This book is concerned with the theory of stochastic processes and the theoretical aspects of statistics for stochastic processes. It combines classic topics such as construction of stochastic processes, associated filtrations, processes with independent increments, Gaussian processes, martingales, Markov properties, continuity and related properties of trajectories with contemporary subjects: integration with respect to Gaussian processes, Ito integration, stochastic analysis, stochastic differential equations, fractional Brownian motion and parameter estimation in diffusion models.

The presentation is made as self-contained as possible, with complete proofs of the facts which are often either omitted from textbooks or are replaced by informal or heuristic arguments. Some auxiliary material, related mainly to different subjects of real analysis and probability theory, is included in the comprehensive appendix. The book is targeted at the widest audience: students of mathematical and related programs, postgraduate students, postdocs, lecturers, researchers and practitioners in any field concerned with the application of stochastic processes will find this book to be a valuable resource.