Completed courses Analysis 2 and Probability 1.
Probability with measure
2025/2026
Programme:
Financial mathematics, First Cycle
Year:
3 year
Semester:
first
Kind:
mandatory
ECTS:
5
Language:
slovenian
Course director:
Lecturer (contact person):
Hours per week – 1. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)
-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.
Readings
- P. Billingsley: Probability and measure, 2nd ed., New York [etc.] : J. Wiley & Sons, cop. 1986.
- D. Khoshnevisan: Probability, Providence : American Mathematical Society, cop. 2007.
- B. Magajna: Osnove teorije mere, Ljubljana : DMFA - založništvo, 2011.
- T. Tao: An introduction to measure theory, Providence : American Mathematical Society, cop. 2011.
- D. Williams: Probability with martingales, Cambridge : Cambridge University Press, 1995.
Objectives and competences
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.
Intended learning outcomes
Understanding the abstract framework of probability and the ability to relate probabilistic concepts to measure theoretical concepts.
Learning and teaching methods
Lectures, problem sessions.
Assessment
2 midterms or written exam, oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)
Lecturer's references
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]
Jaka Smrekar:
SMREKAR, Jaka. Spaces of functions and sections with paracompact domain. Proceedings. Section A, Mathematics. Jun. 2025, vol. 155, iss. 3, str. 893-915.
SMREKAR, Jaka. Rational Reidemeister trace of an outer automorphism of finite order. Journal of Pure and Applied Algebra. [Print ed.]. July 2023, vol. 227, iss. 7, art. 107350 (13 str.).
KIRALY, Peter, SMREKAR, Jaka, JAKI MEKJAVIĆ, Polona. Morphological parameters predicting subthreshold micropulse laser effectiveness in central serous chorioretinopathy. Lasers in medical science. 2022, vol. 37, may, str. 3129-3136.
Matija Vidmar:
VIDMAR, Matija. Fock structure of complete Boolean algebras of type I factors and of unital factorizations. Journal of mathematical analysis and applications. [Print ed.]. Feb. 2025, vol. 542, iss. 1, [article no.] 128780, 35 str. ISSN 0022-247X. https://www.sciencedirect.com/science/article/pii/S0022247X24007029, DOI: 10.1016/j.jmaa.2024.128780.
VIDMAR, Matija. On laws exhibiting universal ordering under stochastic restart. Communications in statistics. theory and methods. 2022, vol. 51, iss. 5, str. 1290-1305. ISSN 0361-0926. https://doi.org/10.1080/03610926.2020.1759640, DOI: 10.1080/03610926.2020.1759640.
VIDMAR, Matija. A couple of remarks on the convergence of σ-fields on probability spaces. Statistics & probability letters. 2018, vol. 134, str. 86-92. ISSN 0167-7152. http://doi.org/10.1016/j.spl.2017.10.017, DOI: 10.1016/j.spl.2017.10.017.
