This first course in the sequence Math 4331-4332 provides a solid introduction to deeper properties of the real numbers, continuous functions, and differentiability needed for advanced study in mathematics, science and engineering. It is assumed that the student is familiar with the material of Math 3333, including an introduction to the real numbers, basic properties of continuous and differentiable functions on the real line, and an ability to do epsilon-delta proofs.

Topics: Metric spaces, open and closed sets, compact and connected sets, convergence of sequences, Cauchy sequences and completeness, properties of continuous functions, contraction mapping principle, countable and uncountable sets, Riemann-Stieltjes integration.

This first course in the sequence Math 4331-4332 provides a solid introduction to deeper properties of the real numbers, continuous functions, and differentiability needed for advanced study in mathematics, science and engineering. It is assumed that the student is familiar with the material of Math 3333, including an introduction to the real numbers, basic properties of continuous and differentiable functions on the real line, and an ability to do epsilon-delta proofs.

Topics: Metric spaces, open and closed sets, compact and connected sets, convergence of sequences, Cauchy sequences and completeness, properties of continuous functions, contraction mapping principle, countable and uncountable sets, Riemann-Stieltjes integration.

Lebesgue Integration on Euclidean Spaces, Frank Jones, Jones & Bartlett. 
However I will be distributing the class notes and they are very complete.

 Recommended reading:
- G.B. Folland, Real Analysis: Modern Techniques and Their Applications (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts).
- Real Analysis, H.L. Royden, (3rd Edition), Prentice Hall. Real and Complex Analysis, W. Rudin, McGraw Hill. Measure Theory, D. L. Cohn, Birkhauser.

This course will be taught in classroom and also can be taken as an online course.

Note: This is the first part of a two semester course sequence.  The second part of this sequence will not have an online version (meeting in classroom only).

 This is an introductory graduate course on probability and statistics. Emphasis will be placed on a thorough understanding of the basic concepts as well as developing problem solving skills. Topics covered include: combinatorial analysis, independence and the Markov property, Markov chains, the major discrete and continuous distributions, joint distributions and conditional proba- bility, modes of convergence. These notions will be examined through examples and applications.

Computers:

There will be a few optional problems which will require a numerical equation solver. You may use Matlab, Mathematica, or any other program you are comfortable with.


Homework:

There will be 7-8 homework sets during the semester. Each homework set will be due two weeks after it was assigned. You are free to work together on the homework sets, however the work you turn in must be your own.

Exams:

There will be one midterm and and a final exam.

This course is an introduction to discrete-time models in finance.

We start with single-period securities markets and discuss arbitrage, risk-neutral probabilities, complete and incomplete markets.  These ideas are then extended to the multiperiod settings.  Valuation of options, futures, and other derivatives on equities, currencies, commodities, and fixed-income securities will be covered under discrete-time paradigms.  The use of binomial trees and the notion of Arrow-Debreu securities will be explored.