2013/2014

2014/2015

DM270

MAT/06 (PROBABILITY AND MATHEMATICAL STATISTICS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

2°

2nd semester (02/03/2015 - 12/06/2015)

6

48 lesson hours

ITALIAN

ORAL TEST

RIGO PIETRO (titolare) - 6 ECTS

The course "Probability" of the Laurea Magistrale. As a consequence, "Stochastic Processes" is not suggested to students of the Laurea Triennale.

This course is the natural continuation of "Probability" (Laurea Magistrale). The characteristic arguments are Markov processes (both in discrete and in continuous time) and weak convergence of probability measures. Some attention will be also paid to large deviations and continuous time martingales. The approach is essentially of the theoretical type. However, the results taken into consideration are fundamental for several applications of probability theory, mainly in mathematical finance, statistical mechanics and dynamical systems.

1. General notions about stochastic processes;

2. Continuous time martingales and Brownian motion;

3. Markov chains (with arbitrary state space);

4. Continuous time Markov processes;

5. Weak convergence of probability measures on metric spaces;

6. Large deviations.

Extended summary

1. General notions about stochastic processes: Definition, paths, equality of processes, filtrations, stopping times. Existence of processes with given finite dimensional distributions.

2. Continuous time martingales and Brownian motion: paths, some inequalities, limit theorems, optional sampling theorem, Doob-Meyer decomposition. Brownian motion.

3. Markov chains (with arbitrary state space): General notions, existence, kernels, strong Markov property, stationary distributions, reversibility, irriducibility, recurrence, ergodicity and its characterizations, random walks, Gibbs sampling, chains with countable state space.

4. Continuous time Markov processes: General notions, existence, kernels, strong Markov property, generators, backward and forward equations, semigroups of operators, some meaningful examples (diffusions).

5. Weak convergence of probability measures on metric spaces: General remarks, portmanteau theorem, some meaningful examples, other modes of convergence, theorems of Alexandrov, Skorohod and Prohorov, empirical processes, Donsker theorems.

6. Large deviations: General remarks on the large deviation principle. Theorems of Cramer, Schilder and Sanov.

Lessons. (Various exercises will be also discussed during such lessons).

1. Kallenberg O.: Foundations of modern probability (Second edition), Springer, 2002.

2. Dudley R.M.: Real analysis and probability, Chapman and Hall, New York, 1993.

3. Meyn S. P. and Tweedie R. L.: Markov Chains and Stochastic Stability, Springer, 1996.

Oral examination. Simple versions of the exercises done in class will be possibly discussed during the examination.

None.