A variational representation for certain functionals of Brownian motion

In this paper we show that the variational representation -log Ee(-f(W)) = inf(v) E{1/2 integral(0)(1)//vs//(2) ds + f(W + integral(0)(.) vs ds)} holds, where W is a standard d-dimensional Brownian motion, f is any bounded measurable function that maps E([0,1]: R-d) into R and the infimum is over all processes v that are progressively measurable with respect to the augmentation of the filtration generated by FV. An application is made to a problem concerned with large deviations, and an extension to unbounded functions is given.