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Lecturer(s)
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Ježek Vladimír, doc. Ing. Ph.D.
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Course content
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1. Basic notions of probability theory-recollection, the concept of conditional expectation and stochastic (random) process. 2.-3. Some frequently used processes-Brownian motion and fractional Brownian motion, processes with jumps, Levy process. 4. Martingales, definition, some properties and applicability of the martingale theory. 5.-6. Continuous-time Markov processes with a general state space, definition and basic properties. Transition densities, examples-SDE 7.-8. Diffusion processes and models, relation to partial differential equations, Kolmogorov and Fokker-Planck equation. Feynman-Kac formula-killing. 9. Random stopping times and the strong Markov property, the Feller property-continuous dependence on initial data 10.-11. Stationary (equilibrium) states ? invariant measures, recurrence and transience, sufficient conditions for solutions to SDE 12-13. Convergence to the stationary state, nondegenerating SDE
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Learning activities and teaching methods
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Interactive lecture, Lecture supplemented with a discussion, Lecture with practical applications, Students' portfolio
- Contact hours
- 52 hours per semester
- Preparation for an examination (30-60)
- 40 hours per semester
- Individual project (40)
- 40 hours per semester
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| prerequisite |
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| Knowledge |
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| Students should have a basic knowledge of probability theory (KMA/PSA), fundamentals of random processes (KMA/ZNP) and of introduction to stochastic analysis (KMA/USA). |
| learning outcomes |
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| Students taking this course will be able to understand the mathematical background of stochastic processes and namely - recognize which stochastic processes are appropriate and needed for modelling randomness in a given research problem - apply stochastic processes to practical problems - analyze the usefulness of stochastic processes in professional area - provide logical and coherent proofs of theoretic results - solve problems via abstract methods - apply correctly formal and rigorous competency in mathematical presentation, both in written and verbal form. |
| teaching methods |
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| Lecture supplemented with a discussion |
| Interactive lecture |
| Students' portfolio |
| assessment methods |
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| Oral exam |
| Written exam |
| Seminar work |
| Individual presentation at a seminar |
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Recommended literature
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Mandl, Petr. Pravděpodobnostní dynamické modely : celost. vysokošk. učebnice pro stud. matematicko-fyz. fakult stud. oboru pravděpodobnost a matem. statistika. Praha : Academia, 1985.
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Maslowski, Bohdan. Stochastic Equations and Stochastic Methods in PDE's. Plzeň, 2006.
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Prášková, Zuzana; Lachout, Petr. Základy náhodných procesů. Praha : Karolinum, 1998. ISBN 80-7184-688-0.
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Štěpán, Josef. Teorie pravděpodobnosti : Matematické základy : Vysokošk. učebnice pro stud. matematicko-fyz. fakult. Praha : Academia, 1987.
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