Master
2021/2022
Modern Methods of Data Analysis: Stochastic Calculus
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Type:
Compulsory course
Area of studies:
Applied Mathematics and Informatics
Delivered by:
Department of Complex System Modelling Technologies
Where:
Faculty of Computer Science
When:
1 year, 1, 2 module
Mode of studies:
offline
Open to:
students of one campus
Master’s programme:
Math of Machine Learning
Language:
English
ECTS credits:
9
Contact hours:
60
Course Syllabus
Abstract
The aim of this course is to provide an introduction to the modern methods of stochastic calculus. The course consists from two parts. The main emphasis of the first part will be on Markov chains. We discuss properties of Markov Chains, study their invariant distributions and convergence to stationary distributions. At the end of the course we discuss Markov Chain Monte-Carlo method (MCMC). The main emphasis of the second part will be in stochastic differential equations, their analytic and numerical solutions. We also briefly recall all necessary facts from the basic of random processes, Wiener process and Martingales.
Learning Objectives
- Students will study how to apply the main modern probabilistic methods in practice and learn important topics from the stochastic calculus.
Expected Learning Outcomes
- Acquaintance with the main aspects of the measure concentration phenomenon
- Be able to apply Markov Chain Monte-Carlo methods in practice
- Be able to apply MCMC methods like ULA or MALA in practice
- Be able to calculate conditional expectations, probabilities and apply their properties (e.g. tower property or total probability property)
- Be able to solve SDE numerically. Know main properties of SDE and their solutions
- Know definition of Markov chains, be able to solve theoretical and practical problems
- Know definition of martingales and its properties
- Know definition of stochastic integral and its properties
- Know definition of Wiener process, know properties of its trajectories.
Course Contents
- Markov chains, discrete state space and discrete time
- Markov chains, continuous time and discrete state spaces
- Conditional probability and conditional distributions
- Markov chains, General state spaces
- Conections with concentration of measure
- MCMC
- Martingales
- Wiener process
- Ito’s integral
- Stochastic differential equations
- Unadjusted Langevin algorithm (ULA), Metropolis adjusted Langevin algorithm (MALA)
Interim Assessment
- 2021/2022 2nd module0.4 * экзамен + 0.3 * письменный экзамен + 0.3 * домашняя работа
Bibliography
Recommended Core Bibliography
- Christophe Andrieu, & Nando De Freitas. (2003). An Introduction to MCMC for Machine Learning. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.C161414B
- Вероятность. Кн. 1: Вероятность - 1: Элементарная теория вероятностей. Математические основания. Предельные теоремы, Ширяев, А. Н., 2004
- Теория случайных процессов, Булинский, А. В., 2003
Recommended Additional Bibliography
- Durmus, A., & Moulines, E. (2016). High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.A78D09BB