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Syllabus

Field of study* : Financial Engineering
Module name
Probability theory and stochastic processes
Module name in english
Probability theory and stochastic processes
Module code Method of evaluation
WIGEFES.21A.12345.18 Exam
Field of study Track Year / semester
Financial Engineering General academic 1 / 1
Specialisation Language of instruction Module
All English Obligatory
Number of hours Number of ECTS points Block
Lectures: 30 Classes: 30 6 A
Level of qualification Mode of studies Education field
Second-cycle programme Full-time Social Sciences
Author Karolina Sobczak
Teachers Karolina Sobczak

Subject’s educational aims

C1 Presentation of fundamental concepts in probability theory and stochastic processes theory
C2 Presentation of basic applications of these theories in economic sciences

Subject’s learning outcomes

Code Outcomes in terms of Learning outcomes within the field
Knowledge
W1 Knows basic probability distributions and fundamental stochastic processes K2_W01, K2_W04
W2 Knows and understands main concepts in probability theory and stochastic processes theory K2_W01, K2_W04
W3 Defines properties of given probability distributions K2_W01, K2_W04
Skills
U1 Calculates basic characteristics of probability distributions K2_U02
U2 Exploits formulas to calculate values of functions of random variables K2_U02
U3 Exploits definitions and theorems to prove given properties of stochastic processes K2_U02
U4 N/A : Stosuje oprogramowanie MATLAB do obliczania wartości: funkcji masy prawdopodobieństwa, funkcji gęstości prawdopdobieństwa, funkcji zmiennych losowych K2_U02
U5 N/A : Posługuje się oprogramowaniem MATLAB w celu wizualizacji wyników obliczeń oraz w celu interpretacji wyników K2_U01, K2_U02
Social competences
K1 Is open-minded to knowledge broading and responsible for his/her educational progress K2_K01, K2_K04
K2 Takes care of good learning environment and good conditions for educational progress of other students K2_K01, K2_K03

Study content

No. Study content Subject’s educational aims Subject’s learning outcomes
1. Events as sets, probability measure and its properties, probability space C1 W2, K1, K2
2. Conditional probability, total probability, Bayes' theorem, independece of events (pairwise, general), conditional independence C1 W2, K1, K2
3. Random variables, distribution function, discrete probability distribution and its examples, indicator function, tails of distribution, law of averages C1 W1, W2, U5, K1, K2
4. Discrete and continuous random variables, probability mass function, density function C1 W1, W2, W3, U4, K1, K2
5. Random vectors, joint distribution function and its properties, marginal distribution functions, joint mass function, joint density function, Monte Carlo simulations C1 W2, U5, K1, K2
6. Examples of discrete distributions (binomial, Bernoulli, hypergeometric, Poisson), independence of random variables C1 W1, W2, W3, K1, K2
7. Expected value of discrete random variable, expected value of discrete random variable's function, properties of expected value C1 W2, U1, U2, K1, K2
8. Moments and central moments of random variable, variance of random variable, standard deviation, properties of variance, unocorrelated random variables C1 W2, U1, U2, K1, K2
9. Examples of expected value and variance for discrete variables (in ditributions: Bernoulli, binomial, trinomial, multinomial, Poisson, hypergeometric, geometric, negative binomial) C2 W1, W3, U1, U2, U4, K1, K2
10. Expected value of continuous random variable, expected value of continuous random variable's function, examples of expected value and variance for continuous variables (in distributions: uniform, exponential, normal, standard normal, gamma, chi-squared, Cauchy, beta, Weilbull) C2 W1, W3, U2, U4, U5, K1, K2
11. Dependence of random variables, covariance and correlation coefficient, conditional probability: distribution function, mass function, density function, conditional expected value C1 W2, K1, K2
12. Stochastic processes, discrete and continuous time, realization of process, examples of stochastic processes (simple random walk, random walk, discrete white noise, Poisson, Wiener, Galton-Watson) C1 W1, W2, U3, U5, K1, K2
13. Simple random walk ans its applications in economic sciences, properties of simple random walk (spatial homogentity, temporal homogeneity, Markov property), reflection principle C2 W1, W2, U3, U5, K1, K2
14. Markov processes, examples of Markov processes, transition matrix, Chapman-Kolmogorov formula C1 W1, W2, U3, K1, K2
15. Classification of states, classification of chains, stationary distributions and the limit theorem, continuous-time Markov chains C1 W1, W2, U3, K1, K2

Bibliography

Obligatory
  1. Grimmett G. R, Stirzaker D. R., 2004, Probability and Random Processes 3rd Edition, Oxford University Press
  2. Grimmett G. R, Stirzaker D. R., 2009, One Thousand Exercises in Probability 1st Edition, Oxford University Press
Recommended
  1. Parzen E., 2015, Stochastic Processes, Dover Books Publications
  2. Gikhman I.I., Skorokhod A.V., 2004, The Theory of Stochastic Processes I, Springer
  3. Stoyanov J.M., 2013, Counterexamples in Probability, Third Edition, Dover Publications
  4. Billingsley P., 2012, Probability and Measure, Anniversary Edition, Jon Wiley & Sons, New York
Entry requirements Znajomość podstawowych pojęć matematycznych, Dobra znajomość języka angielskiego
Teaching methods Lecture with multimedia presentation, Case study, Exercises, Laboratories
Method of evaluation Written exam, Written exam with open questions, Final quiz, Final test, Class participation, Individual project, Group project / Group work, Research

Settlement of ECTS points

Forms of student work Average number of hours for student work*
Preparation for test 20
Participation in classes 30
Consultations with teacher 7
Participation in the exam 2
Preparation for classes 20
Participation in lectures 30
Preparation for exam 25
Data collection 10
Literature research 7
Student work in total
Number of hours
151
ECTS points
6
Contact hours (with the teacher)
Number of hours
69
ECTS points
2.5
Practical-class work
Number of hours
30
ECTS points
1

* one hour of classes = 45 minutes

Methods of evaluating the learning outcomes

Learning-outcome code Methods of evaluation
Written exam Written exam with open questions Final quiz Final test Class participation Individual project Group project / Group work Research
W1 x x x x x x x x
W2 x x x x x x
W3 x x x x x x x
U1 x x x x x x
U2 x x x x x x x x
U3 x x x x x
U4 x x x x
U5 x x x x
K1 x x x x x x x x
K2 x x x x x x x