Abstract
We study a nonlinear filtering problem in which the signal to be estimated is a reflecting diffusion in a random environment. Under the assumption that the observation noise is independent of the signal, we develop a nonparametric functional estimation method for finding workable approximate solutions to the conditional distributions of the signal state. Furthermore, we show that the pathwise average distance, per unit time, of the approximate filter from the optimal filter is asymptotically small in time. Also, we use simulations based upon a particle filter algorithm to show the efficiency of the method.
Highlights
We consider a nonlinear filtering problem in which the signal to be estimated is a reflecting diffusion process in a random environment
Whereas in calm waters the dinghy moves as a diffusion process reflecting at the shores, lake swells provide a random environment potentially altering its motion greatly
The D-valued signal process Xt for the dinghy can be described by the following stochastic differential equation (SDE)
Summary
We consider a nonlinear filtering problem in which the signal to be estimated is a reflecting diffusion process in a random environment. Under the assumption that the observation noise is independent of the signal, we develop a nonparametric estimation method for finding workable approximate solutions to the conditional distributions of the signal state given the back observations. In this connection, we refer the interested reader to Chow, Khasminskii and Liptser (1997, 2001), Elliott (2001), Kouritzin, Remillard and Chan (2001) for some other recent works on nonparametric and parametric estimations for filtering problems.
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