Abstract
In this paper we show that the variational representation $$-\log Ee^{-f(W)} = \inf_v E{1/2 \int_0^1 \parallel v_s \parallel^2 ds + f(W + \int_0^{\cdot} v_s ds)}$$ holds, where $W$ is a standard $d$-dimensional Brownian motion, $f$ is any bounded measurable function that maps $C([0, 1]: \mathbb{R}^d)$ into $\mathbb{R}$ 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.
Citation
Michelle Boué. Paul Dupuis. "A variational representation for certain functionals of Brownian motion." Ann. Probab. 26 (4) 1641 - 1659, October 1998. https://doi.org/10.1214/aop/1022855876
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