Lévy processes, Lévy-Khintchine formula, jump measures, construction of Lévy processes,

Potential theory, solving PDE by means of stochastic processes,

Basic concepts of stochastic differential equations, the Ornstein-Uhlenbeck process.

# Stochastic processes 3

N.V. Krylov: Introduction to the Theory of Random Processes, Graduate Studies in Mathematics, vol. 43, American Mathematical Society, 2002.

D.W. Stroock: Probability Theory: an analytic view, Cambridge University Press, 2003.

R. Bass: Probabilistic Techniques in Analysis, Springer-Verlag, 1995.

R. Durrett: Stochastic Calculus: A Practical Introduction, CRC Press, 1996.

Within the course we present an introduction to the theory of Lévy processes, we learn about the probabilistic approach to the potential theory and partial differential equations, and finally we meet the basics of stochastic differential equations.

Knowledge and understanding:

Deepening of study and rigorous treatment of certain particular features of stochastic processes, probabilistic approach to problems from PDE.

Application:

Basic tools for modelling in many branches of

mathematics and its applications.

Reflection:

Learning about deeper connections between various areas of mathematics, meticulous treatment of jumps.

Transferable skills:

The skills acquired are transferable to other areas of mathematical modelling, among the rest to financial models.

Lectures, exercises, homeworks, seminars

Homework and seminar assignments

Oral exam

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Mihael Perman:

PERMAN, Mihael, PITMAN, Jim, YOR, Marc. Size-biased sampling of Poisson processes and excursions. Probability theory and related fields, ISSN 0178-8051, 1992, 92, no. 1, str. 21-39. [COBISS-SI-ID 12236377]

PERMAN, Mihael, WELLNER, Jon A. On the distribution of Brownian areas. Annals of applied probability, ISSN 1050-5164, 1996, let. 6, št. 4, str. 1091-1111. [COBISS-SI-ID 7101017]

PERMAN, Mihael. An excursion approach to Ray-Knight theorems for perturbed Brownian motion. Stochastic Processes and their Applications, ISSN 0304-4149. [Print ed.], 1996, let. 63, str. 67-74. [COBISS-SI-ID 7621465]

Oliver Dragičević:

CARBONARO, Andrea, DRAGIČEVIĆ, Oliver. Bellman function and linear dimension-free estimates in a theorem of Bakry. Journal of functional analysis, ISSN 0022-1236, 2013, vol. 265, iss. 7, str. 1085-1104. [COBISS-SI-ID 16719705]

DRAGIČEVIĆ, Oliver, PETERMICHL, Stefanie, VOLBERG, Alexander. A rotation method which gives linear L[sup]p estimates for powers of the Ahlfors-Beurling operator. Journal de Mathématiques Pures et Appliquées, ISSN 0021-7824. [Print ed.], 2006, vol. 86, iss. 6, str. 492-509. [COBISS-SI-ID 14157657]

DRAGIČEVIĆ, Oliver, VOLBERG, Alexander. Bellman function, Littlewood-Paley estimates and asymptotics for the Ahlfors-Beurling operator in L[sup]p(C). Indiana University mathematics journal, ISSN 0022-2518, 2005, vol. 54, no. 4, str. 971-996. [COBISS-SI-ID 14139737]