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© 2018

Stochastic Evolution Systems

Linear Theory and Applications to Non-Linear Filtering

Benefits

  • Provides a self-contained development of the theory of generalized solutions of stochastic parabolic equations

  • Explores equations of optimal non-linear filtering, interpolation, and extrapolation of diffusion processes in detail

  • Establishes various connections between diffusions and linear stochastic parabolic equations

Book

Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 89)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Boris L. Rozovsky, Sergey V. Lototsky
    Pages 1-37
  3. Boris L. Rozovsky, Sergey V. Lototsky
    Pages 39-84
  4. Boris L. Rozovsky, Sergey V. Lototsky
    Pages 85-122
  5. Boris L. Rozovsky, Sergey V. Lototsky
    Pages 123-170
  6. Boris L. Rozovsky, Sergey V. Lototsky
    Pages 171-212
  7. Boris L. Rozovsky, Sergey V. Lototsky
    Pages 213-241
  8. Boris L. Rozovsky, Sergey V. Lototsky
    Pages 243-278
  9. Boris L. Rozovsky, Sergey V. Lototsky
    Pages 279-314
  10. Back Matter
    Pages 315-330

About this book

Introduction

This monograph, now in a thoroughly revised second edition, develops the theory of stochastic calculus in Hilbert spaces and applies the results to the study of generalized solutions of stochastic parabolic equations.

The emphasis lies on second-order stochastic parabolic equations and their connection to random dynamical systems. The authors further explore applications to the theory of optimal non-linear filtering, prediction, and smoothing of partially observed diffusion processes. The new edition now also includes a chapter on chaos expansion for linear stochastic evolution systems.

This book will appeal to anyone working in disciplines that require tools from stochastic analysis and PDEs, including pure mathematics, financial mathematics, engineering and physics.

Keywords

MSC (2010): 60H15, 35R60 Boundary value problem Markov property Martingale Sobolev space diffusion process filtering problem local martingale partial differential equation backward diffusion equation chaos solution of parabolic equations interpolation extrapolation Hormander's condition in filtering stochastic characteristics stochastic integration in Hilbert space

Authors and affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

About the authors

Boris Rozovsky earned a Master’s degree in Probability and Statistics, followed by a PhD in Physical and Mathematical Sciences, both from the Moscow State (Lomonosov) University. He was Professor of Mathematics and Director of the Center for Applied Mathematical Sciences at the University of Southern California. Currently, he is the Ford Foundation Professor of Applied Mathematics at Brown University.

Sergey Lototsky earned a Master’s degree in Physics in 1992 from the Moscow Institute of Physics and Technology, followed by a PhD in Applied Mathematics in 1996 from the University of Southern California. After a year-long post-doc at the Institute for Mathematics and its Applications and a three-year term as a Moore Instructor at MIT, he returned to the department of Mathematics at USC as a faculty member in 2000. He specializes in stochastic analysis, with emphasis on stochastic differential equation. He supervised more than 10 PhD students and had visiting positions at the Mittag-Leffler Institute in Sweden and at several universities in Israel and Italy.

Bibliographic information

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Reviews

“The book will be useful for those who are working in the area of applications of stochastic evolution systems in physics, biology and control theory, and require tools from stochastic analysis and partial differential equations.” (Anatoliy Swishchuk, zbMATH 1434.60004, 2020)

“A remarkable quality of this monograph is that the results are stated and proved with a great level of generality and rigor. The reader will find many interesting results, as well as lots of long and technical proofs … .” (Charles-Edouard Bréhier, Mathematical Reviews, October, 2019)