A variational representation for certain functionals of Brownian motion

Abstract

In this paper we show that the variational representation -logEe-f(W) = infυE{1/2∫01∥υ s∥2ds+f(W+∫0̇fυsds)} holds, where W is a standard d-dimensional Brownian motion, f is any bounded measurable function that maps script C sign([0, 1]: ℝd) into ℝ and the infimum is over all processes υ 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.