Verjetnost z mero

Opravljena predmeta Analiza 2 in Verjetnost 1.

-σ-algebre, Borelove množice, merljive funkcije.
-Definicija mere, primeri.
-Abstrakten integral, izrek o monotoni konvergenci, Fatoujeva lema, izrek o dominirani konvergenci.
-Produktne mere, Fubinijev izrek.
-L-prostori.
-Slučajne spremenljivke kot merljive funkcije.
-Porazdelitve kot mere
-Pričakovana vrednost.
-Pogojna pričakovana vrednost, Radon-Nikodýmov izrek.

1. P. Billingsley: Probability and measure, 2nd ed., New York [etc.] : J. Wiley & Sons, cop. 1986.
2. D. Khoshnevisan: Probability, Providence : American Mathematical Society, cop. 2007.
3. B. Magajna: Osnove teorije mere, Ljubljana : DMFA - založništvo, 2011.
4. T. Tao: An introduction to measure theory, Providence : American Mathematical Society, cop. 2011.
5. D. Williams: Probability with martingales, Cambridge : Cambridge University Press, 1995.

Finančna matematika temelji na verjetnosti. Teorija mere daje teoretične temelje za verjetnost v obsegu, ki je potreben za trdne definicije in razvoj metod finančne matematike v zveznem času.

Razumevanje abstraktnega okvira verjetnosti, zmnožnost povezave pojmov teorije mere z verjetnostjo.

Predavanja, vaje.

2 kolokvija ali pisni izpit, ustni izpit
(ocene: 5 (negativno), 6-10 (pozitivno), ob upoštevanju Statuta 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]

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.