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

PDF
 Published Version
379kB 
Abstract
Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuousdiscrete filtering problems. Suppose that t → Xt is a Markov process and we wish to calculate the measurevalued 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 observationdependent 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 Lévystable signals and give rates of convergence for E1/2[µnt  µt2√], ∙ √ is a Sobolev norm, as well as related
convergence results.
Divisions:  Concordia University > Faculty of Arts and Science > Mathematics and Statistics 

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 16:20 
Last Modified:  08 Dec 2010 23:24 
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 277287.
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´evystable 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). Nonlinear filtering and measurevalued processes. Probab. Theory Related Fields 109 217244. Del Moral P. (1996). Non linear filtering: interacting particle resolution. Markov Proc. Rel. Fields 2 555579. Del Moral P., Kouritzin M.A. and Miclo L. (2001). On a class of discrete generation interacting particle systems. Electron. J. Probab. 6 126. Del Moral P. and Miclo L. (2000). Branching and interacting particle systems approximations of FeynmanKac 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 18611872. Kouritzin M.A. (2000). Exact infinite dimensional filters and explicit solutions. Stochastic models(Ottawa, ON, 1998). CMS Conf. Proc. 26 265282. 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 659679. Longnecker M. and Serfling R.J. (1977). General moment and probability inequalities for the maximum partial sum. Acta Math. Acad. Sci. Hungar. 30 129133. 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 2368. Samorodnitsky G. and Taqqu M.S. (1994). Stable NonGaussian 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. 
Repository Staff Only: item control page