Ergodic Properties of Nonlinear Filtering Processes
Hiroshi Kunita · Birkhäuser Boston eBooks · 1991
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 0.1 $$ Y_t = \int_0^t {h\left( {x_s } \right)dt + N_t ,} $$ 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 0.2 $$ \pi _t \left( A \right) = P\left( {x_t \in \left. A \right|\mathcal{G}_t } \right), $$ where A is a Borel subset of S and 0.3 $$ \mathcal{G}_t = \mathop \cap \limits_{\varepsilon > 0} \sigma \left( {Y_s ;s \leqslant t + \varepsilon } \right). $$