Markov Processes From K It S Perspective
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Markov Processes from K. Itô's Perspective
Author: Daniel W. Stroock
language: en
Publisher: Princeton University Press
Release Date: 2003-05-26
Kiyosi Itô's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Itô's program. The modern theory of Markov processes was initiated by A. N. Kolmogorov. However, Kolmogorov's approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, Itô interpreted Kolmogorov's famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Itô's thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes and without explicit use of Itô's stochastic integral calculus. In the second half, the author provides a systematic development of Itô's theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovich's variation on Itô's theme and ends with an application to the characterization of the paths on which a diffusion is supported. The book should be accessible to readers who have mastered the essentials of modern probability theory and should provide such readers with a reasonably thorough introduction to continuous-time, stochastic processes.
Nonlinear Markov Processes and Kinetic Equations
Author: Vassili N. Kolokoltsov
language: en
Publisher: Cambridge University Press
Release Date: 2010-07-15
A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from general vector-valued differential equations and yields a natural link with probability, both in interpreting results and in the tools of analysis. This brilliant book, the first devoted to the area, develops this interplay between probability and analysis. After systematically presenting both analytic and probabilistic techniques, the author uses probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior. The book addresses the most fundamental questions in the theory of nonlinear Markov processes: existence, uniqueness, constructions, approximation schemes, regularity, law of large numbers and probabilistic interpretations. Its careful exposition makes the book accessible to researchers and graduate students in stochastic and functional analysis with applications to mathematical physics and systems biology.
Perspectives of Systems Informatics
Author: Amir Pnueli
language: en
Publisher: Springer Science & Business Media
Release Date: 2010-02-08
This book contains thoroughly refereed and revised papers from the 7th International Andrei Ershov Memorial Conference on Perspectives of System Informatics, PSI 2009, held in Akademgorodok, Novosibirsk, Russia, in June 2009. The 26 revised full papers and 4 revised short papers presented were carefully reviewed and selected from 67 submissions. The volume also contains 5 invited papers covering a range of hot topics in system informatics. The papers address all current aspects of theoretical computer science, programming methodology, and new information technologies, which are among the most important contributions of system informatics.