The Annals of Applied Probability

Measure-valued processes and interacting particle systems. Application to nonlinear filtering problems

P. Del Moral

Full-text: Open access


In the paper we study interacting particle approximations of discrete time and measure-valued dynamical systems. These systems have arisen in such diverse scientific disciplines as physics and signal processing. We give conditions for the so-called particle density profiles to converge to the desired distribution when the number of particles is growing. The strength of our approach is that is applicable to a large class of measure-valued dynamical systems arising in engineering and particularly in nonlinear filtering problems. Our second objective is to use these results to solve numerically the nonlinear filtering equation. Examples arising in fluid mechanics are also given.

Article information

Ann. Appl. Probab., Volume 8, Number 2 (1998), 438-495.

First available in Project Euclid: 9 August 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G57: Random measures 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 93E11: Filtering [See also 60G35]

Nonlinear filtering measure-valued processes interacting and branching particle systems genetic algorithms


Del Moral, P. Measure-valued processes and interacting particle systems. Application to nonlinear filtering problems. Ann. Appl. Probab. 8 (1998), no. 2, 438--495. doi:10.1214/aoap/1028903535.

Export citation


  • [1] Benes, V. E. (1981). Exact finite-dimensional filters for certain diffusions with nonlinear drift. Stochastics 5 65-92.
  • [2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [3] Braun, W. and Hepp, K. (1977). The Vlasov dy namics and its fluctuation in the 1/n limit of interacting particles. Comm. Math. Phy s. 56 101-113.
  • [4] Carvalho, H. (1995). Filtrage optimal non lin´eaire du signal GPS NAVSTAR en racalage de centrales de navigation. Th ese de L'Ecole Nationale Sup´erieure de l'A´eronautique et de l'Espace.
  • [5] Carvalho, H., Del Moral, P., Monin, A. and Salut, G. (1997). Optimal nonlinear filtering in GPS/INS integration. IEEE Trans. Aerospace and Electron. Sy st. 33 835-850.
  • [6] Cerf, R. (1994). Une th´eorie asy mptotique des algorithmes g´en´etiques. Ph.D. dissertation, Univ. Montpellier II, Sciences et techniques du Languedoc.
  • [7] Chaley at-Maurel, M. and Michel, D. (1983). Des r´esultats de non existence de filtres de dimension finie. C.R. Acad. Sci. Paris Ser. I 296. plus 0pt
  • [8] Crisan, D. (1996). The nonlinear filtering problem. Ph.D. dissertation, Univ. Edinburgh.
  • [9] Crisan, D. and Ly ons, T. J. (1996). Nonlinear filtering and measure valued processes. Imperial College, London. Preprint.
  • [10] Crisan, D. and Ly ons, T. J. (1996). Convergence of a branching particle method to the solution of the Zakai equation. Imperial College, London. Preprint.
  • [11] Crisan, D. and Ly ons, T. J. (1996). A particle approximation of the solution of the Kushner- Stratonovitch equation. Imperial College, London. Preprint.
  • [12] Dawson, D. A. (1975). Stochastic evolution equations and related measure-valued processes. J. Multivariate Anal. 5 1-52.
  • [13] Dellacherie, C. and Meyer, P. A. (1978). Probabilit´es et Potentiel. Hermann, Paris.
  • [14] Del Moral, P. (1994). R´esolution particulaire des probl emes d'estimation et d'optimisation nonlin´eaires. Th ese, Univ. Paul Sabatier, Rapport LAAS 94269.
  • [15] Del Moral, P. (1995). Nonlinear filtering using random particles. Theor. Probab. Appl. 40.
  • [16] Del Moral, P. (1996). Asy mptotic properties of nonlinear particle filters. Publ. Laboratoire de Statistiques et Probabilit´es, Univ. Paul Sabatier, 11-96.
  • [17] Del Moral, P. (1996). Nonlinear filtering: Monte Carlo particle resolution. Publ. Laboratoire de Statistiques et Probabilit´es, Univ. Paul Sabatier, 02-96.
  • [18] Del Moral, P. (1996). Nonlinear filtering: interacting particle resolution. Markov Processes and Related Fields 2.
  • [19] Del Moral, P. and Miclo, L. (1996). On the convergence and the applications of the generalized simulated annealing. Publ. Laboratoire de Statistiques et Probabilit´es, Univ. Paul Sabatier, 16-96.
  • [20] Del Moral, P., Noy er, J. C. and Salut, G. (1995). R´esolution particulaire et traitement non lin´eaire du signal: application radar/sonar. Traitement du Signal.
  • [21] Del Moral, P., Rigal, G., Noy er, J. C. and Salut, G. (1993). Traitement non-lin´eaire du signal par reseau particulaire: application radar. Quatorzi eme colloque GRETSI, Juan les Pins.
  • [22] Dobrushin, R. L. (1971). Markov processes with a large number of locally interacting components: existence of a limit process and its ergodicity. Problems Inform. Transmission 7 70-87.
  • [23] Dobrushin, R. L. (1970). Prescribing a sy stem of random variables by conditional distributions. Theory Probab. Appl. 15 459-485.
  • [24] Gordon, N. J., Salmon, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/nonGaussian Bayesian state estimation. IEE Proceedings F 140 107-113.
  • [25] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30.
  • [26] Holland, J. H. (1975). Adaptation in Natural and Artificial Sy stems. Univ. Michigan Press, Ann Arbor.
  • [27] Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory. Academic Press, New York.
  • [28] Kallianpur, G. and Striebel, C. (1967). Stochastic differential equations occuring in the estimation of continuous parameter stochastic processes. Technical Report 103, Dept. Statistics, Univ. Minnesota.
  • [29] Kunita, H. (1971). Asy mptotic behavior of nonlinear filtering errors of Markov processes. J. Multivariate Anal. 1 365-393.
  • [30] Liang, D. and McMillan, J. (1989). Development of a marine integrated navigation sy stem. Kalman Filter Integration of Modern Guidance and Navigation Sy stems, AGARD-LS166, OTAN.
  • [31] Liggett, T. M. (1985). Interacting Particle Sy stems. Springer, New York.
  • [32] M´el´eard, S. and Roelly-Coppoletta, S. (1987). A propagation of chaos result for a sy stem of particles with moderate interaction. Stochastic Process. Appl. 26 317-332.
  • [33] McKean, H. P. (1967). Propagation of chaos for a class of nonlinear parabolic equations. Lecture Series in Differential Equations 7 41-57. Catholic Univ.
  • [34] Ocone, D. L. (1980). Topics in nonlinear filtering theory. Ph.D. dissertation, MIT.
  • [35] Pardoux, E. (1989). Filtrage non-lin´eaire et equations aux d´eriv´es partielles stochastiques associ´ees. Ecole d' ´Et´e de Probabilit´es de Saint-Flour XIX, 1989. Lecture Notes in Math. 1464 1-164. Springer, Berlin.
  • [36] Parthasarathy, K. R. (1968). Probability Measures on Metric Spaces. Academic Press, New York.
  • [37] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • [38] Shiga, T. and Tanaka, H. (1985). Central limit theorem for a sy stem of Markovian particles with mean field interactions. Z. Wahrsch. Verw. Gebiete 69 439-459.
  • [39] Shiry aev, A. N. (1966). On stochastic equations in the theory of conditional Markov processes. Theory Probab. Appl. 11 179-184.
  • [40] Spitzer, F. (1969). Random processes defined through the interaction of an infinite particle sy stem. Lecture Notes in Math. 89 201-223. Springer, Berlin.
  • [41] Stettner, L. (1989). On invariant measures of filtering processes. In Stochastic Differential Sy stems. Lecture Notes in Control and Inform. Sci. 126. Springer, Berlin.
  • [42] Stratonovich, R. L. (1960). Conditional Markov processes. Theory Probab. Appl. 5 156-178.
  • [43] Sznitman, A. S. (1984). Nonlinear reflecting diffusion process and the propagation of chaos and fluctuations associated. J. Functional Anal. 56 311-336.
  • [44] Sznitman, A. S. (1989). Topics in propagation of chaos. Ecole d' ´Et´e de Probabilit´es de SaintFlour XIX, 1989. Lecture Notes in Math. 1464 165-251.
  • [45] Van Dootingh, M., Viel, F., Rakotopara, D. and Gauthier, J. P. (1991). Coupling of nonlinear control with a stochastic filter for state estimation: application on a free radical polimerization reactor. In I.F.A.C. International Sy mposium ADCHEM'91, Toulouse, France.
  • [46] Watanabe, S. (1968). A limit theorem of branching processes and continuous state branching processes. J. Math. Ky oto Univ. 8 141-167.