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The very basic results of the continuous time Markov processes theory will be treated.
Last update: T_KPMS (16.05.2013)
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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. Last update: T_KPMS (16.05.2013)
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Oral exam. Last update: Zichová Jitka, RNDr., Dr. (13.05.2023)
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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.
Last update: T_KPMS (16.05.2013)
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Lecture. Last update: T_KPMS (16.05.2013)
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Oral exam according to sylabus. Last update: Zichová Jitka, RNDr., Dr. (13.05.2023)
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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. Last update: T_KPMS (16.05.2013)
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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. Last update: Seidler Jan, RNDr., CSc. (28.05.2019)
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