Kouritzin, Michael A. and Sun, Wei (2004) *Rates for Branching Particle Approximations of Continuous-Discrete Filters.* Technical Report. Concordia University. Department of Mathematics & Statistics, Montreal, Quebec.

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## Abstract

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that t → Xt is a Markov process and we wish to calculate the measure-valued process t →µt(∙) ≐ P(Xt ∉∙|σ{Ytk , tk ≤ t}), where tk = kε and Ytk is a distorted, corrupted, partial observation of Xtk. Then, one constructs a particle system with observation-dependent branching and n initial particles whose empirical measure at time t;µnt , closely approximates µt. Each particle evolves independently of the other particles according to the law of the signal between observation times tk, and branches with small probability at an observation time. For filtering problems where ε is very small, using the algorithm considered in this paper requires far fewer computations than other algorithms that branch or interact all particles regardless of the value of ε. We analyze the algorithm on Lévy-stable signals and give rates of convergence for E1/2[||µnt - µt||2√], ||∙ ||√ is a Sobolev norm, as well as related

convergence results.

Divisions: | Concordia University > Faculty of Arts and Science > Mathematics and Statistics |
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Item Type: | Monograph (Technical Report) |

Authors: | Kouritzin, Michael A. and Sun, Wei |

Series Name: | Department of Mathematics & Statistics. Technical Report No. 12/04 |

Corporate Authors: | Concordia University. Department of Mathematics & Statistics |

Institution: | Concordia University |

Date: | December 2004 |

Keywords: | Filtering, reference probability measure method, branching particle approximations, rates of convergence, Fourier analysis |

ID Code: | 6661 |

Deposited By: | DIANE MICHAUD |

Deposited On: | 02 Jun 2010 12:20 |

Last Modified: | 08 Dec 2010 18:24 |

References: | Ballantyne D.J., Chan H.Y. and Kouritzin M.A. (2000). A novel branching particle method for tracking. Signal and Data Processing of Small Targets 2000. Proceedings of SPIE 4048 277-287.
Blount D. and Kouritzin M.A. (2001). H¨older continuity for spatial and path processes via spectral analysis. Probab. Theory Related Fields 119 589–603. Cartea À. and Howison S. (2004). Option pricing with L´evy-stable processes. http://www.finance.ox.ac.uk/file links/mf papers/2004mf01.pdf . Crisan D. (2003). Exact rates of convergence for a branching particle approximation to the solution of the Zakai equation. Ann. Probab. 31 693–718. Crisan D., Del Moral P. and Lyons T.J. (1999). Discrete filtering using branching and interacting particle systems. Markov Proc. Rel Fields 5 293–318. Crisan D. and Lyons T.J. (1997). Non-linear filtering and measure-valued processes. Probab. Theory Related Fields 109 217-244. Del Moral P. (1996). Non linear filtering: interacting particle resolution. Markov Proc. Rel. Fields 2 555-579. Del Moral P., Kouritzin M.A. and Miclo L. (2001). On a class of discrete generation interacting particle systems. Electron. J. Probab. 6 1-26. Del Moral P. and Miclo L. (2000). Branching and interacting particle systems approximations of Feynman-Kac Formulae with applications to non linear filtering. Séminaire de Probabilités, XXXIV. Lecture Notes in Math. 1729 1–145. Springer, Berlin. Garroppo R.G., Giordano S., Pagano M. and Procissi G. (2002). Testing ®-stable processes in capturing the queuing behavior of broadband teletraffic. Signal Processing 82 1861-1872. Kouritzin M.A. (2000). Exact infinite dimensional filters and explicit solutions. Stochastic models(Ottawa, ON, 1998). CMS Conf. Proc. 26 265-282. Amer. Math. Soc., Providence, RI. Kouritzin M.A. and Heunis A.J. (1994). A law of the iterated logarithm for stochastic processes defined by differential equations with a small parameter. Ann. Probab. 22 659-679. Longnecker M. and Serfling R.J. (1977). General moment and probability inequalities for the maximum partial sum. Acta Math. Acad. Sci. Hungar. 30 129-133. Marinelli C. and Rachev S.T. (2002). Some applications of stable models in finance. http://www.columbia.edu/»cm788/review.pdf . Mikosch T., Resnick S., Rootzén and Stegeman A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab. 12 23-68. Samorodnitsky G. and Taqqu M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York. Sherman A.S. and Peskin C.S. (1986). A Monte Carlo method for scalar reaction diffusion equations. SIAM J. Sci. Statist. Comput. 7 1360–1372. Stein E. (1970). Singular Integrals and Differentialbility Properties of Functions. Princeton University Press, Princeton. Yor M. (1977). Sur les théories du filtrage et de la prédiction. Séminaire de Probabilités, XI. Lecture Notes in Math. 581 257–297. Springer, Berlin. |

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