Random walks (about 18 hours) Definition, Markovianity, temporal and spatial invariance, random walks with reflecting and absorbing barriers, reflection principle, ballot theorem, distribution of the maximum, hitting time theorem, first and second arc sine law, random walks and generating functions, short introduction on Black–Scholes model. Brownian motion (about 18 hours) Definition and existence, Brownian motion as a limit of a simple random walk, path properties of Brownian motion, Brownian motion as a strong Markov process, transience and recurrence. Branching processes (about 6 hours) Definition, expectation and variance of the population size, geometric branching, probability of extinction of the population. Markov chains (about 18 hours) Definition, homogeneous Markov chains, transition matrix, examples of Markov chains, classification of states, classification of chains, stationary distribution and limit theorem, chains with finitely many states, short introduction on Monte Carlo method, MCMC and search engine algorithms. Poisson processes (about 6 hours) Definition and main properties. Stationary processes (about 6 hours) Definition, variance and covariance function, linear predictions, spectral theorem for autocorrelation functions, ergodic theorem.

Recommended books: - G.R. Grimmett and D.R. Stirzaker. Probability and Random Processes. 3rd edn, OUP, 2001 - P. Mörters and Y. Peres. Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics, 2010 Helpful books: - D. Williams. Probability with Martingales. CUP, 1991 Teaching material is also delivered through e-learning platform "moodle" at the following address https://elearning2.uniroma1.it/course/view.php?id=6395