Stochastic Processes at CEU -- winter semester 2013/14

LIST OF QUESTIONS FOR THE FINAL EXAM

Everybody will get two such questions. According to a wish of the audience, the emphasis will be on continuous time processes.

1. Stochastic processes. Finite dimensional distributions versus pathwise properties.
2. Filtrations, natural filtration of a stochastic process. Markov processes, martingales, processes of independent increments, stationary processes.
3. Poisson process, Wiener process. Definition and basic properties.
4. Convergence to equilibrium for finite state space time homogeneous Markov chains.
5. Transience, recurrence and positive recurrence for countable state space Markov chains. Pólya's theorem about recurrence of the simple symmetric random walk.
6. Discrete state space, time homogeneous Markov chains in continuous time. Phenomenological descriptions with exponential clocks. The infinitesimal generator and its relation to finite time transition probability matrices. Examples.
7. Stochastic integrals: difficulties of a pathwise definition, the problem of choosing the intermediate point. Itô's choice.
8. Definition of the Itô integral for a fixed time, for progressively measurable processes. Motivation, basic properties, Itô isometry
9. The Itô integral as a stochastic process. Continuity and martingale property.
10. Itô processes and the Itô formula. Examples, applications. Examples of stochastic differential equations.
11. Itô representation theorem.
12. The martingale representation theorem.