Stochastic Processes and Stochastic Calculus

This course aims at introducing some important stochastic processes (Markov chains, martingales,
Poisson process, Wiener process) and Ito calculus.
Some proofs are sketched or omitted in order to have more time for examples, applications and
In particular, the course deals with the following topics:
- Markov chains (definitions and basic properties, classification of states, invariant measure, stationary distribution, some convergence results and applications, passage problems, random walks, urn models, introduction to the Markov chain Monte Carlo method),
- conditional expectation,
- martingales (definitions and basic properties, Burkholder transform, stopping theorem and some
applications, predictable compensator and Doob decomposition, some convergence results, game theory, random walks, urn models),
- Poisson process,
- Markov processes,
- Wiener process and Ito calculus,
- Ornstein-Uhlenbeck process.

Prerequisites: The topics of ?Foundations of Probability Theory and Statistical Inference? are supposed to be known.