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Course, academic year 2019/2020
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Stochastic Analysis - NMTP432
Title in English: Stochastická analýza
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2019
Semester: summer
E-Credits: 8
Hours per week, examination: summer s.:4/2 C+Ex []
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: doc. RNDr. Daniel Hlubinka, Ph.D.
Class: Pravděp. a statistika, ekonometrie a fin. mat.
M Mgr. PMSE
M Mgr. PMSE > Povinně volitelné
Classification: Mathematics > Probability and Statistics
Pre-requisite : NMSA405
Is pre-requisite for: NMTP551, NMTP562
In complex pre-requisite: NMTP533, NMTP543
Annotation -
Last update: T_KPMS (16.05.2013)
Stochastic processes. Continuous martingales and Brownian motion. Markov times. Spaces of stochastic processes. Doob Meyer decomposition. Quadratic variation of a continuous martingale. Stochastic integral. Exponential martingales and Lévy characterization of Brownian motion. Trend removing Girsanov theorem for Brownian motion. Brownian representation of a continuous martingale by a stochastic integral. Local time of a continuous martingale. An introduction to the theory of stochastic differential equations. Applications to physics and financial mathematics.
Aim of the course -
Last update: T_KPMS (16.05.2013)

An advanced lecture on Brownian motion and stochastic integral is designed to to complete a student knowledge and abilities to handle a stochastic process both from theoretical and applied view.

Course completion requirements -
Last update: RNDr. Jitka Zichová, Dr. (06.03.2018)

The credits for exercise classes are necessary condition for the exam.

Conditions for the credits:

Attendance in the classes. At most four absences are tolerated during the semester.

The nature of the credits excludes a retry.

Literature - Czech
Last update: T_KPMS (16.05.2013)

Dupačová, J., Hurt, J., Štěpán, J.: Stochastic Modeling in Economics and Finance.

Kluwer Academic Publishers, London, 2002.

O. Kallenberg: Foundations of modern probability. Springer, New York, 2002.

Karatzas, I., Shreve, D.E.: Brownian Motion and Stochastic Calculus.

Springer Verlag, New York, 1991.

Teaching methods -
Last update: T_KPMS (16.05.2013)

Lecture+exercises

Requirements to the exam -
Last update: RNDr. Jitka Zichová, Dr. (06.03.2018)

The exam is oral. Some questions and problems are given to the student. The content of the questions is adapted to the topics covered during the lectures.

Syllabus -
Last update: T_KPMS (16.05.2013)

1. Stochastic processes and their construction.

2. Continuous martingales and Brownian motion.

3. Markov times, martingales stopped by a Markov time.

4. Spaces of stochastic processes.

5. Doob Meyer decomposition. Quadratic variation of a continuous martingale.

6. Stochastic integral and its properties.

7. Exponential martingales and Lévy characterization of Brownian motion.

8. Trend removing Girsanov theorem for Brownian motion.

9. Brownian representation of a continuous martingale by a stochastic integral.

10. Local time of a continuous martingale.

11. An introduction to the theory of stochastic differential equations.

12. Stochastic analysis applied to physics and financial mathematics.

Entry requirements -
Last update: doc. RNDr. Daniel Hlubinka, Ph.D. (10.05.2018)

Knowledge required before enrollment:

conditional probability and conditional expectation

discrete martingales

 
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