Brownian motion:

Basic properties, existence, path properties, natural filtration, first hitting time, Markov properties, strong Markov property, reflection principle, associated processes (running supremum process, Brownian bridge etc.),

quadratic variation.

Continuous time martingales:

Filtrations, stopping times, stopping theorems,

uniform integrability, maximal inequalities, convergence of martingales.

Stochastic integral:

Stochastic integral wrt Brownian motion,

Itô isometry, continuous semimartingales, local martingales, quadratic variation and covariation, stochastic integral wrt continuous semimartingales, Itô's formula, Girsanov Theorem, representation of martingales.

# Stochastic processes 2

S. Resnick: Adventures in Stochastic Processes, Birkhäuser Boston, 2002.

I. Karatzas, S. E. Shreve: Brownian Motion and Stochastic Calculus, 2nd Edition, Springer, 2005.

M. Yor, D. Revuz: Continuous Martingales and Stochastic Calculus, 2nd Edition, Springer, 2004

J. M. Steele: Stochastic Calculus and Financial Applications, Springer,

New York, 2001.

This course is an introduction to the theory of stochastic processes in continuous time with continuous sample paths. It rigorously treats Brownian motion as a basic example and building block, introduces martingales in continuous time, stochastic calculus and Ito's formula.

Knowledge and understanding:

Mathematical tools for rigorous treatment and applications of stochastic processes.

Application:

Basic tools for modelling in many branches of

Mathematics and its applications.

Reflection:

The contents of the course help in retrospect to deepen the understanding of the concepts of probability, dependence and time.

Transferable skills:

The skills acquired are transferable to other areas of mathematical modelling, in particular it is immediately applicable to financial models.

Lectures, exercises, homeworks, consultations

Type (examination, oral, coursework, project):

written exam

Grading: 1-5 (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]

Janez Bernik:

BERNIK, Janez, MASTNAK, Mitja, RADJAVI, Heydar. Realizing irreducible semigroups and real algebras of compact operators. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2008, vol. 348, no. 2, str. 692-707. [COBISS-SI-ID 14899289]

BERNIK, Janez, MASTNAK, Mitja, RADJAVI, Heydar. Positivity and matrix semigroups. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2011, vol. 434, iss. 3, str. 801-812. [COBISS-SI-ID 15745625]

BERNIK, Janez, MARCOUX, Laurent W., RADJAVI, Heydar. Spectral conditions and band reducibility of operators. Journal of the London Mathematical Society, ISSN 0024-6107, 2012, vol. 86, no. 1, str. 214-234. [COBISS-SI-ID 16357721]