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Stochastic Analysis - NMTP432

Title:	Stochastická analýza
Guaranteed by:	Department of Probability and Mathematical Statistics (32-KPMS)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2016 to 2016
Semester:	summer
E-Credits:	8
Hours per week, examination:	summer s.:4/2, C+Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech, English
Teaching methods:	full-time
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
Incompatibility :	NSTP149, NSTP153
Pre-requisite :	NMSA405
Interchangeability :	NSTP149, NSTP153
Is incompatible with:	NSTP153
Is pre-requisite for:	NMTP562, NMTP533, NMTP551
Is interchangeable with:	NSTP153, NSTP149, NSTP168
In complex pre-requisite:	NMTP543

Opinion survey results Examination dates Schedule Noticeboard

Annotation -

Last update: doc. Ing. Marek Omelka, Ph.D. (16.02.2023)

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: RNDr. Petr Čoupek, Ph.D. (16.02.2023)

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.

Literature - Czech

Last update: RNDr. Petr Čoupek, Ph.D. (16.02.2023)

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: RNDr. Petr Čoupek, Ph.D. (16.02.2023)

Lecture+exercises

Syllabus -

Last update: RNDr. Petr Čoupek, Ph.D. (16.02.2023)

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.