A backward particle interpretation of Feynman-Kac formulae

Pierre Del Moral; Arnaud Doucet; Sumeetpal S. Singh

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 5, page 947-975
  • ISSN: 0764-583X

Abstract

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We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals “on-the-fly” as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates, yielding what seems to be the first results of this type for this class of models. We also illustrate these results in the context of filtering of hidden Markov models, as well as in computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to h-processes.

How to cite

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Del Moral, Pierre, Doucet, Arnaud, and Singh, Sumeetpal S.. "A backward particle interpretation of Feynman-Kac formulae." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 947-975. <http://eudml.org/doc/250778>.

@article{DelMoral2010,
abstract = { We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals “on-the-fly” as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates, yielding what seems to be the first results of this type for this class of models. We also illustrate these results in the context of filtering of hidden Markov models, as well as in computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to h-processes. },
author = {Del Moral, Pierre, Doucet, Arnaud, Singh, Sumeetpal S.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Feynman-Kac models; mean field particle algorithms; functional central limit theorems; exponential concentration; non asymptotic estimates; functional central limit theorems; convergence filtering; hidden Markov models; Schrödinger equations; numerical example},
language = {eng},
month = {8},
number = {5},
pages = {947-975},
publisher = {EDP Sciences},
title = {A backward particle interpretation of Feynman-Kac formulae},
url = {http://eudml.org/doc/250778},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Del Moral, Pierre
AU - Doucet, Arnaud
AU - Singh, Sumeetpal S.
TI - A backward particle interpretation of Feynman-Kac formulae
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/8//
PB - EDP Sciences
VL - 44
IS - 5
SP - 947
EP - 975
AB - We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals “on-the-fly” as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates, yielding what seems to be the first results of this type for this class of models. We also illustrate these results in the context of filtering of hidden Markov models, as well as in computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to h-processes.
LA - eng
KW - Feynman-Kac models; mean field particle algorithms; functional central limit theorems; exponential concentration; non asymptotic estimates; functional central limit theorems; convergence filtering; hidden Markov models; Schrödinger equations; numerical example
UR - http://eudml.org/doc/250778
ER -

References

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