Advanced Probability

Probability theory is now applied in a variety of fields including physics, engineering, informatics, biology, economics and social sciences. This course is an introduction to the rigorous theory of probability with perspective theme given by the Doob's theory of martingales and an introduction to the stochastic processes. Given the reference to the basic theory, the relevant concept of the conditional expectation is examined in depth. Stochastic processes and in particular their measurability properties and the construction of the path space are introduced. Two classes of processes are studied into details: the martingales at both discrete and continuous time, and the Markov processes, via its characterizations and the study of the discrete time Markov Chain.

Student learn how to treat and discuss the main properties of principal probabilistic objects.
He learn of the main mathematical properties of stochastic processes. He gain a knowledge of important classes of processes and of advanced techniques in stochastic analysis.