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@article{Cont:2013:10.1214/11-AOP721,
author = {Cont, R and Fournie, D-A},
doi = {10.1214/11-AOP721},
journal = {Annals of Probability},
pages = {109--133},
title = {Functional Ito calculus and stochastic integral representation of martingales},
url = {http://dx.doi.org/10.1214/11-AOP721},
volume = {41},
year = {2013}
}

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TY  - JOUR
AB  - We develop a non-anticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependentfunctionals which possess certain directional derivatives. The construction isbased on a pathwise derivative, introduced by B Dupire, for functionals on thespace of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space ofsquare-integrable martingales. This extension defines a weak derivative whichis shown to be the inverse of the Ito integral and which may be viewed as a non-anticipative "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 non-anticipative quantities which may be computed pathwise.
AU  - Cont,R
AU  - Fournie,D-A
DO  - 10.1214/11-AOP721
EP  - 133
PY  - 2013///
SN  - 0091-1798
SP  - 109
TI  - Functional Ito calculus and stochastic integral representation of martingales
T2  - Annals of Probability
UR  - http://dx.doi.org/10.1214/11-AOP721
UR  - http://arxiv.org/abs/1002.2446v4
UR  - http://projecteuclid.org/euclid.aop/1358951982
VL  - 41
ER  -

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