Focusing on one of the major branches of probability theory, this book treats the large class of processes with continuous sample paths that possess the "Markov property". The exposition is based on the theory of stochastic analysis. The diffusion processes discussed are interpreted as solutions of Itô's stochastic integral equations. The book is designed as a selfcontained introduction, requiring no background in the theory of probability or even in measure theory. In particular, the theory of local continuous martingales is covered without the introduction of the idea of conditional expectation. Krylov covers such subjects as the Wiener process and its properties, the theory of stochastic integrals, stochastic differential equations and their relation to elliptic and parabolic partial differential equations, Kolmogorov's equations, and methods for proving the smoothness of probabilistic solutions of partial differential equations. With many exercises and thoughtprovoking problems, this book would be an excellent text for a graduate course in diffusion processes and related subjects. Readership Graduate students and researchers interested in an understanding of the important features of the theory of diffusion processes and its relationship with the theory of elliptic and parabolic second order partial differential equations. Reviews "For those with a good background in probability theory and analysis, this book is an excellent addition to the already good collection of books. The style is very relaxed but rigorous, written in the great pedagogical tradition of the Russian masters."  Journal of the American Statistical Association "What makes this book different is the presentation of the material. The author starts from scratch, introducing all the necessary concepts and techniques as he needs them. This makes it easy to follow his line of thought and to get to the main topics, stochastic integrals and stochastic differential equations, without detour and without many prerequisites ... invaluable help when studying from this book is a "dual" presentation of the material: All the main concepts and results are accompanied by a discussion of the intuitive idea behind them, and almost all proofs are given in a straightforward and precise manner."  Zentralblatt MATH "An accessible introduction to diffusion processes for working mathematicians and advanced graduate students in analysis ... a provocative, instructive, and refreshing perspective from which probabilists can benefit."  Mathematical Reviews "The book contains ideas of the author that have not been systematically presented in any other standard texts. Tremendous efforts are made to explore the probabilistic solutions of partial differential equations, reflecting the interest of the author. As an "introduction" to the theory, the book is elementary enough, even for those who have not had serious training in probability theory ... But on the other hand, the book is rich enough even for specialists in the field, as it contains many ideas which are different from the classical books on the subject."  Bulletin of the AMS "This is an appealing introduction to the theory of Markov processes with continuous sample paths, based on stochastic analysis by interpreting diffusion processes as solutions of Itô's Stochastic integral equation."  Monatshefte für Mathematik Table of Contents  Elements of measure and integration theory
 The Wiener process
 Itô's stochastic integral
 Some applications of Itô's formula
 Itô's stochastic equations
 Further methods for investigating the smoothness of probabilistic solutions of differential equations
 Appendix A. Proof of Lemma II.2.4
 Appendix B. Proof of Theorem II.8.1
 List of notations
 Comments
 References
 Index
