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We develop a nonanticipative calculus for functionals of a continuoussemimartingale, using an extension of the Ito formula to path-dependentfunctionals which possess certain directional derivatives. The construction isbased on a pathwise derivative, introduced by Dupire, for functionals on thespace of right-continuous functions with left limits. We show that thisfunctional 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 anonanticipative "lifting" of the Malliavin derivative. These results lead to aconstructive martingale representation formula for Ito processes. By contrastwith the Clark-Haussmann-Ocone formula, this representation only involvesnonanticipative quantities which may be computed pathwise.

Functional It\^{o} calculus and stochastic integral representation of martingales