Description
Book SynopsisThis work contains a collection of lectures on stochastic processes. The material is arranged in such a way to fit a lecture time of one-and-a-half hours, in order to make the book convenient for lecturers and students. The book can be used in the preparation of courses in stochastic processes for which an understanding of basic notions of mathematical analysis, theory of complex functions, theory of differential equations and probability theory is required. The subjects in the book have different levels of abstraction.
Table of ContentsFrontmatter -- Contents -- Preface -- Lecture 1. Stochastic processes. definitions. examples -- Lecture 2. The kolmogorov consistency theorem. classification of processes -- Lecture 3. Random walks. recurrence. renewal theorem -- Lecture 4. Martingales. inequalities for martingales -- Lecture 5. Theorems on the limit of a martingale -- Lecture 6. Stationary sequences. ergodic theorem -- Lecture 7. Ergodic theorem. metric transitivity -- Lecture 8. Regularization of a process. continuity -- Lecture 9. Processes without discontinuities of the second kind -- Lecture 10. Continuity of processes with independent increments. martingales with continuous time -- Lecture 11. Measurable processes -- Lecture 12. Stopping times. associated tr-algebras -- Lecture 13. Completely measurable processes -- Lecture 14. L2-theory -- Lecture 15. Stochastic integrals -- Lecture 16. Stationary processes. spectral representations -- Lecture 17. Stationary sequences. regularity and singularity -- Lecture 18. The prediction of a stationary sequence -- Lecture 19. Markov processes -- Lecture 20. Homogeneous markov processes and associated semigroups -- Lecture 21. Homogeneous purely discontinuous processes. conditions for their regularity -- Lecture 22. Processes with adenumerable set of states -- Lecture 23. Simple birth and death processes -- Lecture 24. Branching processes with particles of only one kind -- Lecture 25. Homogeneous processes and strongly continuous semigroups. resolvent operator and generator -- Lecture 26. The hille-iosida theorem -- Lecture 27. Processes with independent increments. representation of the discontinuous part -- Lecture 28. General representation of a stochastically continuous process with independent increments -- Lecture 29. Diffusion processes -- Lecture 30. Stochastic integrals -- Lecture 31. Existence, uniqueness, and properties of solutions of stochastic differential equations -- Lecture 32. Itô's formula with some corollaries -- Bibliography
