Functional Itō calculus and stochastic integral representation of martingales
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Publication:1942112
DOI10.1214/11-AOP721zbMATH Open1272.60031arXiv1002.2446OpenAlexW2012244785MaRDI QIDQ1942112
Author name not available (Why is that?)
Publication date: 15 March 2013
Published in: (Search for Journal in Brave)
Abstract: 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.
Full work available at URL: https://arxiv.org/abs/1002.2446
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