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Course, academic year 2019/2020
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Markov Processes - NMTP562
Title in English: Markovské procesy
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:4/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Additional information: http://simu0292.utia.cas.cz/seidler/teaching.html
Guarantor: RNDr. Jan Seidler, CSc.
Class: M Mgr. PMSE
M Mgr. PMSE > Volitelné
Classification: Mathematics > Probability and Statistics
Pre-requisite : NMTP432
Annotation -
Last update: T_KPMS (16.05.2013)
The very basic results of the continuous time Markov processes theory will be treated.
Aim of the course -
Last update: T_KPMS (16.05.2013)

The subject is aimed at the study of basic properties of continuous time

Markov processes taking values in general state spaces and the emphasis is

put on Feller processes and large-time behaviour.

Course completion requirements - Czech
Last update: RNDr. Jitka Zichová, Dr. (19.04.2018)

Složení ústní zkoušky.

Literature - Czech
Last update: T_KPMS (16.05.2013)

L.C.G. Rogers, D. Williams: Diffusion Markov processes and martingales. Vol. 1., Cambridge univ. press, 1994.

S.N. Ethier, T.G. Kurtz: Markov processes, Wiley, 1986.

Teaching methods -
Last update: T_KPMS (16.05.2013)

Lecture.

Requirements to the exam - Czech
Last update: RNDr. Jan Seidler, CSc. (11.10.2017)

Zkouška je ústní, požadavky odpovídají sylabu předmětu v rozsahu, který byl presentován na přednášce.

Syllabus -
Last update: T_KPMS (16.05.2013)

1. The Markov property, transition functions and operators associated with them, construction

of a process with a given transition function, shift operators and homogenous prpocesses.

2. Feller processes in locally compact spaces, their C0 semigroups, resolvents and generators,

the Hille-Yosida theorem, properties of sample paths, strong Markov processes.

3. Jump processes, processes with independent increments, Lévy processes, the Lévy-

Khinchin formula.

4. Diffusion processes: local characteristics, construction via stochastic differential equations,

the Kolmogorov equation.

5. Elementary ergodic theory: invariant measures, transient and recurrent processes, basic

results on existence of an invariant measure, (Krylov-Bogolyubov, Sunyach), strong Feller

processes, uniqueness and stability of invariant measures.

Entry requirements -
Last update: RNDr. Jan Seidler, CSc. (28.05.2019)

Students should have mastered the basics of probability theory and have some idea about Markov chains. A knowledge of stochastic analysis, or even stochastic differential equations, is desirable but not absolutely necessary – contact the lecturer.

 
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