1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2020/2021
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, mathematical analysis, functional analysis.

Scope

Lecture:
Introduction (5 h.)
1. Stochastic processes in practical problems
2. Elements of stochastic analysis, stochastic processes, definition and properties, Kołmogorov’s theorem
3. Wiener process: existence and properties
Stochastic square-mean analysis (13 h.):
1. Hilbert process and different types of its convergences
2. Square-mean continuity and differentiability of Hilbert processes
3. Square-mean integrals of Riemann and Lebesgue type
4. Square-mean integrability
5. Variation of stochastic processes, existence of Riemann-Stieltjes and Lebesgue-Stieltjes trajectory integrals
Stochastic Itô integral (7 h.):
1. Wiener filtration and adapted processes
2. Simple processes and their Wiener integrals
3. Convergence of simple processes to process from M[a,b] and convergence of their integrals in L2 (Ω)
4. Stochastic Itô integral and its properties
5. Itô formula and its applications
6. Stochastic Itô differential equations

Class
Properties of random variables
Properties of stochastic processes
Convergence of stochastic processes
continuity and differentiability of Hilbert processes
Stochastic differentials of different processes
Applications of Itô formula
Solving of stochastic Itô differential equations

Teaching methods

Conventional lecture; problem lecture
Auditorium exercises – solving standard problems enlightening the significance of the theory, exercises on applications, solving problems.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Final exam and grade.

Recommended reading

1. R. Lipcer, A. Sziriajew, Statystyka procesów stochastycznych, PWN 1981.
2. K. Sobczyk, Stochastyczne równania różniczkowe, WNT 1996.
3. M. Fisz, Rachunek prawdopodobieństwa i statystyka matematyczna, PWN 1958.

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 Alina Szelecka (last modification: 18-09-2020 13:46)