Stochastic processes 2

There are no prerequisites.

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.

1. I. Karatzas, S. E. Shreve: Brownian motion and stochastic calculus, 2nd ed., New York : Springer, cop. 1998.
2. S. Resnick: Adventures in stochastic processes, Boston : Birkhäuser, cop. 1992.
3. D. Revuz, M. Yor: Continuous martingales and Brownian motion, Berlin : Springer, 1991.
4. J. M. Steele: Stochastic calculus and financial applications, New York : Springer, cop. 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

Written 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]

Matija Vidmar:
MIJATOVIĆ, Aleksandar, VIDMAR, Matija, JACKA, Saul D. Markov chain approximations for transition densities of Lévy processes. Electronic journal of probability. 2014, vol. 19, paper no. 21 (str. 1-37). ISSN 1083-6489. http://dx.doi.org/10.1214/EJP.v19-2208.
VIDMAR, Matija. Another characterization of homogeneous Poisson processes. Stochastics. 2018, vol. 90, iss. 6, str. 876-885. ISSN 1744-2508. https://doi.org/10.1080/17442508.2018.1457674, DOI: 10.1080/17442508.2018.1457674
VIDMAR, Matija. A temporal factorization at the maximum for certain positive self-similar Markov processes. Journal of Applied Probability. Dec 2020, vol. 57, iss. 4, str. 1045-1069. ISSN 0021-9002. https://doi.org/10.1017/jpr.2020.62, DOI: 10.1017/jpr.2020.62