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Обычная версия сайта
2019/2020

Научно-исследовательский семинар "Введение в теорию случайных процессов"

Статус: Дисциплина общефакультетского пула
Когда читается: 3, 4 модуль
Язык: английский
Кредиты: 6

Course Syllabus

Abstract

The course is a continuation of the standard course in probability theory (associated mainly with combinatorics) and is intended for an initial introduction to the theory of random processes. Special attention is paid to the connection of this theory with functional analysis and the general measure theory. The course is aimed at bachelors 2–4 courses, undergraduates and graduate students.
Learning Objectives

Learning Objectives

  • This course is aimed at providing students with a solid working knowledge in the basic concepts, important techniques and examples in theory of random processes. Learn probabilistic methods for analyzing random processes and proving mathematical theorems. Develop probabilistic intuition.
Expected Learning Outcomes

Expected Learning Outcomes

  • The student is expected be able to use measure-theoretic and analytic techniques for the derivation of equations describing Markov and diffusion processes
  • The students are expected to be able to outline proofs of important theorems of continuous-time martingale processes
  • Student is expected to be able to construct probability measures on infinite dimensional spaces and, in particular, function spaces
  • Student would learn to apply stochastic calculus to derive solutions of stochastic differential equations and to study their properties
Course Contents

Course Contents

  • The concept of a random process.
  • Elements of random analysis.
  • Correlation theory of random processes.
  • Markov processes with discrete and continuous time
  • Wiener and Poisson processes.
  • Stochastic integral. Ito’s formula.
  • (sub/super) martingales.
  • Infinitesimal semigroup operator.
  • Stochastic stability of dynamical systems.
  • Large deviations in Markov processes and chaotic dynamics.
  • Nonlinear Markov processes.
Assessment Elements

Assessment Elements

  • non-blocking Cumulative assessment
  • non-blocking Final exam
    Oral exam
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    0.4 * Cumulative assessment + 0.6 * Final exam
Bibliography

Bibliography

Recommended Core Bibliography

  • Grimmett, G., & Welsh, D. J. A. (2014). Probability : An Introduction (Vol. 2nd ed). Oxford: OUP Oxford. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=852090
  • Krylov, N. V. (2002). Introduction to the Theory of Random Processes. Providence: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971029

Recommended Additional Bibliography

  • Chaumont, L., & Yor, M. (2012). Exercises in Probability : A Guided Tour From Measure Theory to Random Processes, Via Conditioning (Vol. 2nd ed). Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=466664
  • Stirzaker, D. (2003). Elementary Probability (Vol. 2nd ed). Cambridge, UK: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=125155