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October 1998 A variational representation for certain functionals of Brownian motion
Michelle Boué, Paul Dupuis
Ann. Probab. 26(4): 1641-1659 (October 1998). DOI: 10.1214/aop/1022855876


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.


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Michelle Boué. Paul Dupuis. "A variational representation for certain functionals of Brownian motion." Ann. Probab. 26 (4) 1641 - 1659, October 1998.


Published: October 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0936.60059
MathSciNet: MR1675051
Digital Object Identifier: 10.1214/aop/1022855876

Primary: 60H99
Secondary: 60F10 , 60J60 , 60J65

Keywords: Brownian motion , large deviations , Relative entropy , variational representation

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 4 • October 1998
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