Boué–Dupuis formula

In stochastic calculus, the Boué–Dupuis formula is variational representation for Wiener functionals. The representation has application in finding large deviation asymptotics.

The theorem was proven in 1998 by Michelle Boué and Paul Dupuis.{{cite journal|first1=Michelle|last1=Boué|first2=Paul|last2=Dupuis|publisher=Institute of Mathematical Statistics|title=A variational representation for certain functionals of Brownian motion|journal=The Annals of Probability|volume=26 |number=4|date=1998 |pages=1641–1659|doi=10.1214/aop/1022855876|doi-access=free}} In 2000{{cite journal|first1=Amarjit|last1=Budhiraja|first2=Paul|last2=Dupuis|title=A variational representation for positive functionals of infinite dimensional Brownian motion|journal=Probability and Mathematical Statistics|number=20 |date=2000|pages=39–61}} the result was generalized to infinite-dimensional Brownian motions and in 2009{{cite journal|first1=Xicheng|last1=Zhang|publisher=Duke University Press|title=A variational representation for random functionals on abstract Wiener spaces|journal=Journal of Mathematics of Kyoto University|volume=49|number=3|date=2009|pages=475–490|doi=10.1215/kjm/1260975036|doi-access=free}} extended to abstract Wiener spaces.

Boué–Dupuis formula

Let $C([0,1],\mathbb{R}^d)$ be the classical Wiener space and $B$ be a $d$ -dimensional standard Brownian motion. Then for all bounded and measurable functions

$f:C([0,1],\mathbb{R}^d)\to\mathbb{R}$ we have the following variational representation

: $-\log \mathbb{E}\left[e^{-f(B)}\right]=\inf\limits_{V}\mathbb{E}\left[\frac{1}{2}\int_0^1\|V_t\|^2\mathrm{d}t + f\left(B+\int_0^{\cdot}V_t\mathrm{d}t\right)\right],$

where:

The expectation is with respect to the probability space of $B$ .
The infimum runs over all processes which are progressively measurable with respect to the augmented filtration $\mathcal{F}^B$ generated by $B$
$\|\cdot \|$ denotes the $d$ -dimensional Euclidean norm.

References

Category:Stochastic calculus

Category:Wiener process

Category:Theorems in probability theory

Category:Calculus of variations