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Boué–Dupuis formula

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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.[1] In 2000[2] the result was generalized to infinite-dimensional Brownian motions and in 2009[3] extended to abstract Wiener spaces.

Boué–Dupuis formula

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Let be the classical Wiener space and be a -dimensional standard Brownian motion. Then for all bounded and measurable functions we have the following variational representation

where:

References

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  1. ^ Boué, Michelle; Dupuis, Paul (1998). "A variational representation for certain functionals of Brownian motion". The Annals of Probability. 26 (4). Institute of Mathematical Statistics: 1641–1659. doi:10.1214/aop/1022855876.
  2. ^ Budhiraja, Amarjit; Dupuis, Paul (2000). "A variational representation for positive functionals of infinite dimensional Brownian motion". Probability and Mathematical Statistics (20): 39–61.
  3. ^ Zhang, Xicheng (2009). "A variational representation for random functionals on abstract Wiener spaces". Journal of Mathematics of Kyoto University. 49 (3). Duke University Press: 475–490. doi:10.1215/kjm/1260975036.