A backward particle interpretation of Feynman-Kac formulae
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, p. 947-975

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.

DOI : https://doi.org/10.1051/m2an/2010048
Classification:  65C05,  65C35,  60G35,  47D08
Keywords: Feynman-Kac models, mean field particle algorithms, functional central limit theorems, exponential concentration, non asymptotic estimates
@article{M2AN_2010__44_5_947_0,
author = {Del Moral, Pierre and Doucet, Arnaud and Singh, Sumeetpal S.},
title = {A backward particle interpretation of Feynman-Kac formulae},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {44},
number = {5},
year = {2010},
pages = {947-975},
doi = {10.1051/m2an/2010048},
zbl = {pre05798939},
mrnumber = {2731399},
language = {en},
url = {http://www.numdam.org/item/M2AN_2010__44_5_947_0}
}

Del Moral, Pierre; Doucet, Arnaud; Singh, Sumeetpal S. A backward particle interpretation of Feynman-Kac formulae. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, pp. 947-975. doi : 10.1051/m2an/2010048. http://www.numdam.org/item/M2AN_2010__44_5_947_0/

[1] D. Bakry, L'hypercontractivitée et son utilisation en théorie des semigroupes, in Lecture Notes in Math. 1581, École d'été de St. Flour XXII, P. Bernard Ed. (1992). | Zbl 0856.47026

[2] P. Billingsley, Probability and Measure. Third edition, Wiley series in probability and mathematical statistics (1995). | Zbl 0649.60001

[3] E. Cancès, B. Jourdain and T. Lelièvre, Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation. ESAIM: M2AN 16 (2006) 1403-1449. | Zbl 1098.81095

[4] F. Cerou, P. Del Moral and A. Guyader, A non asymptotic variance theorem for unnormalized Feynman-Kac particle models. Ann. Inst. Henri Poincaré (to appear).

[5] P.-A. Coquelin, R. Deguest and R. Munos, Numerical methods for sensitivity analysis of Feynman-Kac models. Available at http://hal.inria.fr/inria-00336203/en/, HAL-INRIA Research Report 6710 (2008).

[6] D. Crisan, P. Del Moral and T. Lyons, Interacting Particle Systems. Approaximations of the Kushner-Stratonovitch Equation. Adv. Appl. Probab. 31 (1999) 819-838. | Zbl 0947.60040

[7] P. Del Moral, Feynman-Kac formulae. Genealogical and interacting particle systems with applications. Probability and its Applications, Springer Verlag, New York (2004). | Zbl 1130.60003

[8] P. Del Moral and A. Doucet, Particle motions in absorbing medium with hard and soft obstacles. Stoch. Anal. Appl. 22 (2004) 1175-1207. | Zbl 1071.60100

[9] P. Del Moral and L. Miclo, Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering, in Séminaire de Probabilités XXXIV, Lecture Notes in Math. 1729, Springer, Berlin (2000) 1-145. | Numdam | Zbl 0963.60040

[10] P. Del Moral and L. Miclo, Particle approximations of Lyapunov exponents connected to Schroedinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | Zbl 1040.81009

[11] P. Del Moral and E. Rio, Concentration inequalities for mean field particle models. Available at http://hal.inria.fr/inria-00375134/fr/, HAL-INRIA Research Report 6901 (2009).

[12] P. Del Moral, J. Jacod and P. Protter, The Monte Carlo Method for filtering with discrete-time observations. Probab. Theory Relat. Fields 120 (2001) 346-368. | Zbl 0979.62072

[13] P. Del Moral, A. Doucet and S.S. Singh, Forward smoothing using sequential Monte Carlo. Cambridge University Engineering Department, Technical Report CUED/F-INFENG/TR 638 (2009).

[14] G.B. Di Masi, M. Pratelli and W.G. Runggaldier, An approximation for the nonlinear filtering problem with error bounds. Stochastics 14 (1985) 247-271. | Zbl 0566.60046

[15] R. Douc, A. Garivier, E. Moulines and J. Olsson, On the forward filtering backward smoothing particle approximations of the smoothing distribution in general state spaces models. Technical report, available at arXiv:0904.0316.

[16] A. Doucet, N. De Freitas and N. Gordon Eds., Sequential Monte Carlo Methods in Pratice. Statistics for engineering and Information Science, Springer, New York (2001). | Zbl 1056.93576

[17] M. El Makrini, B. Jourdain and T. Lelièvre, Diffusion Monte Carlo method: Numerical analysis in a simple case. ESAIM: M2AN 41 (2007) 189-213. | Numdam | Zbl 1135.81379

[18] M. Émery, Stochastic calculus in manifolds. Universitext, Springer-Verlag, Berlin (1989). | Zbl 0697.60060

[19] S.J. Godsill, A. Doucet and M. West, Monte Carlo smoothing for nonlinear time series. J. Am. Stat. Assoc. 99 (2004) 156-168. | Zbl 1089.62517

[20] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes 24. Second edition, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam (1989). | Zbl 0684.60040

[21] M. Kac, On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65 (1949) 1-13. | Zbl 0032.03501

[22] G. Kallianpur and C. Striebel, Stochastic differential equations occurring in the estimation of continuous parameter stochastic processes. Tech. Rep. 103, Department of Statistics, University of Minnesota, Minneapolis (1967). | Zbl 0195.44503

[23] N. Kantas, A. Doucet, S.S. Singh and J.M. Maciejowski, An overview of sequential Monte Carlo methods for parameter estimation in general state-space models, in Proceedings IFAC System Identification (SySid) Meeting, available at http://publications.eng.cam.ac.uk/16156/ (2009).

[24] H. Korezlioglu and W.J. Runggaldier, Filtering for nonlinear systems driven by nonwhite noises: an approximating scheme. Stoch. Stoch. Rep. 44 (1983) 65-102. | Zbl 0786.60058

[25] J. Picard, Approximation of the nonlinear filtering problems and order of convergence, in Filtering and control of random processes, Lecture Notes in Control and Inf. Sci. 61, Springer (1984) 219-236. | Zbl 0539.93078

[26] G. Poyiadjis, A. Doucet and S.S. Singh, Sequential Monte Carlo computation of the score and observed information matrix in state-space models with application to parameter estimation. Technical Report CUED/F-INFENG/TR 628, Cambridge University Engineering Department (2009).

[27] D. Revuz, Markov chains. North-Holland (1975). | Zbl 0539.60073

[28] M. Rousset, On the control of an interacting particle approximation of Schroedinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844. | Zbl 1174.60045

[29] A.N. Shiryaev, Probability, Graduate Texts in Mathematics 95. Second edition, Springer (1986). | Zbl 0835.60002

[30] D.W. Stroock, Probability Theory: an Analytic View. Cambridge University Press, Cambridge (1994). | Zbl 0960.60001

[31] D.W. Stroock, An Introduction to Markov Processes, Graduate Texts in Mathematics 230. Springer (2005). | Zbl 1068.60003