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Lecturer(s)
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Ježek Vladimír, doc. Ing. Ph.D.
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Course content
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1. Basic notions of probability theory - recollection, the concept of stochastic process, some useful tests and results 2.-3. Einstein-Smoluchowski model of Brownian motion as a motivation for the definition of the Wiener process, Wiener process and its basic features 4.-5. Motivating remarks on the white noise and stochastic integral. Stochastic integral and its basic properties. 6. Stochastic differential, the Ito formula and some useful results 7. Stochastic differential equation, introduction and basic results on existence and uniqueness of solutions. 8. Linear and bilinear equations as the simplest continuous stochastic models and their applications. 9.-10. Large time behaviour, exponential stability, Lyapunov stability and instability, stabilization and destabilization by noise, examples 11. Girsanov formula and the notion of weak solution, applications to problems with friction 12.-13. Examples of applications of stochastic equations in physics, mathematical biology and finance mathematics- some basic models and their analysis.
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Learning activities and teaching methods
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Interactive lecture, Lecture supplemented with a discussion, Lecture with practical applications, Students' portfolio
- Preparation for an examination (30-60)
- 40 hours per semester
- Individual project (40)
- 40 hours per semester
- Contact hours
- 52 hours per semester
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| prerequisite |
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| Knowledge |
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| Students should have a basic knowledge of probability theory (KMA/PSA) and of ordinary differential equations (KMA/ODR). |
| learning outcomes |
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| Students taking this course will be able to appreciate the basic problems of stochastic analysis and namely - recognize which stochastic process is appropriate for modelling randomness in a given research problem, - apply stochastic analysis tools to practical problems, - analyze the usefulness of stochastic differential equations in professional area, - provide logical and coherent proofs of theoretic results - solve problems via abstract methods, - apply correctly formal and rigorous competency in mathematical presentation, both in written and verbal form. |
| teaching methods |
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| Lecture supplemented with a discussion |
| Interactive lecture |
| Students' portfolio |
| assessment methods |
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| Oral exam |
| Written exam |
| Seminar work |
| Individual presentation at a seminar |
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Recommended literature
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Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. 2nd ed. New York : Springer, 1991. ISBN 0-387-97655-8.
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Mandl, Petr. Pravděpodobnostní dynamické modely : celost. vysokošk. učebnice pro stud. matematicko-fyz. fakult stud. oboru pravděpodobnost a matem. statistika. Praha : Academia, 1985.
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Maslowski, Bohdan. Stochastic Equations and Stochastic Methods in PDE's. Plzeň, 2006.
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Oksendal, Bernt. Stochastic differential equations : an introduction with applications. 6th ed. Berlin : Springer, 2003. ISBN 3-540-04758-1.
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