Fall 2022: Math 500, Applied Analysis.
Recently developed courses
Quantitative Risk Management (Math 588, Illinois Tech)
The course covers the major concepts and ideas from the modern risk management . It builds upon general theory of risk measures and performance measures and addresses the current regulatory requirements for market participants.
Significant part of the course will be devoted to practical aspects of the risk management, including: working with real market data, implementing the theoretical concepts by developing Python libraries that perform a comprehensive risk analysis of portfolios of various financial instruments.
The course should be accessible to students with knowledgeable of general probability (Math 475) and elements of financial markets (Math 548 or Math 485).
1. Basic concepts in risk management
History of risk management and contemporary regulatory framework
Modeling portfolio value and its change (basic concepts)
Value-at-risk, expected shortfall and other risk measures
2. Modeling portfolio value and its change
Empirical properties of financial data
Financial time series
Extreme value theory (optional)
Multivariate models and copulas
3. Theory of risk and performance measures
Coherent and convex risk measures
Aggregation and capital allocation
4. Applications to Market risk and Credit Risk
5. Central Clearing Counterparties (CCP)
Textbook: Alexander J. McNeil, RÃijdiger Frey & Paul Embrechts, Quantitative Risk Management Concepts, Techniques and Tools, Princeton University Press, First Revised Edition, 2015, ISBN-13: 9781400873210 .
Course Materials: Lecture Notes (for registered students only). All materials will be posted on the BlackBoard.
Stochastic Partial Differential Equations (Math 545, Illinois Tech)
This course aims to give a fair introduction to general theory of Stochastic Partial Differential Equations (SPDEs), and their applications to various areas of applied mathematics. Starting with simple examples of SPDEs, such as stochastic heat equation, the courses advances to general theory of existence, uniqueness and regularity of the solutions for a SPDEs (mainly parabolic). In the second part of the course, students will learn fundamentals of statistical analysis for stochastic processes with main focus on parameters estimation problems for parabolic SPDEs.
The course is designed for graduate students with general research interests in stochastic methods. Students are expected to have some basic knowledge of PDEs and stochastic calculus.
Martingales, Stochastic Integral, Kolmogorov criterion, Gronwall and BDG inequalities
Some deterministic PDEs
Brownian Motion and Ito’s calculus in infinite dimensions
2. Examples of SPDEs based on applications
3. Stochastic Parabolic Equations
Heat equation and reaction-diffusion equations
Energy estimates in Sobolev spaces
Existence and uniqueness of the solutions
Regularity of the solutions
4. Numerical Solutions for SPDE
5. Statistical Inference for SPDEs
Statistical inference for stochastic ordinary differential equations
Statistical inference for diagonalizable SPDE
Statistical inference for non-diagonalizable SPDE
Textbook: S. V. Lototsky and B. L. Rozovsky, Stochastic Partial Differential Equations, Springer (2017), ISBN 978-3-319-58647-2.
Courses taught at Illinois Tech
MATH 485 (Intro to Math Finance) Fall 2008, 2009, 2011, 2012, 2013, 2014, 2015
MATH 475 (Probability), Fall 2009, 2013
MATH 474 (Probability and Statistics), Spring 2012, 2014
MATH 252 (Introduction to Differential Equations), Fall 2007
MATH 500 (Applied Analysis I), Fall 2008, 2010, 2012, 2022
MATH 501 (Applied Analysis II), Spring 2009, 2011, 2013
MATH 543 (Stochastic Analysis), Fall 2007, 2011
MATH 545 (Stochastic Partial Differential Equations), Spring 2012, 2016, 2019
MATH 548 (Mathematical Finance I), Fall 2008, 2009, 2011-2018, 2020, 2021, 2022
MATH 582 (Mathematical Finance II), Spring 2008, 2010, 2011, 2014, 2017, 2018
MATH 588 (Advanced Quantitative Risk Management), Spring 2019 -2021
MATH 491 (Reading and Research),
MATH 593 (Seminar in Applied Math), Fall 2009, Spring 2010
MATH 591 (Research and Thesis for M.S.)
MATH 691 (Research and Thesis for Ph.D.)
Courses taught at University of Moldova
Mathematical Analysis I
Mathematical Analysis II