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Ergodic Properties of Nonlinear Filtering Processes

  • Hiroshi Kunita
Part of the Progress in Probability book series (PRPR, volume 19)

Abstract

Let x t be a temporally homogenous Markov process with state space S, called a system process in this paper. Suppose that we want to observe the sample path x t , but what we can observe is a stochastic process Y t of the form
$$ Y_t = \int_0^t {h\left( {x_s } \right)dt + N_t ,} $$
(0.1)
where h is a continous function on S and N t is a standard Brownian motion independent of x s . The filtering of the system based on the observation data Y t is defined by a conditional distribution
$$ \pi _t \left( A \right) = P\left( {x_t \in \left. A \right|\mathcal{G}_t } \right), $$
(0.2)
where A is a Borel subset of S and
$$ \mathcal{G}_t = \mathop \cap \limits_{\varepsilon > 0} \sigma \left( {Y_s ;s \leqslant t + \varepsilon } \right). $$
(0.3)

Keywords

Markov Process Invariant Measure Stochastic Differential Equation Nonnegative Solution Standard Brownian Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Hiroshi Kunita
    • 1
  1. 1.Department of Applied ScienceKyushu University 36FukuokaJapan

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