The Annals of Probability
- Ann. Probab.
- Volume 41, Number 1 (2013), 109-133.
Functional Itô calculus and stochastic integral representation of martingales
We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Itô 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 Itô integral and which may be viewed as a nonanticipative “lifting” of the Malliavin derivative.
These results lead to a constructive martingale representation formula for Itô processes. By contrast with the Clark–Haussmann–Ocone formula, this representation only involves nonanticipative quantities which may be computed pathwise.
Ann. Probab. Volume 41, Number 1 (2013), 109-133.
First available in Project Euclid: 23 January 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus 60G44: Martingales with continuous parameter 60H25: Random operators and equations [See also 47B80]
Cont, Rama; Fournié, David-Antoine. Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41 (2013), no. 1, 109--133. doi:10.1214/11-AOP721. https://projecteuclid.org/euclid.aop/1358951982