Probability with measure

2023/2024

Programme:

Financial mathematics, First Cycle

Year:

3 year

Semester:

first

Kind:

mandatory

ECTS:

Language:

slovenian

Course director:

Assoc. Prof. Dr. Janez Bernik, Assoc. Prof. Dr. Mihael Perman

Lecturer (contact person):

Assist. Prof. Dr. Matija Vidmar

Hours per week – 1. semester:

Lectures

Seminar

Tutorial

Lab

Completed courses Analysis 2 and Probability 1.

-σ-algebras, Borel sets, measurable functions.
-Definition of measure, examples.
-Abstract integral, monotone konvergence theorem, Fatou's lemma, dominated convergence theorem.
-Product measures, Fubini's theorem.
- L-spaces.
-Random variables as measurable functions.
-Distributions.
-Expected value.
-Conditional expectation, Radon-Nikodým theorem.

B. Magajna, Osnove teorije mere, DMFA, 2001.
T. Tao, An introduction to measure theory, American Mathematical Society, 2011.
D. Khoshnevisan, Probability, American Mathematical Society, 2007.
D. Williams, Probability with Martingales, Cambridge Univeristy Press, 1991.
P. Billingsley, Probability and Measure, Wiley, 1979.

Finacial mathematics is based on probability theory. Measure theory provides the theoretical foundations for probability which are necessary for rigorus definitions and development of continuous time finance.

Understanding the abstract framework of probability and the ability to relate probabilistic concepts to measure theoretical concepts.

Lectures, problem sessions.

2 midterms or written exam, oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Mihael Perman:
AHČAN, Aleš, MASTEN, Igor, POLANEC, Sašo, PERMAN, Mihael. Quantile approximations in auto-regressive portfolio models. Journal of Computational and Applied Mathematics, ISSN 0377-0427. [Print ed.], Feb 2011, vol. 235, iss. 8, str. 1976-1983. [COBISS-SI-ID 19878630]
KOMELJ, Janez, PERMAN, Mihael. Joint characteristic functions construction via copulas. Insurance. Mathematics & economics, ISSN 0167-6687, 2010, vol. 47, iss. 2, str. 137-143. [COBISS-SI-ID 16242777]
HUZAK, Miljenko, PERMAN, Mihael, ŠIKIĆ, Hrvoje, VONDRAČEK, Zoran. Ruin probabilities and decompositions for general perturbed risk processes. Annals of applied probability, ISSN 1050-5164, 2004, vol. 14, no. 3, str. 1378-1397. [COBISS-SI-ID 13168985]
Janez Bernik:
BERNIK, Janez, MASTNAK, Mitja. Lie algebras acting semitransitively. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2013, vol. 438, iss. 6, str. 2777-2792. [COBISS-SI-ID 16553561]
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]
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]