Introduction to Stochastic Processes Field of study: Mathematics
Programme code: W4-S1MT19.2023

Module name: Introduction to Stochastic Processes
Module code: W4-MT-S1-23-WPSt
Programme code: W4-S1MT19.2023
Semester: summer semester 2025/2026
Language of instruction: Polish
Form of verification: exam
ECTS credits: 5
Purpose and description of the content of education:
Moduł Wstęp do procesów stochastycznych ma na celu wykształcenie umiejętności swobodnego posługiwania się podstawowymi pojęciami z i narzędziami teorii procesów stochastycznych. Przewiduje się realizację następujących treści programowych: 1. Warunkowa wartość oczekiwana. 2. Podstawowe pojęcia z zakresu teorii procesów stochastycznych. 3. Twierdzenie Kołmogorowa o istnieniu procesu. 4. Momenty stopu. 5. Martyngały z czasem ciągłym. 6. Proces Wienera - definicja, konstrukcja i podstawowe własności. 7. Proces Poissona (informacyjnie). 8. Całka stochastyczna i formuła Ito.
List of modules that must be completed before starting this module (if necessary): Probability Theory A [W4-MT-S1-23-RPraA]
Learning outcome of the module Codes of the learning outcomes of the programme to which the learning outcome of the module is related [level of competence: scale 1-5]
potrafi podać różne przykłady dyskretnych i ciągłych rozkładów prawdopodobieństwa i omówić wybrane eksperymenty losowe oraz modele matematyczne, w jakich te rozkłady występują; zna zastosowania praktyczne podstawowych rozkładów [WPSt_1]
 K_U31 [4/5]
potrafi wyznaczyć parametry rozkładu zmiennej losowej o rozkładzie dyskretnym i ciągłym; potrafi wykorzystać twierdzenia graniczne i prawa wielkich liczb do szacowania prawdopodobieństw [WPSt_2]
 K_U33 [4/5]
rozumie budowę teorii matematycznych, potrafi użyć formalizmu matematycznego do budowy i analizy prostych modeli matematycznych w innych dziedzinach nauk [WPSt_3]
 K_W03 [3/5]
potrafi praktycznie wykorzystać wiedzę matematyczną [WPSt_4]
 K_U38 [4/5]
zna podstawowe przykłady zarówno ilustrujące konkretne pojęcia matematyczne, jak i pozwalające obalić błędne hipotezy lub nieuprawnione rozumowania [WPSt_5]
 K_W05 [3/5]
potrafi w sposób zrozumiały, w mowie i na piśmie, przedstawiać poprawne rozumowania matematyczne, formułować twierdzenia i definicje [WPSt_6]
 K_U01 [4/5]
Form of teaching Number of hours Methods of conducting classes Assessment of the learning outcomes Learning outcomes
lecture [WPSt_fs_1] 30 Formal lecture/ course-related lecture [a01] 
Explanation/clarification [a05] 
Lecture-discussion [b02] 
Activating method – discussion / debate [b04]  		exam WPSt_1 WPSt_2 WPSt_3 WPSt_5
discussion classes [WPSt_fs_2] 15 Explanation/clarification [a05] 
Activating method – discussion / debate [b04] 
Working with a programmed textbook [d02] 
Laboratory exercise / experiment [e01] 
Individual work with a text [f02]  		course work WPSt_1 WPSt_2 WPSt_3 WPSt_4 WPSt_6
The student's work, apart from participation in classes, includes in particular:
Name Category Description
Search for materials and review activities necessary for class participation [a01] Preparation for classes
reviewing literature, documentation, tools and materials as well as the specifics of the syllabus and the range of activities indicated in it as required for full participation in classes
Literature reading / analysis of source materials [a02] Preparation for classes
reading the literature indicated in the syllabus; reviewing, organizing, analyzing and selecting source materials to be used in class
Developing practical skills [a03] Preparation for classes
activities involving the repetition, refinement and consolidation of practical skills, including those developed during previous classes or new skills necessary for the implementation of subsequent elements of the curriculum (as preparation for class participation)
Consulting materials complementary to those indicated in the syllabus [a04] Preparation for classes
agreeing on materials complementary to those indicated in the syllabus, supporting the implementation of tasks resulting from or necessary for class participation
Getting acquainted with the syllabus content [b01] Consulting the curriculum and the organization of classes
reading through the syllabus and getting acquainted with its content
Verification / adjustment / discussion of syllabus provisions [b02] Consulting the curriculum and the organization of classes
consulting the content of the syllabus, possibly in the presence of the year tutor or members of the class group, and, if necessary, reassessing the provisions concerning special conditions for class participation, e.g., space and time requirements, technical and other requirements, including conditions for participation in classes outside the walls of the university, classes organized in blocks, organized online, etc.
Determining the stages of task implementation contributing to the verification of learning outcomes [c01] Preparation for verification of learning outcomes
devising a task implementation strategy embracing the division of content, the range of activities, implementation time and/or the method(s) of obtaining the necessary materials and tools, etc.
Studying the literature used in and the materials produced in class [c02] Preparation for verification of learning outcomes
exploring the studied content, inquiring, considering, assimilating, interpreting it, or organizing knowledge obtained from the literature, documentation, instructions, scenarios, etc., used in class as well as from the notes or other materials/artifacts made in class
Analysis of the corrective feedback provided by the academic teacher on the results of the verification of learning outcomes [d01] Consulting the results of the verification of learning outcomes
reading through the academic teacher’s comments, assessments and opinions on the implementation of the task aimed at checking the level of the achieved learning outcomes
Development of a corrective action plan as well as supplementary/corrective tasks [d02] Consulting the results of the verification of learning outcomes
reviewing and selecting tasks and activities enabling the elimination of errors indicated by the academic teacher, their verification or correction resulting in completing the task with at least the minimum passing grade
Attachments
Module description (PDF)
Information concerning module syllabuses might be changed during studies.
Syllabuses (USOSweb)
Semester Module Language of instruction
(no information given)