# Hypoellipticity and Ergodicity of the Wonham Filter as a Diffusion Process

Hypoellipticity and Ergodicity of the Wonham Filter as a Diffusion Process The ergodicity problem of the Wonham filter as a diffusion process is discussed. Since the Wonham equation is degenerate, we apply an ergodic theorem of degenerate Markov diffusions to the problem. Under a certain condition, the Wonham equation satisfies Hörmander’s condition, and the Wonham filter has a continuous transition density. From these results, we obtain that the Wonham filter is uniformly ergodic as a diffusion process. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Hypoellipticity and Ergodicity of the Wonham Filter as a Diffusion Process

, Volume 64 (1) – Aug 1, 2011
24 pages

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Publisher
Springer-Verlag
Copyright
Copyright © 2011 by Springer Science+Business Media, LLC
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Systems Theory, Control; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-010-9129-2
Publisher site
See Article on Publisher Site

### Abstract

The ergodicity problem of the Wonham filter as a diffusion process is discussed. Since the Wonham equation is degenerate, we apply an ergodic theorem of degenerate Markov diffusions to the problem. Under a certain condition, the Wonham equation satisfies Hörmander’s condition, and the Wonham filter has a continuous transition density. From these results, we obtain that the Wonham filter is uniformly ergodic as a diffusion process.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Aug 1, 2011

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