In this paper we show that the variational representation -log Ee-f(W) = infᵥ E1/2 ∫₀¹ ∥ vₛ ∥² ds + f(W + ∫₀· vₛ ds) holds, where W is a standard d-dimensional Brownian motion, f is any bounded measurable function that maps C([0, 1]: Rᵈ) 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.
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Boué et al. (1998) studied this question.
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