This fall term graduate course has traditionally covered the material of my book Stochastic Calculus and Financial Applications. In Fall 2015, I expect to address the same basic material, but there will be new wrinkles. Extra attention will be given this year to the famous Feynman-Kac formula which is now understood to offer a very powerful window on many problems from finance to nano-technology.

Fall 2015 (Blog ORDER)

Is this course for you?

This course should be useful for well-prepared students who are in the fields of finance, economics, statistics, or mathematics, but it is definitely directed toward students who also have a genuine interest in fundamental mathematics. Naturally, we deal with financial theory to a serious extent, but, in this course, financial theory and financial practice are the salad and desert --- not the main course.
Our work requires a high level of comfort with the tools of real analysis, including uniform continuity, Cauchy's convergence criterion, notions of integrability, and calculations in inner product spaces. Knowledge of function spaces (L-one, L-two, Hilbert space, etc) may not be explicitly assumed, but many function spaces will be introduced and used in the course and students who have not seen these before face heavy sledding.
"Measure-theoretic probability theory" enters the conversation regularly, but, with a reasonable amount of work, it can also be picked up as the course progresses. Basic mathematical analysis is the core prerequisite, there is no denying that some knowledge of measure theory will be useful --- at least to the level of having understood the Borel-Cantelli lemmas, the definition of convergence with probability one, and the Dominated Convergence Theorem.
Students who have had Statistics 530-531 are perfectly well prepared, as are students with a graduate course in real analysis. Many students with lesser backgrounds have taken the course and done well. It is substantially a matter of priorities and motivation. Still, never has a MBA had a successful experience with this class; only MBA's with plenty of Ph.D. level quant experience should consider this. Undergraduates have succeed in this class, but it is a very tough challenge; it's only feasible if you have already had 530-531.
We are after the absolute core of stochastic calculus, and we are going after it in the simplest way that we can possibly muster. If we are honest at each turn, this challenge is plenty hard enough.
There is a syllabus for 955 but this page is the place to come for up-to-date information about the course content and procedures.

Course Policies

House Keeping --- Please no cell phones, no IM'ing, no open lap tops, no newspapers, no hoagies, no Au Bon Pain Salad Boxes, etc. A coffee or a soft drink is OK, but please be kind to your neighbor --- we have a bounded space.
Homework --- It is important to solve problems and to discuss the solutions of problems. This is a critical step to genuine learning. We do not have a grader, so the grade will rest on the final.
Grading. The final will a take-home and it will be longish. It will be a central part of your learning experience. On the final, you can consult any book, but you may not discuss these exam problems with any other person.
Auditing --- Certainly. You are most welcome but please follow the standing rules (no phones, no lap-tops, no food).

Course Topics

On the first day of class, I will draw a mind map that puts the topics of the course into a frame that I believe to be much more meaningful than a simple list. The picture is based on a rectangle with vertices: a=martingales b=Brownian motion c=Ito calculus d=arbitrage . The rest of our topics hang naturally off of these vertices.

What the "big picture" does not show directly --- but which I try to underscore at every turn --- is the importance of problem solving. There really are "techniques for solving problems," and one finds a different "place to stand" once even a modest mastery of these techniques has been attained.

Random walks and first step analysis
First martingale steps
Brownian motion
Martingales: The next steps
Richness of paths
Itô integration
Localization and Itô's integral
Itô's formula
Stochastic differential equations
Arbitrage and SDEs
The diffusion equation
Representation theorems
Girsanov theory
Arbitrage and martingales
The Feynman-Kac connection

Stochastic Calculus and Financial Applications

Stat 955: Core Theory and Honest Problem Solving

Resources

e-HandoutS

Related LINKS

Fall 2015 (Blog ORDER)

Is this course for you?

We are after the absolute core of stochastic calculus, and we are going after it in the simplest way that we can possibly muster. If we are honest at each turn, this challenge is plenty hard enough.

There is a syllabus for 955 but this page is the place to come for up-to-date information about the course content and procedures.

Course Policies

House Keeping --- Please no cell phones, no IM'ing, no open lap tops, no newspapers, no hoagies, no Au Bon Pain Salad Boxes, etc. A coffee or a soft drink is OK, but please be kind to your neighbor --- we have a bounded space.

Homework --- It is important to solve problems and to discuss the solutions of problems. This is a critical step to genuine learning. We do not have a grader, so the grade will rest on the final.

Grading. The final will a take-home and it will be longish. It will be a central part of your learning experience. On the final, you can consult any book, but you may not discuss these exam problems with any other person.

Auditing --- Certainly. You are most welcome but please follow the standing rules (no phones, no lap-tops, no food).

Course Topics

What the "big picture" does not show directly --- but which I try to underscore at every turn --- is the importance of problem solving. There really are "techniques for solving problems," and one finds a different "place to stand" once even a modest mastery of these techniques has been attained.