1. SylabUZ
2. Faculty of Exact and Natural Sciences
3. winter term 2022/2023
4. Mathematics - Second-cycle studies leading to MS degree
5. Stochastic Processes 2

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Stochastic Processes 2 - course description

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

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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)