View source for Boué–Dupuis formula

{{Orphan|date=February 2023}}

In [[stochastic calculus]], the '''Boué–Dupuis formula''' is [[Calculus of variations|variational]] representation for [[Wiener functional]]s. The representation has application in finding [[large deviation theory|large deviation]] asymptotics.

The theorem was proven in 1998 by [[Michelle Boué]] and [[Paul Dupuis (Mathematician)|Paul Dupuis]].<ref>{{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}}</ref> In 2000<ref>{{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}}</ref> the result was generalized to infinite-dimensional [[Brownian motion]]s and in 2009<ref>{{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}}</ref> extended to [[abstract Wiener space]]s.

==Boué–Dupuis formula==
Let <math>C([0,1],\mathbb{R}^d)</math> be the [[classical Wiener space]] and <math>B</math> be a <math>d</math>-dimensional standard Brownian motion. Then for all [[bounded function|bounded]] and [[Measurable function|measurable]] functions
<math>f:C([0,1],\mathbb{R}^d)\to\mathbb{R}</math> we have the following variational representation
:<math>-\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],</math>
where:
*The [[expectation value|expectation]] is with respect to the [[probability space]] of <math>B</math>.
*The [[infimum]] runs over all [[Stochastic Process|processes]] which are [[progressively measurable]] with respect to the [[filtration (probability theory)#Augmented filtration|augmented filtration]] <math>\mathcal{F}^B</math> generated by <math>B</math>
*<math>\|\cdot \|</math> denotes the <math>d</math>-dimensional [[Euclidean norm]].

==References==
<references />

[[Category:Stochastic calculus]]
[[Category:Wiener process]]
[[Category:Probability theorems]]
[[Category:Calculus of variations]]