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Course, academic year 2019/2020
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Probability Theory 2 - NMSA405
Title in English: Teorie pravděpodobnosti 2
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2019
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Guarantor: doc. RNDr. Zbyněk Pawlas, Ph.D.
Class: M Mgr. PMSE
M Mgr. PMSE > Povinné
Classification: Mathematics > Probability and Statistics
Is pre-requisite for: NMTP436, NMTP434, NMTP438, NMTP450, NMST450, NMFM535, NMTP432
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. (11.10.2017)

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 (at most 3 absences), one homework presented at the blackboard.

Attempt to receive the credit from exercise class cannot be repeated.

Literature - Czech
Last update: T_KPMS (15.05.2013)

Štěpán J.: Teorie pravděpodobnosti. Matematické základy. Academia, Praha, 1987

Kallenberg, O.: Foundations of modern probability. Springer, 1997.

Lachout, P.: Diskrétní martingaly. Karolinum, Praha, 2007.

Teaching methods -
Last update: T_KPMS (15.05.2013)

Lecture+exercises.

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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