavin calculus; FORMULA

摘要

We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension defines a weak derivative which is shown to be the inverse of the Ito integral and which may be viewed as a nonanticipative lifting of the Malliavin derivative. These results lead to a constructive martingale representation formula for Ito processes. By contrast with the Clark-Haussmann-Ocone formula, this representation only involves nonanticipative quantities which may be computed pathwise.