Introduction to Probability
Prerequisites
The lectures are addressed to PhD students, mainly oriented to applications. Previous knowledge of the basics of Probability are very useful, although not strictly necessary. No previous knowledge of stochastic processes is required.
Programme
Foundations and elementary examples: Probability space, conditional probability and independence,
expected values and main results of calculus, discrete and continuous examples. Conditional expectation.
Markov chains: transition matrix and graph, Markov process, construction of the process, invariant measures,
their existence, uniqueness results and convergence to equilibrium.
Continuous time Markov chains: some elements of theory, infinitesimal generators, useful rules.
Brownian motion, Kolmogorov regularity theorem, quadratic variation.
Elements of martingale theory. Examples.
Elements of stochastic integration and stochastic differential equations. Links with Partial Differential Equations.
Educational aims
Goal of the lectures is to introduce students to basic topics of Probability theory and stochastic processes, useful in several fields and applications, including Mathematical Finance.
Bibliographical references
Notes of the teachers.