Ergodic Properties of Nonlinear Filtering Processes

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)