(in Polish) Procesy Levy'ego i procesy stabilne 1000-1M13PLS
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.
Type of course
Prerequisites (description)
Bibliography
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.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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