Skip to main content
SpringerLink
Log in

Branching and interacting particle systems approximations of feynman-kac formulae with applications to non-linear filtering

  • Cours Spécialisé
  • Chapter
  • First Online:
Book cover Séminaire de Probabilités XXXIV

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1729))

Abstract

This paper focuses on interacting particle systems methods for solving numerically a class of Feynman-Kac formulae arising in the study of certain parabolic differential equations, physics, biology, evolutionary computing, nonlinear filtering and elsewhere. We have tried to give an “exposé” of the mathematical theory that is useful for analyzing the convergence of such genetic-type and particle approximating models including law of large numbers, large deviations principles, fluctuations and empirical process theory as well as semigroup techniques and limit theorems for processes.

In addition, we investigate the delicate and probably the most important problem of the long time behavior of such interacting measure valued processes.

We will show how to link this problem with the asymptotic stability of the corresponding limiting process in order to derive useful uniform convergence results with respect to the time parameter.

Several variations including branching particle models with random population size will also be presented. In the last part of this work we apply these results to continuous time and discrete time filtering problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Abela, D. Abramson, A. De Silval, M. Krishnamoorthy, and G. Mills. Computing optimal schedules for landing aircraft. Technical report, Department of Computer Systems Eng. R.M.I.T., Melbourne, May 1993.

    Google Scholar 

  2. J.-M. Alliot, D. Delahaye, J.-L. Farges, and M. Schoenauer. Genetic algorithms for automatic regrouping of air traffic control sectors. In J. R. McDonnell, R. G. Reynolds, and D. B. Fogel, editors, Proceedings of the 4th Annual Conference on Evolutionary Programming, pages 657–672. MIT Press, March 1995.

    Google Scholar 

  3. G. Ben Arous and M. Brunaud. Methode de laplace: étude variationnelle des fluctuations de diffusions de type “champ moyen”. Stochastics, 31–32:79–144, 1990.

    MATH  Google Scholar 

  4. R. Atar and O. Zeitouni. Exponential stability for nonlinear filtering. Annales de l'Institut Henri Poincaré, 33(6):697–725, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Atar and O. Zeitouni. Lyapunov exponents for finite state space nonlinear filtering. Society for Industrial and Applied Mathematics. Journal on Control and Optimization, 35(1):36–55, January 1997.

    MathSciNet  MATH  Google Scholar 

  6. J. Baker. Adaptative selection methods for genetic algorithms. In J. Grefenstette, editor, Proc. International Conf. on Genetic Algorithms and their Applications. L. Erlbaum Associates, 1985.

    Google Scholar 

  7. J. Baker. Reducing bias and inefficiency in the selection algorithm. In J. Grefenstette, editor, Proc. of the Second International Conf. on Genetic Algorithms and their Applications. L. Erlbaum Associates, 1987.

    Google Scholar 

  8. A. Bakirtzis, S. Kazarlis, and V. Petridis. A genetic algorithm solution to the economic dispatch problem. http://www.dai.ed.ac.uk/groups/evalg/eag_local_copies_of_papers.body.html.

    Google Scholar 

  9. D. Bakry. L'hypercontractivité et son utilisation en théorie des semigroupes. In P. Bernard, editor, Lectures on Probability Theory. Ecole d'Eté de Probabilités de Saint-Flour XXII-1992, Lecture Notes in Mathematics 1581. Springer-Verlag, 1994.

    Google Scholar 

  10. P. Barbe and P. Bertail. The Weighted Bootstrap. Lecture Notes in Statistics 98. Springer-Verlag, 1995.

    Google Scholar 

  11. E. Beadle and P. Djuric. A fast weighted Bayesian bootstrap filter for nonlinear model state estimation. Institute of Electrical and Electronics Engineers. Transactions on Aerospace and Electronic Systems, AES-33:338–343, January 1997.

    Google Scholar 

  12. B.E. Benes. Exact finite-dimensional filters for certain diffusions with nonlinear drift. Stochastics, 5:65–92, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  13. E. Bolthausen. Laplace approximation for sums of independent random vectors i. Probability Theory and Related Fields, 72:305–318, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  14. C.L. Bridges and D.E. Goldberg. An analysis of reproduction and crossover in a binary-coded genetic algorithm. In J.J. Grefenstette, editor, Proc. of the Second International Conf. on Genetic Algorithms and their Applications. L. Erlbaum Associates, 1987.

    Google Scholar 

  15. R.S. Bucy. Lectures on discrete time filtering, Signal Processing and Digital Filtering. Springer-Verlag, 1994.

    Google Scholar 

  16. A. Budhiraja and D. Ocone. Exponential stability of discrete time filters for bounded observation noise. Systems and Control Letters, 30:185–193, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  17. Z. Cai, F. Le Gland, and H. Zhang. An adaptative local grid refinement method for nonlinear filtering. Technical report, INRIA, October 1995.

    Google Scholar 

  18. H. Carvalho, P. Del Moral, A. Monin, and G. Salut. Optimal nonlinear filtering in gps/ins integration. Institute of Electrical and Electronics Engineers. Transactions on Aerospace and Electronic Systems, 33(3):835–850, July 1997.

    Google Scholar 

  19. R. Cerf. Une théorie asymptotique des algorithmes génétiques. Thèse de doctorat, Université Montpellier II, March 1994.

    Google Scholar 

  20. M. Chaleyat-Maurel and D. Michel. Des résultats de non existence de filtres de dimension finie. Comptes Rendus de l'Académie des Sciences de Paris. Série I. Mathématique, 296, 1983.

    Google Scholar 

  21. D. Crisan, J. Gaines, and T.J. Lyons. A particle approximation of the solution of the kushner-stratonovitch equation. Society for Industrial and Applied Mathematics. Journal on Applied Mathematics, 58(5):1568–1590, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  22. D. Crisan and M. Grunwald. Large deviation comparison of branching algorithms versus resampling algorithm. Preprint, 1998.

    Google Scholar 

  23. D. Crisan and T.J. Lyons. Nonlinear filtering and measure valued processes. Probability Theory and Related Fields, 109:217–244, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  24. D. Crisan, P. Del Moral, and T.J. Lyons. Interacting particle systems approximations of the Kushner-Stratonovitch equation. Advances in Applied Probability, 31(3), September 1999.

    Google Scholar 

  25. D. Crisan, P. Del Moral, and T.J. Lyons. Non linear filtering using branching and interacting particle systems. Markov Processes and Related Fields, 5(3):293–319, 1999.

    MathSciNet  MATH  Google Scholar 

  26. G. Da Prato, M. Furhman, and P. Malliavin. Asymptotic ergodicity for the Zakai filtering equation. Comptes Rendus de l'Académie des Sciences de Paris. Série I. Mathématique, 321(5):613–616, 1995.

    MathSciNet  MATH  Google Scholar 

  27. E.B. Davies. Heat Kernels and Spectral Theory. Cambridge University Press, 1989.

    Google Scholar 

  28. M. Davis. New approach to filtering for nonlinear systems. Institute of Electrical and Electronics Engineers. Proceedings, 128(5):166–172, 1981. Part D.

    MathSciNet  Google Scholar 

  29. D. Dawson. Measure-valued Markov processes. In P.L. Hennequin, editor, Lectures on Probability Theory. Ecole d'Eté de Probabilités de Saint-Flour XXI-1991, Lecture Notes in Mathematics 1541. Springer-Verlag, 1993.

    Google Scholar 

  30. P. Del Moral. Non-linear filtering: interacting particle resolution. Markov Processes and Related Fields, 2(4):555–581, 1996.

    MathSciNet  MATH  Google Scholar 

  31. P. Del Moral. Filtrage non linéaire par systèmes de particules en interaction. Comptes Rendus de l'Académie des Sciences de Paris. Série I. Mathématique, 325:653–658, 1997.

    Google Scholar 

  32. P. Del Moral. Maslov optimization theory: optimality versus randomness. In V.N. Kolokoltsov and V.P. Maslov, editors, Idempotency Analysis and its Applications, Mathematics and its Applications 401, pages 243–302. Kluwer Academic Publishers, Dordrecht/Boston/London, 1997.

    Google Scholar 

  33. P. Del Moral. Measure valued processes and interacting particle systems. Application to non linear filtering problems. The Annals of Applied Probability, 8(2):438–495, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Del Moral. A uniform convergence theorem for the numerical solving of non linear filtering problems. Journal of Applied Probability, 35:873–884, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  35. P. Del Moral and A. Guionnet. Large deviations for interacting particle systems. Applications to non linear filtering problems. Stochastic Processes and their Applications, 78:69–95, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  36. P. Del Moral and A. Guionnet. On the stability of measure valued processes. Applications to non linear filtering and interacting particle systems. Publications du Laboratoire de Statistique et Probabilités, no 03-98, Université Paul Sabatier, 1998.

    Google Scholar 

  37. P. Del Moral and A. Guionnet. A central limit theorem for non linear filtering using interacting particle systems. The Annals of Applied Probability, 9(2):275–297, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  38. P. Del Moral and A. Guionnet. On the stability of measure valued processes with applications to filtering. Comptes Rendus de l'Académie des Sciences de Paris. Série I. Mathématique, 329:429–434, 1999.

    MathSciNet  MATH  Google Scholar 

  39. P. Del Moral and J. Jacod. Interacting particle filtering with discrete observations. Publications du Laboratoire de Statistiques et Probabilités, no 11–99, 1999.

    Google Scholar 

  40. P. Del Moral and J. Jacod. The monte-carlo method for filtering with discrete time observations. central limit theorems. Publications du Laboratoire de Probabilités, no 515, 1999.

    Google Scholar 

  41. P. Del Moral, J. Jacod, and P. Protter. The Monte Carlo method for filtering with discrete time observations. Publications du Laboratoire de Probabilités, no 453, June 1998.

    Google Scholar 

  42. P. Del Moral and M. Ledoux. Convergence of empirical processes for interacting particle systems with applications to nonlinear filtering. to appear in Journal of Theoretical Probability, January 2000.

    Google Scholar 

  43. P. Del Moral and L. Miclo. Asymptotic stability of non linear semigroup of Feynman-Kac type. Préprint, publications du Laboratoire de Statistique et Probabilités, no 04-99, 1999.

    Google Scholar 

  44. P. Del Moral and L. Miclo. On the convergence and the applications of the generalized simulated annealing. SIAM Journal on Control and Optimization, 37(4):1222–1250, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  45. P. Del Moral and L. Miclo. A Moran particle system approximation of Feynman-Kac formulae. to appear in Stochastic Processes and their Applications, 2000.

    Google Scholar 

  46. P. Del Moral, J.C. Noyer, and G. Salut. Résolution particulaire et traitement non-linéaire du signal: application radar/sonar. In Traitement du signal, September 1995.

    Google Scholar 

  47. P. Del Moral, G. Rigal, and G. Salut. Estimation et commande optimale non linéaire. Technical Report 2, LAAS/CNRS, March 1992. Contract D.R.E.T.-DIGILOG.

    Google Scholar 

  48. B. Delyon and O. Zeitouni. Liapunov exponents for filtering problems. In M.H.A. Davis and R.J. Elliot, editors, Applied Stochastic Analysis, pages 511–521. Springer-Verlag, 1991.

    Google Scholar 

  49. A. Dembo and O. Zeitouni. Large Deviations Techniques and Application. Jones and Bartlett, 1993.

    Google Scholar 

  50. A.N. Van der Vaart and J.A. Wellner. Weak Convergence and Empirical Processes with Applications to Statistics. Springer Series in Statistics. Springer, 1996.

    Google Scholar 

  51. J.-D. Deuschel and D.W. Stroock. Large Deviations. Pure and applied mathematics 137. Academic Press, 1989.

    Google Scholar 

  52. P. Diaconis and B. Efron. Méthodes de calculs statistiques intensifs sur ordinateurs. Pour la Science, 1983. translation of the American Scientist.

    Google Scholar 

  53. R.L. Dobrushin. Central limit theorem for nonstationnary Markov chains, i,ii. Theory of Probability and its Applications, 1(1 and 4):66–80 and 330–385, 1956.

    Google Scholar 

  54. E.B. Dynkin and A. Mandelbaum. Symmetric statistics, Poisson processes and multiple Wiener integrals. The Annals of Statistics, 11:739–745, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  55. S. Ethier and T. Kurtz. Markov Processes, Characterization and Convergence. Wiley series in probability and mathematical statistics. John Wiley and Sons, New York, 1986.

    Book  MATH  Google Scholar 

  56. M. Fujisaki, G. Kallianpur, and H. Kunita. Stochastic differential equations for the non linear filtering problem. Osaka J. Math., 1:19–40, 1972.

    MathSciNet  MATH  Google Scholar 

  57. F. Le Gland. Monte-Carlo methods in nonlinear filtering. In Proceedings of the 23rd IEEE Conference on Decision and Control, pages 31–32, Las Vegas, December 1984.

    Google Scholar 

  58. F. Le Gland. High order time discretization of nonlinear filtering equations. In 28th IEEE CDC, pages 2601–2606, Tampa, 1989.

    Google Scholar 

  59. F. Le Gland, C. Musso, and N. Oudjane. An analysis of regularized interacting particle methods for nonlinear filtering. In Proceedings of the 3rd IEEE European Workshop on Computer-Intensive Methods in Control and Signal Processing, Prague, September 1998.

    Google Scholar 

  60. D.E. Goldberg. Genetic algorithms and rule learning in dynamic control systems. In Proceedings of the First International Conference on Genetic Algorithms, pages 8–15, Hillsdale, NJ, 1985. L. Erlbaum Associates.

    Google Scholar 

  61. D.E. Goldberg. Simple genetic algorithms and the minimal deceptive problem. In L. Davis, editor, Genetic Algorithms and Simulated Annealing. Pitman, 1987.

    Google Scholar 

  62. D.E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA, 1989.

    MATH  Google Scholar 

  63. D.E. Goldberg and P. Segrest. Finite Markov chain analysis of genetic algorithms. In J.J. Grefenstette, editor, Proc. of the 2nd Int. Conf. on Genetic Algorithms. L. Erlbaum Associates, 1987.

    Google Scholar 

  64. N.J. Gordon, D.J. Salmon, and A.F.M. Smith. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F, 140:107–113, 1993.

    Google Scholar 

  65. C. Graham and S. Méléard. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. The Annals of Probability, 25(1):115–132, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  66. A. Guionnet. About precise Laplace's method; applications to fluctuations for mean field interacting particles. Preprint, 1997.

    Google Scholar 

  67. J.H. Holland. Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, 1975.

    Google Scholar 

  68. J. Jacod and A.N. Shiryaev. Limit Theorems for Stochastic Processes. A Series of Comprehensive Studies in Mathematics 288. Springer-Verlag, 1987.

    Google Scholar 

  69. J.M. Johnson and Y. Rahmat-Samii. Genetic algorithms in electromagnetics. In IEEE Antennas and Propagation Society International Symposium Digest, volume 2, pages 1480–1483, 1996.

    Google Scholar 

  70. F. Jouve, L. Kallel, and M. Schoenauer. Mechanical inclusions identification by evolutionary computation. European Journal of Finite Elements, 5(5–6):619–648, 1996.

    MathSciNet  MATH  Google Scholar 

  71. F. Jouve, L. Kallel, and M. Schoenauer. Identification of mechanical inclusions. In D. Dagsgupta and Z. Michalewicz, editors, Evolutionary Computation in Engeneering, pages 477–494. Springer Verlag, 1997.

    Google Scholar 

  72. G. Kallianpur and C. Striebel. Stochastic differential equations occuring in the estimation of continuous parameter stochastic processes. Tech. Rep. 103, Department of statistics, Univ. of Minnesota, September 1967.

    Google Scholar 

  73. G. Kitagawa. Monte-Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal on Computational and Graphical Statistics, 5(1):1–25, 1996.

    MathSciNet  Google Scholar 

  74. H. Korezlioglu. Computation of filters by sampling and quantization. Technical Report 208, Center for Stochastic Processes, University of North Carolina, 1987.

    Google Scholar 

  75. H. Korezlioglu and G. Maziotto. Modelization and filtering of discrete systems and discrete approximation of continuous systems. In Modélisation et Optimisation des Systèmes, VI Conférence INRIA, Nice, 1983.

    Google Scholar 

  76. H. Korezlioglu and W.J. Runggaldier. Filtering for nonlinear systems driven by nonwhite noises: an approximating scheme. Stochastics and Stochastics Reports, 44(1–2):65–102, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  77. H. Kunita. Asymptotic behavior of nonlinear filtering errors of Markov processes. Journal of Multivariate Analysis, 1(4):365–393, 1971.

    Article  MathSciNet  MATH  Google Scholar 

  78. H. Kunita. Ergodic properties nonlinear filtering processes. In K.C. Alexander and J.C. Watkins, editors, Spatial Stochastic Processes. Birkhaüser Boston, Boston, MA, 1991.

    Google Scholar 

  79. S. Kusuoda and Y. Tamura. Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math, 31, 1984.

    Google Scholar 

  80. J.W. Kwiatkowski. Algorithms for index tracking. Technical report, Department of Business Studies, The University of Edinburgh, 1991.

    Google Scholar 

  81. R.S. Liptser and A.N. Shiryayev. Theory of Martingales. Dordrecht: Kluwer Academic Publishers, 1989.

    Book  MATH  Google Scholar 

  82. G.B. Di Masi, M. Pratelli, and W.G. Runggaldier. An approximation for the nonlinear filtering problem with error bounds. Stochastics, 14(4):247–271, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  83. S. Méléard. Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. In D. Talay and L. Tubaro, editors, Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995, Lecture Notes in Mathematics 1627. Springer-Verlag, 1996.

    Google Scholar 

  84. C. Musso and N. Oudjane. Regularization schemes for branching particle systems as a numerical solving method of the nonlinear filtering problem. In Proceedings of the Irish Signals Systems Conference, Dublin, June 1998.

    Google Scholar 

  85. C. Musso and N. Oudjane. Regularized particle schemes applied to the tracking problem. In International Radar Symposium, Munich, Proceedings, September 1998.

    Google Scholar 

  86. Y. Nishiyama. Some central limit theorems for l −-valued semimartingales and their applications. Probability Theory and Related Fields, 108:459–494, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  87. A. Nix and M.D. Vose. Modelling genetic algorithms with Markov chains. Annals of Mathematics and Artificial Intelligence, 5:79–88, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  88. D. Ocone and E Pardoux. Asymptotic stability of the optimal filter with respect to its initial condition. Society for Industrial and Applied Mathematics. Journal on Control and Optimization, 34:226–243, 1996.

    MathSciNet  MATH  Google Scholar 

  89. D.L. Ocone. Topics in nonlinear filtering theory. Phd thesis, MIT, Cambridge, 1980.

    Google Scholar 

  90. E. Pardoux. Filtrage non linéaire et équations aux dérivés partielles stochastiques associées. In P.L. Hennequin, editor, Ecole d'Eté de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Mathematics 1464. Springer-Verlag, 1991.

    Google Scholar 

  91. E. Pardoux and D. Talay. Approximation and simulation of solutions of stochastic differential equations. Acta Applicandae Mathematicae, 3(1):23–47, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  92. J. Picard. Approximation of nonlinear filtering problems and order of convergence. In Filtering and Control of Random Processes, Lecture Notes Control and Inf. Sc. 61. Springer, 1984.

    Google Scholar 

  93. J. Picard. An estimate of the error in time discretization of nonlinear filtering problems. In C.I. Byrnes and A. Lindquist, editors, Proceedings of the 7th MTNS — Theory and Applications of nonlinear Control Systems, Stockholm, 1985, pages 401–412, Amsterdam, 1986. North-Holland Pub.

    Google Scholar 

  94. J. Picard. Nonlinear filtering of one-dimensional diffusions in the case of a high signal-to-noise ratio. Society for Industrial and Applied Mathematics. Journal on Applied Mathematics, 16:1098–1125, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  95. S.T. Rachev. Probability Metrics and the Stability of Stochastic Models. Wiley, New York, 1991.

    MATH  Google Scholar 

  96. D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer-Verlag, 1991.

    Google Scholar 

  97. J. Shapcott. Index tracking: genetic algorithms for investment portfolio selection. Technical Report SS92-24, EPCC, September 1992.

    Google Scholar 

  98. T. Shiga and H. Tanaka. Central limit theorem for a system of markovian particles with mean field interaction. Zeitschrift für Wahrscheinlichkeitstheorie verwandte Gebiete, 69, 1985. 439–459.

    Article  MathSciNet  MATH  Google Scholar 

  99. A.N. Shiryaev. On stochastic equations in the theory of conditional Markov processes. Theor. Prob. Appl., 11:179–184, 1966.

    Google Scholar 

  100. A.N. Shiryaev. Probability. Number 95 in Graduate Texts in Mathematics. Springer-Verlag, New-York, second edition, 1996.

    Book  MATH  Google Scholar 

  101. B. Simon. Trace ideals and their applications. London Mathematical Society Lecture Notes Series 35. Cambridge University Press, 1977.

    Google Scholar 

  102. L. Stettner. On invariant measures of filtering processes. In K. Helmes and N. Kohlmann, editors, Stochastic Differential Systems, Proc. 4th Bad Honnef Conf., Lecture Notes in Control and Inform. Sci., pages 279–292, 1989.

    Google Scholar 

  103. L. Stettner. Invariant measures of pair state/approximate filtering process. In Colloq. Math. LXII, pages 347–352, 1991.

    Google Scholar 

  104. R.L. Stratonovich. Conditional Markov processes. Theor. Prob. Appl., 5:156–178, 1960.

    Article  MATH  Google Scholar 

  105. D. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Springer-Verlag, 1979.

    Google Scholar 

  106. D. W. Stroock. An Introduction to the Theory of Large Deviations. Universitext. Springer-Verlag, New-York, 1984.

    Book  MATH  Google Scholar 

  107. M. Talagrand. Sharper bounds for Gaussian and empirical processes. The Annals of Probability, 22:28–76, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  108. D. Talay. Efficient numerical schemes for the approximation of expectations of functionals of the solution of s.d.e. and applications. In Filtering and Control of random processes (Paris, 1983), Lecture Notes in Control and Inform. Sci. 61, pages 294–313, Berlin-New York, 1984. Springer.

    Chapter  Google Scholar 

  109. H. Tanaka. Limit theorems for certain diffusion processes. In Proceedings of the Taniguchi Symp., Katata, 1982, pages 469–488, Tokyo, 1984. Kinokuniya.

    Google Scholar 

  110. D. Treyer, D.S. Weile, and E. Michielsen. The application of novel genetic algorithms to electromagnetic problems. In Applied Computational Electromagnetics, Symposium Digest, volume 2, pages 1382–1386, Monterey, CA, March 1997.

    Google Scholar 

  111. M. D. Vose. The Simple Genetic Algorithm, Foundations and Theory. The MIT Press Books, August 1999.

    Google Scholar 

  112. D. Williams. Probability with Martingales. Cambridge Mathematical Textbooks, 1992.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Statistiques et Probabilités CNRS UMR C5583, Université Paul Sabatier, 118, route de Narbonne, 31062, Toulouse cedex, France

    P. Del Moral & L. Miclo

Authors
  1. P. Del Moral
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Miclo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jacques Azéma Michel Ledoux Michel Émery Marc Yor

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag

About this chapter

Cite this chapter

Del Moral, P., Miclo, L. (2000). Branching and interacting particle systems approximations of feynman-kac formulae with applications to non-linear filtering. In: Azéma, J., Ledoux, M., Émery, M., Yor, M. (eds) Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics, vol 1729. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103798

Download citation

  • DOI: https://doi.org/10.1007/BFb0103798

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67314-9

  • Online ISBN: 978-3-540-46413-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation