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 self-contained 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 thought-provoking 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.