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On the convergence of the wavelet-Galerkin method for nonlinear filtering

Nowak Ł. D., Pasławska-Południak M., Twardowska K.

International Journal of Applied Mathematics and Computer Science

2010

Vol. 20, no 1

93-108

The aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.

On the Existence of Solutions of Some Stochastic Functional Integral Equations

Pasławska-Południak M.

Commentationes Mathematicae

2010

Vol. 50, [Z] 1

49-60

The integral equation of Urysohn type is considered, for the deterministic and stochastic cases . We show, using the fixed point theorem of Darbo type that under some assumptions the equations have solutions belonging to the space of continuous functions. The main tool used in our paper is the technique associated with measures of noncompactness.

Approximation of the Zakai Equation in a Nonlinear Filtering Problem With Delay

Twardowska K., Marnik T., Pasławska-Południak M.

International Journal of Applied Mathematics and Computer Science

2003

Vol. 13, no 2

151-160

A nonlinear filtering problem with delays in the state and observation equations is considered. The unnormalized conditional probability density of the filtered diffusion process satisfies the so-called Zakai equation and solves the nonlinear filtering problem. We examine the solution of the Zakai equation using an approximation result. Our theoretical deliberations are illustrated by a numerical example.