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We start with the notions of sub-, super-, martingale. The lecture is mainly devoted to discrete time martingales. The detailed
technical explanation serves as basics for extended courses, e.g. for stochastic analysis.
Last update: T_KPMS (15.05.2013)
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To explain basics of the martingale theory. Last update: T_KPMS (15.05.2013)
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The course is finalized by a credit from exercise class and by a final exam.
The credit from exercise class is necessary for taking part in the final exam.
Requirements for receiving the credit from exercise class: active participation (attendance at least 75% during in-person classes), elaboration of two homeworks.
Attempt to receive the credit from exercise class cannot be repeated. Last update: Pawlas Zbyněk, doc. RNDr., Ph.D. (30.09.2021)
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J. Jacod, P. Protter (2004): Probability Essentials, 2nd edition, Springer, Berlin.
O. Kallenberg (2002): Foundations of Modern Probability, 2nd edition, Springer, New York.
J. Štěpán (1987): Teorie pravděpodobnosti - matematické základy, Academia, Praha. Last update: Pawlas Zbyněk, doc. RNDr., Ph.D. (28.10.2019)
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Lecture+exercises. Last update: Pawlas Zbyněk, doc. RNDr., Ph.D. (29.09.2021)
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The final exam is oral. All material covered during the course may be part of the exam. Last update: Pawlas Zbyněk, doc. RNDr., Ph.D. (11.10.2017)
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1. random sequence, finite-dimensional distributions, Daniell's theorem
2. filtration, stopping times, martingale (submartingale, supermartingale) with discrete time
3. optional stopping and optional sampling theorem, maximal inequalities
4. convergence of submartingales
5. limit theorems for martingale differences Last update: T_KPMS (24.04.2015)
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Basics of probability theory - probability space, random vectors, independence, convergence, conditional expectation, characteristic function, law of large numbers, central limit theorem. Last update: Pawlas Zbyněk, doc. RNDr., Ph.D. (18.05.2018)
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