CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2024/2025

# Stochastic Methods

The course is not on the list Without time-table
Code Completion Credits Range Language
01STOM KZ 2 2+0 Czech
Garant předmětu:
Lecturer:
Tutor:
Supervisor:
Department of Mathematics
Synopsis:

Keywords:

Markov processes, transition probabilities, stationary distribution, hitting probabilities, transition rates, Poisson process, queuing theory.

Requirements:
Syllabus of lectures:

1.Introduction to stochastic systems, homogeneity, stationarity, simulation of Bernoulli process and simple random walk.

2.Analysis of random walk and simulation of gambler?s ruin problem.

3. Discrete-time Markov chains I, transition probability, Chapman-Kolmogorov theorem, classification of states, recurrent and transient states.

4.Discrete-time Markov chains II, Ergodic theorem, stationary distribution.

5.Discrete-time Markov chains III, hitting probabilities, Reversibility, Branching processes, Simulation of Ehrenfest and Bernoulli processes of diffusion.

6.Continuous-time Markov Chain I, transition functions

7.Continuous-time Markov Chain II, Kolmogorov backward and forward equations, Limiting probabilities and stationary distribution, Balance equations.

8.Birth and Death process:

9.Poisson process.

10.Renewal process.

11.Queueing theory

12.Markov Chain Monte Carlo.

Syllabus of tutorials:
Study Objective:

Acquired knowůledge:

Understanding the limit behavior of stochastic systems in the connection with the state classification and will be able to construct the transition probabilities matrix (transition rates) based on given information.

Acquired skills:

Application of given methods in particular examples in physics and engineering.

Study materials:

Key references:

[1] Grimmett, G., Stirzaker, D.: Probability and Random Processes, Oxford Uni. press, 2001.

[2] Lefebvre, M.: Applied Stochastic Processes, Springer, 2000.

Recommended references:

[1] Prášková, Z., Lachout, P.: Základy náhodných procesů, Karolinum 1998.

[2] Norris, J. R.: Markov Chains, Cambridge Uviversity Press 1997.

[3] Häggström, O.: Finite Markov chains and algorithmic applications, Cambridge Uviversity Press 2002.

[4] Ching, Wai-Ki: Markov chains: models, algorithms and applications, Springer 2006.

Working environment:

R, Matlab

Note:
Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2024-05-18
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