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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.
Last update: Omelka Marek, doc. Ing., Ph.D. (16.02.2023)
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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. Last update: Čoupek Petr, RNDr., Ph.D. (16.02.2023)
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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. Last update: Hlubinka Daniel, doc. RNDr., Ph.D. (19.02.2020)
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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.
Last update: Čoupek Petr, RNDr., Ph.D. (16.02.2023)
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Lecture+exercises Last update: Čoupek Petr, RNDr., Ph.D. (16.02.2023)
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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. Last update: Čoupek Petr, RNDr., Ph.D. (16.02.2023)
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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.
Last update: Čoupek Petr, RNDr., Ph.D. (16.02.2023)
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Knowledge required before enrollment: conditional probability and conditional expectation discrete martingales Last update: Čoupek Petr, RNDr., Ph.D. (16.02.2023)
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