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Магистратура 2019/2020

Стохастические модели

Направление: 01.04.02. Прикладная математика и информатика
Когда читается: 2-й курс, 3 модуль
Формат изучения: Full time
Прогр. обучения: Прикладная статистика с методами сетевого анализа
Язык: английский
Кредиты: 4

Course Syllabus

Abstract

Mathematical models based on probability theory prove to be extremely useful in describing and analyzing complex systems that exhibit random components. The goal of this course is to introduce several classes of stochastic processes, analyze their behavior over a finite or infinite time horizon, and help students enhance their problem solving skills. The course combines classic topics such as martingales, Markov chains, renewal processes, and queuing systems with approaches based on Stein’s method and on concentration inequalities. The course focuses mostly on discrete-time models and explores a number of applications in operations research, finance, and engineering. This is an elective course, offered to MASNA students, and examples used in class may differ depending on students’ interests.
Learning Objectives

Learning Objectives

  • The course gives students an important foundation to develop and conduct their own research as well as to evaluate research of others.
Expected Learning Outcomes

Expected Learning Outcomes

  • Know the theoretical foundation of stochastic processes.
  • Be able to explore the advantages and disadvantages of stochasticity in the models and demonstrate how it contributes to the analysis.
  • Know modern extensions to stochastic modeling.
  • Be able to work with major linear modeling programs, especially R, so that they can use them and interpret their output.
  • Know the basic principles behind working with all types of data for using stochastic components in models.
  • Be able to develop and/or foster critical reviewing skills of published empirical research using applied statistical methods.
  • Be able to criticize constructively and determine existing issues with applied linear models in published work .
  • Have an understanding of the basic principles of stochastic models and lay the foundation for future learning in the area.
  • Have the skill to meaningfully develop an appropriate model for the research question
  • Have the skill to work with statistical software, required to analyze the data.
Course Contents

Course Contents

  • Understanding randomness
    The first session will focus on understanding randomness and the statistical relationships between random events. It will also review expectation and integration, almost sure convergence and the dominated convergence theorem, convergence in probability and in distribution, the e law of large number and the ergodic theorem.
  • Stein’s method and central limit theorems
    The session introduces the notions of coupling, Poisson approximation and Le Cam’s theorem, the Stein-Chen method, and Stein’s method for the geometric and the normal distribution.
  • Conditional expectation and martingales
    The focus of this session will be on conditional expectation, martingales, the martingale stopping theorem, the Hoeffding-Azuma inequality, the martingale convergence theorem, and the uniform integrability.
  • Probability inequalities
    This sessions builds the understanding of Jensen’s inequality, probability bounds via the im-portance sampling identity, Chernoff bounds, second moment and conditional expectation ine-qualities.
  • Discrete-time Markov chains
    This session covers the Chapman-Kolmogorov equations and classification of states, the strong Markov property, stationary and limiting distributions, transition among classes, the gambler’s ruin problem, and mean times in transient states; branching processes and time reversibility.
  • Renewal theory
    This session will introduce the limit theorems, renewal reward processes, and Blackwell’s therem.
  • Queueing theory (multiple class meetings)
    This session will focus on the Poisson process, and a range of queueing systems: M/M/1, M/G/1, G/M/1, and G/G/1.
Assessment Elements

Assessment Elements

  • non-blocking Final In-Class or Take-home exam (at the discretion of the instructor)
  • non-blocking Homework Assignments (5 x Varied points)
  • non-blocking In-Class Labs (9-10 x Varied points)
  • non-blocking Quizzes (Best 9 of 10, Varied points)
Interim Assessment

Interim Assessment

  • Interim assessment (3 module)
    0.5 * Final In-Class or Take-home exam (at the discretion of the instructor) + 0.2 * Homework Assignments (5 x Varied points) + 0.2 * In-Class Labs (9-10 x Varied points) + 0.1 * Quizzes (Best 9 of 10, Varied points)
Bibliography

Bibliography

Recommended Core Bibliography

  • Medhi, J. (2003). Stochastic Models in Queueing Theory (Vol. 2nd ed). Amsterdam: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=205403
  • Meerschaert, M. M., & Sikorskii, A. (2011). Stochastic Models for Fractional Calculus. Berlin: De Gruyter. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=430094
  • Ruggeri, F., Ríos Insua, D., & Wiper, M. M. (2012). Bayesian Analysis of Stochastic Process Models. Hoboken, New Jersey: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=443018
  • Vickson, R. G., & Ziemba, W. T. (2006). Stochastic Optimization Models In Finance (2006 Edition) (Vol. 2006 ed). Hackensack, NJ: World Scientific. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=210801

Recommended Additional Bibliography

  • Li, Q.-L. (2010). Constructive Computation in Stochastic Models with Applications : The RG-Factorizations. Beijing: Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=374057
  • Rachev, S. T., Fabozzi, F. J., & Stoyanov, S. V. (2008). Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization : The Ideal Risk, Uncertainty, and Performance Measures. Hoboken, N.J.: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=219812
  • Tan, W. Y. (2002). Stochastic Models With Applications To Genetics, Cancers, Aids And Other Biomedical Systems. River Edge, N.J.: World Scientific. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=210588