## The Annals of Applied Probability

### Rates for branching particle approximations of continuous-discrete filters

#### Abstract

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that tXt is a Markov process and we wish to calculate the measure-valued process tμt(⋅)≐P{Xt∈⋅|σ{Ytk, tkt}}, where tk=kɛ 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, μtn, 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 ɛ 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 ɛ. We analyze the algorithm on Lévy-stable signals and give rates of convergence for E1/2{‖μntμtγ2}, where ‖⋅‖γ is a Sobolev norm, as well as related convergence results.

#### Article information

Source
Ann. Appl. Probab., Volume 15, Number 4 (2005), 2739-2772.

Dates
First available in Project Euclid: 7 December 2005

https://projecteuclid.org/euclid.aoap/1133965778

Digital Object Identifier
doi:10.1214/105051605000000539

Mathematical Reviews number (MathSciNet)
MR2187310

Zentralblatt MATH identifier
1177.93081

Subjects

#### Citation

Kouritzin, Michael A.; Sun, Wei. Rates for branching particle approximations of continuous-discrete filters. Ann. Appl. Probab. 15 (2005), no. 4, 2739--2772. doi:10.1214/105051605000000539. https://projecteuclid.org/euclid.aoap/1133965778

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