A backward particle interpretation of Feynman-Kac formulae

¹ 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
² Department of Statistics & Department of Computer Science, University of British Columbia, 333-6356 Agricultural Road, Vancouver, BC, V6T 1Z2, Canada. arnaud@stat.ubc.ca
³ The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan.
⁴ Department of Engineering, University of Cambridge, Trumpington Street, CB2 1PZ, UK. sss40@cam.ac.uk

Received: 19 July 2009
Revised: 15 February 2010

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.