Stochastic evolution systems : linear theory and applications to non-linear filtering

by B.L. Rozovskii

Covering the general theory of linear stochastic evolution systems with unbounded drift and diffusion operators, this book sureys Ito's second-order parabolic equations and explores filtering problems for processes whose trajectories can be described by them.

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  • 1 Examples and Auxiliary Results.- 1.0. Introduction.- 1.1. Examples of Stochastic Evolution Systems.- 1.2. Measurability and Integrability in Banach Spaces.- 1.3. Martingales in ?1.- 1.4. Diffusion Processes.- 2 Stochastic Integration in a Hilbert Space.- 2.0. Introduction.- 2.1. Martingales and Local Martingales.- 2.2. Stochastic Integrals with Respect to Square Integrable Martingale.- 2.3. Stochastic Integrable with Respect to a Local Martingale.- 2.4. An Energy Equality in a Rigged Hilbert Space.- 3 Linear Stochastic Evolution Systems in Hilbert Spaces.- 3.0. Introduction.- 3.1. Coercive Systems.- 3.2. Dissipative Systems.- 3.3. Uniqueness and the Markov Property.- 3.4. The First Boundary Problem for Ito's Partial Differential Equations.- 4 Ito'S Second Order Parabolic Equations.- 4.0. Introduction.- 4.1. The Cauchy Problem for Superparabolic Ito's Second Order Parabolic Equations.- 4.2. The Cauchy Problem for Ito's Second Order Equations.- 4.3. The Forward Cauchy Problem and the Backward One in Weighted Sobolev Spaces.- 5 Ito's Partial Differential Equations and Diffusion Processes.- 5.0. Introduction.- 5.1. The Method of Stochastic Characteristics.- 5.2. Inverse Diffusion Processes, the Method of Variation of Constants and the Liouville Equations.- 5.3. A Representation of a Density-valued Solution.- 6 Filtering Interpolation and Extrapolation of Diffusion Processes.- 6.0. Introduction.- 6.1. Bayes' Formula and the Conditional Markov Property.- 6.2. The Forward Filtering Equation.- 6.3. The Backward Filtering Equation Interpolation and Extrapolation.- 7 Hypoellipticity of Ito's Second Order Parabolic Equations.- 7.0. Introduction.- 7.1. Measure-valued Solution and Hypoellipticity under Generalized Hormander's Condition.- 7.2. The Filtering Transition Density and a Fundamental Solution of the Filtering Equation in Hypoelliptic and Superparabolic Cases.- Notes.- References.

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書名 Stochastic evolution systems : linear theory and applications to non-linear filtering
著作者等 Rozovskiĭ, Boris Lʹvovich
Yarkho A.
Rozovskii B.L.
書名別名 Эволюционные стохастические системы

Ėvoli︠u︡t︠s︡ionnye stokhasticheskie sistemy
シリーズ名 Mathematics and its applications
出版元 Kluwer Academic Publishers
刊行年月 c1990
ページ数 xviii, 315 p.
大きさ 25 cm
ISBN 0792300378
NCID BA11466013
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言語 英語
原文言語 ロシア語
出版国 オランダ