SylabUZ

Generate PDF for this page

Stochastic Processes 2 - course description

General information
Course name Stochastic Processes 2
Course ID 11.1-WK-MATED-SP2-S22
Faculty Faculty of Exact and Natural Sciences
Field of study Mathematics
Education profile academic
Level of studies Second-cycle studies leading to MS degree
Beginning semester winter term 2022/2023
Course information
Semester 3
ECTS credits to win 7
Available in specialities Mathematics and computer science in finance and insurance
Course type optional
Teaching language english
Author of syllabus
  • prof. dr hab. Jerzy Motyl
Classes forms
The class form Hours per semester (full-time) Hours per week (full-time) Hours per semester (part-time) Hours per week (part-time) Form of assignment
Lecture 30 2 - - Exam
Class 30 2 - - Credit with grade

Aim of the course

After the course of “stochastic processes 2” students should be able to solve themselves practical and theoretical problems on the topic.

Prerequisites

Probability theory,

Scope

Lecture:

Introduction (5 h.)

1. Stochastic processes in practical problems, population growth, Brownian motion, theory of signals (2 h)
2. Elements of theory of probability, stochastic analysis (1 h),

3. Stochastic processes, definition and properties, Kołmogorov’s theorem (1 h),
4. Wiener process: existence and properties (1 h)

Stochastic square-mean analysis (15 h.):

1. Hilbert process, its interpretation in functional analysis and different types of its convergences (2h)
2. Square-mean continuity and differentiability of Hilbert processes (4 h)
3. Square-mean integrals of Riemann and Lebesgue type (2 h)
4. Square-mean integrability (3 h)
5. Variation of stochastic processes (2 h),

6. Existence of Riemann-Stieltjes and Lebesgue-Stieltjes trajectory integrals (2 h)

Stochastic Itô integral (10 h.):

1. Wiener filtration and adapted processes (1 h)
2. Simple processes and their Wiener integrals (1 h)
3. Convergence of simple processes to process from M[a,b] and convergence of their integrals in L2 (Ω) (2 h)
4. Stochastic Itô integral processesof process from M[a,b] and its properties (2 h)
5. Itô formula and its applications (2 h)
6. Stochastic Itô differential equations (2 h)

Classes
1. Properties of random variables (4 h)
2. Properties of stochastic processes (4 h)
3. Convergence of stochastic processes (4 h)

4. Continuity and differentiability of Hilbert processes (4 h)
5. Stochastic differentials of different processes (4 h)
6. Applications of Itô formula (4 h)
7. Solving of stochastic Itô differential equations (4 h)

8. Test of competition (2 h)

Teaching methods

Conventional lecture; classes – solving standard problems,  exercises 

Learning outcomes and methods of theirs verification

Outcome description Outcome symbols Methods of verification The class form

Assignment conditions

Test of competition with problems of the different level of difficulty controlling if student got the minimal efect of efficiency.

The course completion grade consists of the classes grades (40%) and examination grades (60%). A positive grade of the classes is required to sit for the exam.  A positive grade of the classes and exam is required to credit for the course.

Recommended reading

1. R Liptser, A. Shiryaev, Statistics of Random Processes I General Theory. Springer 1977.

2. K. Sobczyk, Stochastic differential equations, Springer 2001.
 

Further reading

1. E. Parzen, Stochastic processes, Holden-Day Inc. 1962.
2. C.W. Gardiner, Handbook of stochastic methods for Physics, Chemistry and the Natural Sciences, Springer-Verlag 1985.

Notes


Modified by dr Ewa Sylwestrzak-Maślanka (last modification: 19-01-2024 20:20)