A representation theorem for smooth Brownian martingales

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, 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.