Lectures on the Theory of Stochastic Processes

Frontmatter -- 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