SylabUZ
Nazwa przedmiotu | Stochastic Processes 1 |
Kod przedmiotu | 11.1-WK-MATED-SP1-S22 |
Wydział | Wydział Matematyki, Informatyki i Ekonometrii |
Kierunek | Mathematics |
Profil | ogólnoakademicki |
Rodzaj studiów | drugiego stopnia z tyt. magistra |
Semestr rozpoczęcia | semestr zimowy 2022/2023 |
Semestr | 2 |
Liczba punktów ECTS do zdobycia | 7 |
Występuje w specjalnościach | Mathematical modelling, Mathematics and computer science in finance and insurance |
Typ przedmiotu | obieralny |
Język nauczania | angielski |
Sylabus opracował |
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Forma zajęć | Liczba godzin w semestrze (stacjonarne) | Liczba godzin w tygodniu (stacjonarne) | Liczba godzin w semestrze (niestacjonarne) | Liczba godzin w tygodniu (niestacjonarne) | Forma zaliczenia |
Wykład | 30 | 2 | - | - | Egzamin |
Ćwiczenia | 30 | 2 | - | - | Zaliczenie na ocenę |
The students get acquainted with the basic definitions and theorems of stochastic processes and their applications.
Mathematical analysis 1 i 2, Linear Algebra, Probability Theory.
Lecture
I. Uniform Markov Chains:
1. Transition matrix. Chapman-Kołmogorov equation. (2 hours.)
2. Classification of states. (2 hours)
3. Random walk. Gambler's ruin. (2 hours)
4. Stationarity and ergodicity of Markov chain. (2 hours)
II. Poisson Process:
1. Construction of Poisson Process. (2 hours)
2. Compound and conditional Poisson process. (2 hours)
3. Applications of such processes. (4 hours)
III. Continuous-time Markov chains:
1. The birth–death process. (2 hours.)
2. The extinction process. (2 hours)
3. Examples of applications of Poisson processes. (2 hours)
IV. General properties of stochastic processes:
1. Existence of process with given distributions. (2 hours)
2. Stochastic equivalence and separability of processes. (2 hours.)
V. Wiener Process:
1. Properties of trajectory. (2 hours)
2. Law of the iterated logarithm. (2 hours.)
Class
I. Uniform Markov Chains:
1. Examples of transition probabilities. (2 hours)
2. Classification of states. (2 hours)
3. Random walkes. Problems. (3 hours)
4. Stationarity and ergodicity of Markov chains. Examples. (3 hours)
II. Poisson Process:
1. Properties of Poisson process. Problems. (2 hours)
2. Compound and conditional Poisson process. Problems. (3 hours)
3. Applications of such processes. (3 hours)
III. Continuous-time Markov Process:
1. The birth-death process. (2 hours)
2. Examples of applications. (3 hours)
IV. General properties of stochastic processes:
1. Existence of process with given distributions. (1 hour)
2. Stochastic equivalence and separability of processes. (1 hour)
V. Wiener Process:
1. Properties of trajectory. Correlation function. (1 hour.)
VI. Test and summary: (4 hours).
Conventional lecture; conversational lecture. Class -solving mathematical problems, the analysis of classic examples of games in economics and other applications.
Opis efektu | Symbole efektów | Metody weryfikacji | Forma zajęć |
A positive evaluation of the class is a prerequisite for passing the exam. The evaluation of the course consists of the assessment of the class (40%) and the evaluation of the exam (60%). The prerequisite to passing the course is a positive evaluation of the exam.
1. Iwanik, A. & Misiewicz, J. K. (2015). Lectures on stochastic processes with objectives. The first part: Markov processes, Warsaw: SCRIPT
2. Feller, W., (1971). An introduction to Probability Theory and its Applications, Vol.1, 2. John Willy&Sons, New York, London 1966.
Zmodyfikowane przez dr hab. inż. Łukasz Balbus, prof. UZ (ostatnia modyfikacja: 28-02-2024 13:00)