The Annals of Probability

A variational representation for certain functionals of Brownian motion

Michelle Boué and Paul Dupuis

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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.

Article information

Ann. Probab., Volume 26, Number 4 (1998), 1641-1659.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60H99: None of the above, but in this section
Secondary: 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65] 60F10: Large deviations

Variational representation Brownian motion large deviations relative entropy


Boué, Michelle; Dupuis, Paul. A variational representation for certain functionals of Brownian motion. Ann. Probab. 26 (1998), no. 4, 1641--1659. doi:10.1214/aop/1022855876.

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