Introduction to Probability

Period of duration of course
Course info
Number of course hours
Number of hours of lecturers of reference
Number of hours of supplementary teaching

Type of exam

Written and oral exam


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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.


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.