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2 International Journal of Applied Mathematics and Computer Science

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Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions

100%

Twardowska K.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS 1. Introduction...........................................................................................................................................5 1.1. The Wong-Zakai theorem and its generalizations.............................................................................5 1.2. Approximation methods for stochastic differential equations.............................................................7 1.3. Extensions of the Wong-Zakai theorem and their applications..........................................................9 2. Approximation theorem of Wong-Zakai type for functional stochastic differential equations................10 2.1. Introductory remarks........................................................................................................................10 2.2. Definitions and notation...................................................................................................................10 2.3. Description of the model..................................................................................................................11 2.4. Approximation theorem....................................................................................................................15 2.5. Examples.........................................................................................................................................24 3. An extension of the Wong-Zakai theorem to stochastic evolution equations in Hilbert spaces............26 3.1. Introductory remarks.......................................................................................................................26 3.2. Definitions and notation..................................................................................................................26 3.3. Description of the model.................................................................................................................27 3.4. The main theorem...........................................................................................................................31 3.5. Examples.........................................................................................................................................41 3.5.1. Equations satisfying the assumptions of Theorem 3.4.1.............................................................41 3.5.2. Stochastic delay equations.........................................................................................................43 3.5.3. Stochastic wave equations..........................................................................................................45 4. Comparison of the results...................................................................................................................46 4.1. Finite-dimensional case..................................................................................................................46 4.2. Stochastic delay equations.............................................................................................................47 5. On the relation between the Itô and Stratonovich integrals in Hilbert spaces.....................................47 6. Conclusions........................................................................................................................................49 References.............................................................................................................................................50

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

47%

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

tom 20

nr 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.

Approximation of the Zakai equation in a nonlinear filtering problem with delay

47%

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

International Journal of Applied Mathematics and Computer Science

2003

tom 13

nr 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.