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A backward particle interpretationof Feynman-Kac formulae

Published online by Cambridge University Press:  26 August 2010

Pierre Del Moral,
Arnaud Doucet and
Sumeetpal S. Singh
Pierre Del Moral
Affiliation:
Centre INRIA Bordeaux et Sud-Ouest & Institut de Mathématiques de Bordeaux, Université de Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France. Pierre.Del-Moral@inria.fr
Arnaud Doucet
Affiliation:
Department of Statistics & Department of Computer Science, University of British Columbia, 333-6356 Agricultural Road, Vancouver, BC, V6T 1Z2, Canada. arnaud@stat.ubc.caThe Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan.
Sumeetpal S. Singh
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, CB2 1PZ, UK. sss40@cam.ac.uk
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Abstract

We design a particle interpretation of Feynman-Kac measures on path spacesbased on a backward Markovian representation combined with a traditionalmean field particle interpretation of the flow of their final timemarginals. In contrast to traditional genealogical tree based models, thesenew particle algorithms can be used to compute normalized additivefunctionals “on-the-fly” as well as theirlimiting occupation measures with a given precision degree that does notdepend on the final time horizon.We provide uniform convergence results w.r.t. the time horizon parameter aswell as functional central limit theorems and exponential concentrationestimates, yielding what seems to be the first results of this type for thisclass of models. We also illustrate these results in the context offiltering of hidden Markov models, as well as in computational physics andimaginary time Schroedinger type partial differential equations, with aspecial interest in the numerical approximation of the invariant measureassociated to h-processes.

Keywords

Feynman-Kac modelsmean field particle algorithmsfunctionalcentral limit theoremsexponential concentrationnon asymptotic estimates

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