  We show that, under certain smoothness conditions, a Brownian martingale,
when evaluated at a fixed time, can be represented via an exponential formula
at a later time. The time-dependent generator of this exponential operator only
depends on the second order Malliavin derivative operator evaluated along a
"frozen path". The exponential operator can be expanded explicitly to a series
representation, which resembles the Dyson series of quantum mechanics. Our
continuous-time martingale representation result can be proven independently by
two different methods. In the first method, one constructs a time-evolution
equation, by passage to the limit of a special case of a backward Taylor
expansion of an approximating discrete time martingale. The exponential formula
is a solution of the time-evolution equation, but we emphasize in our article
that the time-evolution equation is a separate result of independent interest.
In the second method, which we only highlight in this article, we use the
property of denseness of exponential functions. We provide several applications
of the exponential formula, and briefly highlight numerical applications of the
backward Taylor expansion.
