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.
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.
Arvustused
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)
1 Examples and Auxiliary Results.- 2 Stochastic Integration in a Hilbert Space.- 3 Linear Stochastic Evolution Systems in Hilbert Spaces.- 4 Ito's Second Order Parabolic Equations.- 5 Ito's Partial Differential Equations and Diffusion Processes.- 6 Filtering, Interpolation and Extrapolation of Diffusion Processes.- 7 Hypoellipticity of Ito's Second Order Parabolic Equations.- 8 Chaos Expansion for Linear Stochastic Evolution Systems.- Notes.- References.- Index.
Boris Rozovsky earned a Masters 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 Masters 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.
