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.

Part 1.Theory of Stochastic Processes.

1. Stochastic Processes. General Properties. Trajectories, Finite-dimensional Distributions.

2. Stochastic Processes with Independent Increments.

3. Gaussian Processes. Integration with Respect to Gaussian Processes.

4. Construction, Properties and Some Functionals of the Wiener Process and Fractional Brownian Motion.

5. Martingales and Related Processes.

6. Regularity of Trajectories of Stochastic Processes.

7. Markov and Diffusion Processes.

8. Stochastic Integration.

9. Stochastic Differential Equations.

Part 2. Statistics of Stochastic Processes.

10. Parameter Estimation.

11. Filtering Problem. Kalman-Bucy Filter.

Yuliya Mishura is Full Professor and Head of the Department of Probability Theory, Statistics and Actuarial Mathematics at Taras Shevchenko National University of Kyiv. Her research interests include martingale theory, fractional processes, stochastic differential equations, the statistics of stochastic processes and financial mathematics.

Georgiy Shevchenko is Full Professor in the Department of Probability Theory, Statistics and Actuarial Mathematics at Taras Shevchenko National University of Kyiv. His research interests include stochastic analysis, stochastic differential equations, models with long memory, the statistics of stochastic processes and financial mathematics.