A backward particle interpretation of Feynman-Kac formulae

Abstract

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.Mathematics Subject Classification. 65C05, 65C35, 60G35, 47D08.

An introduction to probabilistic methods with applications

“…In a different context, in [8] these models are used to describe the complex stochastic evolution of assets and wealth portfolio processes in financial markets. In [16], these diffusion equations are rather used to model stochastic signal-observation evolutions in nonlinear filtering problems, or some reference Markov process in Feynman-Kac-Schrödinger semigroups. Another important application of these stochastic models are the Langevin dynamics defined by…”

“…These Feynman-Kac functional models play a central role in physics and engineering sciences, including non linear filtering, information theory, Bayesian inference, and mathematical finance. Five articles in this volume are more or less directly concerned with the stochastic analysis of these Feynman-Kac models in different application domains: Specially, we refer the reader to [8] on applications in mathematical finance, to [37] on the approximation of Dirichlet and "Fermion" ground state energies, to [2] on filtering problems of atmospheric fluid velocities along discrete acquisition systems and [16] on stochastic particle approximation of the full FeynmanKac path measures Q t , with applications in numerical physics and advanced signal processing, as well as in parameter estimation in hidden Markov chain problems. Finally, in [7] the reader will find some very nice extended Feynman-Kac formulae for linearized Poisson-Boltzmann equations arising in molecular dynamics.…”

“…Other applications of these models in signal processing and hidden Markov chain models are also provided in [16]. In this contribution, the authors design a mean field particle approximation of integrals of additive functions F n (x 0 , .…”

“…As we mentioned above, the article by Del Moral et al [16] provides an application of these models in Bayesian inference and hidden Markov chain analysis. In this situation, the potential functions represent the likelihood functions of current observations, and the functional derivatives defined above represent the gradient of the log-likelihood of an observation sequence.…”

“…The article by Del Moral et al [16] is concerned with an interacting particle approximation of FeynmanKac semigroups in path space, with applications in numerical physics, Bayesian statistics, and advanced signal processing. These new stochastic models are based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of the final time marginals.…”