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.
Publié le : 1998-10-14
Classification:
Variational representation,
Brownian motion,
large deviations,
relative entropy,
60H99,
60J60,
60J65,
60F10
@article{1022855876,
author = {Bou\'e, Michelle and Dupuis, Paul},
title = {A variational representation for certain functionals of Brownian
motion},
journal = {Ann. Probab.},
volume = {26},
number = {1},
year = {1998},
pages = { 1641-1659},
language = {en},
url = {http://dml.mathdoc.fr/item/1022855876}
}
Boué, Michelle; Dupuis, Paul. A variational representation for certain functionals of Brownian
motion. Ann. Probab., Tome 26 (1998) no. 1, pp. 1641-1659. http://gdmltest.u-ga.fr/item/1022855876/