(in Polish) Procesy Levy'ego i procesy stabilne

elective monographs

It is assumed that the students are familiar with the subjects covered by the courses: Probability Theory I and II (or equivalent). Moreover, basic knowledge of the general theory of stochastic processes is also necessary (construction of a stochastic process, Brownian motion, Poisson process, and their properties, properties of sample paths, notions of a filtration, stopping time, etc.)

The course will cover selected topics on two important (not disjoint) classes of processes: Levy processes, i.e. processes with stationary independent increments, and stable processes. Properties and various representations of these processes, as well as limit theorems will be discussed.

We plan to cover the following topics:

Poisson random measure, its construction and properties, integration.

Infinitely divisible laws. Levy-Khintchine theorem. Theorem on convergence to an infinitely divisible law.

Levy processes (processes with stationary independent increments). Levy-Ito decomposition - representation of a Levy process with help of a Poisson random measure and an independent Brownian motion. Properties (e.g. properties of sample paths, moments, recurrence and transience, asymptotic behaviour for small times). Subordination. Wiener-Hopf factorization. Levy processes without positive jumps. First passage times as subordinators. Examples of limit theorems in which the limits are Levy processes.

Stable processes. The general form of stable laws on a real line. Series representation of stable random variables. Multidimensional stable laws. Characteristic function and spectral measure. Measuring dependence - covariation and codifference. Independently scattered stable random measure and stable processes which have a representation as an integral with respect to this random measure. Examples (fractional stable processes, stable Ornstein Uhlenbeck process). Limit theorems.

Self-similar stable processes. Special case - Gaussian processes: fractional Brownian motion and other processes of "fractional type". Properties and representations. Fractional Brownian motion as a limit of functionals related to alpha-stable Levy processes.

We reserve the right to modify this program, depending on the students preferences.

Applebaum, D.:Levy processes and stochastic calculus

Bertoin, J. Levy processes

Kallenberg, O.: Foundations of modern probability, Springer, 2001.

Kyprianou. A.E.: Introductory lectures on Fluctuations of Levy processes, Springer, 2006.

Sato, K-I.: Levy processes and infinitely divisible distributions,

Cambridge University Press, 2005.

Samorodnitsky G., Taqqu: Stable non-Gaussian random processes, Chapman & Hall/CRC, 2000.