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Main menu for Browse IS/STAG
Course info
KMA / USA-A
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Course description
Department/Unit / Abbreviation
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KMA
/
USA-A
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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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Introduction to Stochastic Analysis
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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,
6
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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English
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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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Winter semester
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Semester taught
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Winter 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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English
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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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N/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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N/A
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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 the basic ideas and tools of the stochastic analysis.
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Requirements on student
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Students have to write an assignment on given topic (4-8 pages long) and present the results (15 minutes talk) 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, especially, the ability to provide logical and coherent proofs of results and specific problems related to stochastic analysis.
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 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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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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52
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Preparation for an examination (30-60)
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52
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Total
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156
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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) and of ordinary differential equations (KMA/ODR). |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
Students taking this course will be able to grasp the basic problems of stochastic analysis and namely
- recognize which stochastic analysis tools are appropriate and suitable 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.
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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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