Introduction to Stochastic Filtering Theory.
Title:
Introduction to Stochastic Filtering Theory.
Author:
Xiong, Jie.
ISBN:
9780191551390
Personal Author:
Physical Description:
1 online resource (285 pages)
Series:
Oxford Graduate Texts in Mathematics ; v.No. 18
Oxford Graduate Texts in Mathematics
Contents:
Contents -- 1 Introduction -- 1.1 Examples -- 1.2 Basic definitions and the filtering equation -- 1.3 An overview -- 2 Brownian motion and martingales -- 2.1 Martingales -- 2.2 Doob-Meyer decomposition -- 2.3 Meyer's processes -- 2.4 Brownian motion -- 3 Stochastic integrals and Itô's formula -- 3.1 Predictable processes -- 3.2 Stochastic integral -- 3.3 Itô's formula -- 3.4 Martingale representation in terms of Brownian motion -- 3.5 Change of measures -- 3.6 Stratonovich integral -- 4 Stochastic differential equations -- 4.1 Basic definitions -- 4.2 Existence and uniqueness of a solution -- 4.3 Martingale problem -- 4.4 A stochastic flow -- 4.5 Markov property -- 5 Filtering model and Kallianpur-Striebel formula -- 5.1 The filtering model -- 5.2 The optimal filter -- 5.3 Filtering equation -- 5.4 Particle-system representation -- 5.5 Notes -- 6 Uniqueness of the solution for Zakai's equation -- 6.1 Hilbert space -- 6.2 Transformation to a Hilbert space -- 6.3 Some useful inequalities -- 6.4 Uniqueness for Zakai's equation -- 6.5 A duality representation -- 6.6 Notes -- 7 Uniqueness of the solution for the filtering equation -- 7.1 An interacting particle system -- 7.2 The uniqueness of the system -- 7.3 Uniqueness for the filtering equation -- 7.4 Notes -- 8 Numerical methods -- 8.1 Monte-Carlo method -- 8.2 A branching particle system -- 8.3 Convergence of V[sup(n)][sub(t)] -- 8.4 Convergence of V[sup(n)] -- 8.5 Notes -- 9 Linear filtering -- 9.1 Gaussian system -- 9.2 Kalman-Bucy filtering -- 9.3 Discrete-time approximation of the Kalman-Bucy filtering -- 9.4 Some basic facts for a related deterministic control problem -- 9.5 Stability for Kalman-Bucy filtering -- 9.6 Notes -- 10 Stability of non-linear filtering -- 10.1 Markov property of the optimal filter -- 10.2 Ergodicity of the optimal filter -- 10.3 Finite memory property.
10.4 Asymptotic stability for non-linear filtering with compact state space -- 10.5 Exchangeability of union intersection for σ-fields -- 10.6 Notes -- 11 Singular filtering -- 11.1 A special example -- 11.2 A general singular filtering model -- 11.3 Optimal filter with discrete support -- 11.4 Optimal filter supported on manifolds -- 11.5 Filtering model with Ornstein-Uhlenbeck noise -- 11.6 Notes -- Bibliography -- List of Notations -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- W -- Z.
Abstract:
Stochastic filtering theory is a field that has seen a rapid development in recent years and this book, aimed at graduates and researchers in applied mathematics, provides an accessible introduction covering recent developments. - ;Stochastic Filtering Theory uses probability tools to estimate unobservable stochastic processes that arise in many applied fields including communication, target-tracking, and mathematical finance. As a topic, Stochastic Filtering Theory has progressed rapidly in recent years. For example, the (branching) particle system representation of the optimal filter has been extensively studied to seek more effective numerical approximations of the optimal filter; the stability of the filter with "incorrect" initial state, as well as the long-term behavior of the optimal filter, has attracted the attention of many researchers; and although still in its infancy, the study of singular filtering. models has yielded exciting results. In this text, Jie Xiong introduces the reader to the basics of Stochastic Filtering Theory before covering these key recent advances. The text is written in a style suitable for graduates in mathematics and engineering with a background in basic probability. -.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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