Completed course Probability 1.

# Stochastic processes 1

Stochasic processes in discrete and continuous time.

Counting processes: homogeneous Poisson processes (definition, interarrival times property, ordered statistics property), inhomogeneous Poisson processes (characterization, construction), renewal processes (definition, elementary renewal theorem, renewal equations).

Brownian motion: construction, path properties, strong Markov property, reflection principle.

Discrete time Markov chains: classification of states, classification of chains, Markov properties, ergodic properties.

Continuous time Markov chains: strong Markov property, forward and backward Kolmogorov equations, birth-death processes, branching processes, ergodic properties.

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

J. R. Norris: Markov Chains, Cambridge University Press, 1999.

J. F. C Kingman: Poisson Processes, Oxford Science Publications, 1993.

Z. Brzeźniak, T. Zastawniak: Basic Stochastic Processes, Springer, 1999.

D. Williams: Probabilty with Martingales, Cambridge University Press, 1995.

B. Øksendal: Stochastic Differential Equations: An Introduction with Applications, 6th Edition, Springer, 2005.

Introduction to the theory of stochastic processes in continuous time and the

basic examples of stochastic processes such as

Poisson processes, renewal processes, Markov chains and Brownian motion.

Knowledge and understanding: Understanding of the interplay between randomness and time evolution and development of the necessary mathematical concepts and tools.

Application: Stochastic processes form a foundation for various kinds of modelling, particulary in insurance and finance.

Reflection: The nature of the course implies that the basic concepts of probability, which were introduced in earlier courses, are now

used, thus broadening and deepening their understanding.

Transferable skills: The skills acquired are directly transferable not only to other branches of mathematics, but to direct modelling of real world phenomena, especially in finance.

Lectures, exercises, homework, consultations

2 midterm exams instead of written exam, 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]