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Main menu for Browse IS/STAG
Course info
KMA / SP
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Course description
Department/Unit / Abbreviation
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KMA
/
SP
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Academic Year
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2023/2024
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Academic Year
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2023/2024
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Title
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Stochastic Processes
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Form of course completion
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Exam
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Form of course completion
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Exam
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Accredited / Credits
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Yes,
5
Cred.
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Type of completion
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Combined
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Type of completion
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Combined
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Time requirements
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Lecture
2
[Hours/Week]
Tutorial
2
[Hours/Week]
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Course credit prior to examination
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Yes
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Course credit prior to examination
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Yes
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Automatic acceptance of credit before examination
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No
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Included in study average
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YES
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Language of instruction
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Czech
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Occ/max
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Automatic acceptance of credit before examination
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No
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Summer semester
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0 / -
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0 / -
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0 / -
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Included in study average
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YES
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Winter semester
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0 / -
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0 / -
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0 / -
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Repeated registration
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NO
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Repeated registration
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NO
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Timetable
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Yes
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Semester taught
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Summer semester
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Semester taught
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Summer semester
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Minimum (B + C) students
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1
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Optional course |
Yes
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Optional course
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Yes
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Language of instruction
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Czech
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Internship duration
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0
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No. of hours of on-premise lessons |
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Evaluation scale |
1|2|3|4 |
Periodicity |
každý rok
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Evaluation scale for credit before examination |
S|N |
Periodicita upřesnění |
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Fundamental theoretical course |
No
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Fundamental course |
No
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Fundamental theoretical course |
No
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Evaluation scale |
1|2|3|4 |
Evaluation scale for credit before examination |
S|N |
Substituted course
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None
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Preclusive courses
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KMA/SP-A
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Prerequisite courses
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N/A
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Informally recommended courses
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N/A
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Courses depending on this Course
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KMA/SZMZ
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Histogram of students' grades over the years:
Graphic PNG
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XLS
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Course objectives:
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The aim of this course is to introduce basic stochastic processes with continuous time and a general state space (in particular, Markov processes) emplozing the methods of stochastic analysis.
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Requirements on student
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Students have to write an assignment on given topic (cca 4-8 pages long) or elaborate corresponding number (cca 6) of homeworks and present the results (15 minutes talk) in the last week of the term. Assesment of the written assignment and its oral presentation will create 40% of the final mark.
Final examination consists of written test (30%) and oral examination (30%).
Knowledge and abilities assessed: All assessment tasks will assess the learning outcomes, especically, the ability to provide logical and coherent proofs of results and specific problems related to stochastic processes.
Assessment criteria: The main criteria for marking will be clear and logical formulation of solution methods and correctness of obtained results.
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Content
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1. Basic notions of probability theory-recollection, the concept of conditional expectation and stochastic (random) process.
2.-3. Some frequently used processes-Brownian motion and fractional Brownian motion, processes with jumps, Levy process.
4. Martingales, definition, some properties and applicability of the martingale theory.
5.-6. Continuous-time Markov processes with a general state space, definition and basic properties. Transition densities, examples-SDE
7.-8. Diffusion processes and models, relation to partial differential equations, Kolmogorov and Fokker-Planck equation. Feynman-Kac formula-killing.
9. Random stopping times and the strong Markov property, the Feller property-continuous dependence on initial data
10.-11. Stationary (equilibrium) states ? invariant measures, recurrence and transience, sufficient conditions for solutions to SDE
12-13. Convergence to the stationary state, nondegenerating SDE
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Activities
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Fields of study
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Guarantors and lecturers
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Literature
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Contact hours
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52
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Individual project (40)
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40
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Preparation for an examination (30-60)
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40
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Total
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132
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
Students should have a basic knowledge of probability theory (KMA/PSA), fundamentals of random processes (KMA/ZNP) and of introduction to stochastic analysis (KMA/USA). |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
Students taking this course will be able to understand the mathematical background of stochastic processes and namely
- recognize which stochastic processes are appropriate and needed for modelling randomness in a given research problem
- apply stochastic processes to practical problems
- analyze the usefulness of stochastic processes 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.
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Oral exam |
Written exam |
Seminar work |
Individual presentation at a seminar |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Lecture supplemented with a discussion |
Interactive lecture |
Students' portfolio |
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