  We show that, under certain smoothness conditions, a Brownian martingale at a
fixed time can be represented as an exponential of its value at a later time.
The time-dependent generator of this exponential operator is equal to one half
times the Malliavin derivative. This result can also be seen as a
generalization of the semi-group theory of parabolic partial differential
equations to the parabolic path-dependent partial differential equations
introduced by Dupire (2009) and Cont and Founi\'e (2011). The exponential
operator can be calculated explicitly in a series expansion, which resembles
the Dyson series of quantum mechanics. Our continuous-time martingale
representation result is proved by a passage to the limit of a special case of
a backward Taylor expansion of an approximating discrete-time martingale. The
latter expansion can also be used for numerical calculations.
