  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.
