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In this paper we show that the variational representation - log Ee -f(W) = inf E{1/2∫ 0 1 ∥v s ∥ 2 ds + f(w + ∫ 0 v s ds)} holds, where W is a standard d-dimensional Brownian motion, f is any bounded measurable function that maps C([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 W. An application is made to a problem concerned with large deviations, and an extension to unbounded functions is given.

A variational representation for certain functionals of Brownian motion