1. SylabUZ
2. Faculty of Exact and Natural Sciences
3. winter term 2023/2024
4. Mathematics - Second-cycle studies leading to MS degree
5. Stochastic Processes 1

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Stochastic Processes 1 - course description

Aim of the course

The students get acquainted with the basic definitions and theorems of stochastic processes and their applications.

Prerequisites

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Mathematical analysis 1 i 2, Linear Algebra, Probability Theory.

 

 

Scope

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).

Teaching methods

Conventional lecture; conversational lecture. Class -solving mathematical problems, the analysis of classic examples of games in economics and other applications.

 

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

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.   

Recommended reading

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.

Further reading

1. Billingsley, P., Convergence of Probability Measures. Second Edition. JOHN WILEY & SONS, INC.

Notes

Modified by dr Ewa Sylwestrzak-Maślanka (last modification: 10-04-2024 16:04)