Spatial Stochastic Processes pp 233-256 | Cite as

# Ergodic Properties of Nonlinear Filtering Processes

Chapter

## Abstract

Let where where

*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)

*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)

*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
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## Copyright information

© Springer Science+Business Media New York 1991