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Course, academic year 2023/2024
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Probability Theory 2 - NMSA405
Title: Teorie pravděpodobnosti 2
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. RNDr. Zbyněk Pawlas, Ph.D.
RNDr. Petr Čoupek, Ph.D.
Class: M Mgr. PMSE
M Mgr. PMSE > Povinné
Classification: Mathematics > Probability and Statistics
Is pre-requisite for: NMTP436, NMTP434, NMST511, NMTP450, NMTP438, NMTP432
Is interchangeable with: NSTP145, NSTP051
In complex pre-requisite: NMFM505, NMFM535, NMFP505
Annotation -
Last update: T_KPMS (15.05.2013)
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.
Aim of the course -
Last update: T_KPMS (15.05.2013)

To explain basics of the martingale theory.

Course completion requirements -
Last update: doc. RNDr. Zbyněk Pawlas, Ph.D. (30.09.2021)

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.

Literature -
Last update: doc. RNDr. Zbyněk Pawlas, Ph.D. (28.10.2019)

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.

Teaching methods -
Last update: doc. RNDr. Zbyněk Pawlas, Ph.D. (29.09.2021)


Requirements to the exam -
Last update: doc. RNDr. Zbyněk Pawlas, Ph.D. (11.10.2017)

The final exam is oral. All material covered during the course may be part of the exam.

Syllabus -
Last update: T_KPMS (24.04.2015)

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

Entry requirements -
Last update: doc. RNDr. Zbyněk Pawlas, Ph.D. (18.05.2018)

Basics of probability theory - probability space, random vectors, independence, convergence, conditional expectation, characteristic function, law of large numbers, central limit theorem.

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