Kiyosi Ito'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 Ito'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...
Kiyosi Ito's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Fel...
This edition develops the basic theory of Fourier transform. Stroock's approach is the one taken originally by Norbert Wiener and the Parseval's formula, as well as the Fourier inversion formula via Hermite functions. New exercises and solutions have been added for this edition.
This edition develops the basic theory of Fourier transform. Stroock's approach is the one taken originally by Norbert Wiener and the Parseval's formu...
To some extent, it would be accurate to summarize the contents of this book as an intolerably protracted description of what happens when either one raises a transition probability matrix P (i. e., all entries (P)j are n- negative and each row of P sums to 1) to higher and higher powers or one exponentiates R(P -- I), where R is a diagonal matrix with non-negative entries. Indeed, when it comes right down to it, that is all that is done in this book. However, I, and others of my ilk, would take offense at such a dismissive characterization of the theory of Markov chains and processes with...
To some extent, it would be accurate to summarize the contents of this book as an intolerably protracted description of what happens when either one r...
This book deals with equations that have played a central role in the interplay between partial differential equations and probability theory. Most of this material has been treated elsewhere, but it is rarely presented in a manner that makes it readily accessible to people whose background is probability theory. Many results are given new proofs designed for readers with limited expertise in analysis. The author covers the theory of linear, second order partial differential equations of parabolic and elliptic type. Many of the techniques have antecedents in probability theory, although the...
This book deals with equations that have played a central role in the interplay between partial differential equations and probability theory. Most of...
The second printing of a book first published in 1988. The first four chapters of the volume are based on lectures given by Stroock at MIT in 1987. They form an introduction to the basic ideas of the theory of large deviations and make a suitable package on which to base a semester-length course for advanced graduate students with a strong background in analysis and some probability theory. A large selection of exercises presents important material and many applications. The last two chapters present various non-uniform results and outline the analytic approach that allows one to test and...
The second printing of a book first published in 1988. The first four chapters of the volume are based on lectures given by Stroock at MIT in 1987. Th...
These notes are based on a course which I gave during the academic year 1983-84 at the University of Colorado. My intention was to provide both my audience as well as myself with an introduction to the theory of 1arie deviations - The organization of sections 1) through 3) owes something to chance and a great deal to the excellent set of notes written by R. Azencott for the course which he gave in 1978 at Saint-Flour (cf. Springer Lecture Notes in Mathematics 774). To be more precise: it is chance that I was around N. Y. U. at the time'when M. Schilder wrote his thesis. and so it may be...
These notes are based on a course which I gave during the academic year 1983-84 at the University of Colorado. My intention was to provide both my aud...
This book provides a rigorous but elementary introduction to the theory of Markov Processes on a countable state space. It should be accessible to students with a solid undergraduate background in mathematics, including students from engineering, economics, physics, and biology. Topics covered are: Doeblin's theory, general ergodic properties, and continuous time processes.Applications are dispersed throughout the book. In addition, a whole chapter is devoted to reversible processes and the use of their associated Dirichlet forms to estimate the rate of convergence to equilibrium. These...
This book provides a rigorous but elementary introduction to the theory of Markov Processes on a countable state space. It should be accessible to ...
