Math 3603: Advanced analysis #

Lebesgue integration and measure theory, the basic constructions in this class, are foundational in the study of analysis and play central roles in functional and Fourier analysis and mathematical physics. In this course, we will focus on their applications to probability theory, whose precise mathematical foundations co-evolved in the beginning of the twentieth century.

Learning goals: Upon completion of this course, a student should be able to:

• define and be able to work with the concepts of measure and the Lebesgue integral;
• use measure theory to address and answer questions in probability;
• understand models for random motion including random walks and Brownian motion;
• define and work with the Itō stochastic integral;
• appreciate and critique the applications of the above topics in mathematical finance.

Books and materials #

We will use a variety of sources during the course of the semester. The material for the lectures and class activities will be drawn from a variety of textbooks and research articles. While there are no required books for the course, the following may be useful during the semester. All will be available in the math department non-circulating library in Searles 214.

For elementary real analysis:

• Principles of Mathematical Analysis by Walter Rudin,
• Introductory Real Analysis by A. Kolomogorov and S. Fomin, and
• Introduction to Analysis by Maxwell Rosenlicht

For measure theory and integration:

• Measure theory and probability by Adams and Guillemin,
• Real and complex analysis by Walter Rudin, and
• Measure theory by Paul Halmos

For financial mathematics:

• Stochastic calculus for finance II by Steven Shreve

Reading and watching mathematics: Learning mathematics is not a spectator sport. Reading mathematics is not like reading a novel; watching mathematics is not like watching an action thriller. Some paragraphs are easy to digest, but you may find yourself looking at one line of text for five or more minutes trying to understand what the author is trying to say.  Use the pause button with reckless abandon when watching a video. As you read or watch, take notes, just as you do in class. This is crucial! If questions arise, write them down and ask during office hours or class.

Course meetings and office hours #

We will meet in Searles 213 every Tuesday and Thursday at 11:40am. Whenever possible and appropriate, lectures will be recorded for your enjoyment and to help you review your notes. When necessary, we will meet over Zoom. The link is available on the main course website.

I will hold formal office hours on:

There may be times in the semester where I will have to reschedule office hours due to a conflict. I will let you know by email beforehand. I am available throughout the week for additional meetings. To schedule either individually or as a group, please send me an email with a couple of times that will work. And please don’t hesitate to set these up, I am very happy to see you.

Course assessment #

Your grade will be based on homework and exam grades.

• Homework will be assigned at every class meeting and will be due at 9pm the following Wednesday unless otherwise indicated. You are permitted, and in fact, encouraged, to work with others on the homework. However, you must write up the solutions independently and cite your collaborators.

• There will be two exams during the course of the semester, both take-home assignments. The first will be due before our spring break, and the second will be due during finals week. Precise details will be announced in class.

Your final grade will be based on the total number of points earned in the class on the homework and exams.

Your homework will be graded using [GradeScope]. To begin, you will need to set up an account.  I will send you the code for our class by email.  Each homework assignment will need to be submitted as a .pdf file.  If you edit your homework electronically, make sure you can save or export your work in this format.  If you write-up your homework the old-fashioned way using pencil and paper, use a scanner or a phone scanner app.  See the GradeScope help document for a list of suggestions [pdf]. Please let me know if this does not work for you; I will come up with an alternative.

Some axioms #

Federico Ardila enunciated the following axioms in his Todos Cuendan. They form a lens through with I view both teaching and doing mathematics.

• Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
• Everyone can have joyful, meaningful, and empowering mathematical experiences.
• Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
• Every student deserves to be treated with dignity and respect.

Outline of course topics #

Part 1: Measure Theory

• Introduction: measures on rings of sets, outer measure, Lebesgue measure
• Measure Theory and Probability: probability measure, Bernoulli sequences and random walks, Infinite Monkey Theorem
• Measure Theory and Integration: measurable functions and random variables,i ntegration of simple functions, Lebesgue integral, Tchebyshev’s Theorem and convergence theorems

Part 2: Random Variables

• Random variables and probability distributions
• Expected value, variance, and higher moments
• Independence
• Law of Large Numbers
• Central Limit Theorem and applications

Part 3: Stochastic Processes

• Discrete Brownian Motion
• Continuous Brownian Motion: existence of continuous Brownian Motion, properties of continuous Brownian Motion, variation and quadratic variation
• Gaussian processes

Part 4: Stochastic Calculus

• Motivation
• Itō integral and general stochastic integrals
• Itō processes and the Itō formula

Part 5: Finance and the Black-Scholes-Merton Formula

• Introduction to options and financial derivatives
• A portfolio hedging model
• Derivation of the Black-Scholes-Merton formula
• Applications and Nassim Taleb’s critique