Rates for branching particle approximations of continuous-discrete filters

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

2006-02-22

arxiv.org/abs/math/0602488v1

Annals of Applied Probability 2005, Vol. 15, No. 4, 2739-2772

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051605000000539 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051605000000539

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that $t\to X_t$ is a Markov process and we wish to calculate the measure-valued process $t\to\mu_t(\cdot)\doteq P\{X_t\in \cdot|\sigma\{Y_{t_k}, t_k\leq t\}\}$, where $t_k=k\epsilon$ and $Y_{t_k}$ is a distorted, corrupted, partial observation of $X_{t_k}$. Then, one constructs a particle system with observation-dependent branching and $n$ initial particles whose empirical measure at time $t$, $\mu_t^n$, closely approximates $\mu_t$. Each particle evolves independently of the other particles according to the law of the signal between observation times $t_k$, and branches with small probability at an observation time. For filtering problems where $\epsilon$ 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 $\epsilon$. We analyze the algorithm on L\'{e}vy-stable signals and give rates of convergence for $E^{1/2}\{\|\mu^n_t-\mu_t\|_{\gamma}^2\}$, where $\Vert\cdot\Vert_{\gamma}$ is a Sobolev norm, as well as related convergence results.

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