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Fall 2025: Math 475 Probability

Recently developed courses

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.

Course Plan:

1. Prelimenaries

- - 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.

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).

Course Plan:

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
  - Performance 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.

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
- Functional Analysis
- Measure Theory
- Set Theory

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