《《泛函Ito微积分与鞅的随机积分表示（英文版）》.pdfVIP

Functional Ito calculus and stochastic integral representation 0 of martingales 1 0 Rama Cont David-Antoine Fourni´e 2 n First draft: June 2009. This version: June 2010.∗ u J 8 ] Abstract R We develop a non-anticipative calculus for functionals of a continuous semimartingale, using P. a notion of pathwise functional derivative. A functional extension of the Ito formula is derived h and used to obtain a constructive martingale representation theorem for a class of continuous t martingales verifying a regularity property. By contrast with the Clark-Haussmann-Ocone for- a mula, this representation involves non-anticipative quantities which can be computed pathwise. m These results are used to construct a weak derivative acting on square-integrable martingales, [ which is shown to be the inverse of the Ito integral, and derive an integration by parts formula for 3 Ito stochastic integrals. We show that this weak derivative may be viewed as a non-anticipative v “lifting” of the Malliavin derivative. 6 Regular functionals of an Ito martingale which have the local martingale property are char- 4 acterized as solutions of a functional differential equation, for which a uniqueness result is given. 4 2 Keywords: stochastic calculus, functional calculus, Ito formula, integration by parts, Malliavin . derivative, martingale representation, semimartingale, Wiener functionals, functional Feynman-Kac 2 0 formula, Kolmogorov equation, Clark-Ocone formula. 0 1 : v i X r a ∗We thank Bruno Dupire for sharing his original ideas wi