The course provides a broad introduction to stochastic processes.

In particular the aim is

- to give a rigorous introduction to the theory of stochastic processes,

- to discuss the most important stochastic processes in some depth with examples and applications,

- to give the flavour of more advanced work and applications,

- to apply these ideas to answer basic questions in several applied situations including biology, finance and search engine algorithms.

At the end of the course the students

- will be familiar with the basic concepts of the theory of stochastic processes in discrete and continuous time and will be able to apply various techniques to study stochastic models that appear in applications,

- will have the tools to grasp and formalize, in the language of stochastic processes, phenomena that evolve in time and space and to solve simple applied problems in new environments and broader contexts,

- will have the tools to evaluate critically and choose between different stochastic models to model phenomena that evolve in time and space,

- will acquire the necessary language skills to read academic books on the topic and research papers,

- will acquire the rationale behind the stochastic model studied (e.g. the ideas of Markovianity, transience, recurrence, equilibrium, stationarity, long and short-time behaviour...) that is necessary to communicate to specialist and non-specialist audiences.

- will acquire the methodology and the language to study in a manner that may be largely autonomous and to apply the methodology to the subsequent studies in the area of statistics and finance.

Program:

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

The examination consists of three questions on three different topics studied during the course.

The students are required to answer the questions showing that they have acquired the intuition and the technical language, and that they are able to present the topics and the proofs with all necessary details.

- Teacher: VALENTINA CAMMAROTA