## The Annals of Probability

### A variational representation for certain functionals of Brownian motion

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022855876

Digital Object Identifier
doi:10.1214/aop/1022855876

Mathematical Reviews number (MathSciNet)
MR1675051

Zentralblatt MATH identifier
0936.60059

#### Citation

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. https://projecteuclid.org/euclid.aop/1022855876

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• AMHERST, MASSACHUSETTS 01003 BROWN UNIVERSITY E-MAIL: boue@math.umass.edu PROVIDENCE, RHODE ISLAND 02912 E-MAIL: dupuis@cfm.brown.edu