This fall term graduate course has traditionally covered the material of my book Stochastic Calculus and Financial Applications. In Fall 2013, I expect to address the same basic material, but there will be new wrinkles. Extra attention will be given this year to the famous FeynmanKac formula which is now understood to offer a very powerful window on many problems from finance to nanotechnology.
Fall 2013 (Blog ORDER)
Final Exam
Please recall note that you are required to use the cover sheet. The exam is due Monday December 16 at NOON and you need to provide hardcopy to my office mail box and a PDF to my email. Your PDF should have a file name like StudentName955.pdf. If you hand in your paper early (i.e. before Monday 9:30 am) please put it in a sealed envelop. Thanks.
12/11 5PM. In problem 7, you can assume that M_0=0. I changed this on the PDF, but did not change the version number change. You can also assume that the process is continuous.
I mentioned the KMT Strong Embedding theorem in the previous class. I have a little bit more to say about it, and I hope to motivate you to take some time to look at a beautiful paper by Sourav Chatterjee that makes nice progress on the embedding problem using variations on Stein's method.
12/5 5PM. Version 8.1. A constant has been fixed in Problem 15, and a note is made that this section of Krylov has some typos. ( PDF and Source)
12/4 10:00 PM. Version 8.0 is posted. The exam is complete ( PDF and Source). Please report any bugs and look back here periodically for bug reports. References to Krylov (2002) refer to the ArXiv version Krylov on Novikov. Good luck to all. Please recall note that you are required to use the cover sheet. The exam is due Monday December 16 at NOON and you need to provide hardcopy to my office mail box and a PDF to my email. Your PDF should have a file name like StudentName955.pdf.
11/30 6:00 AM. Version 7.0 is posted. You now have 14 problems ( PDF and Source). You can expect one more problem to be posted between now and Monday December 9, our last day of class.
11/29 (9:30 PM) Dummy variable in 11b fixed. No logical change, so no change in version number. Hope to get another problem up tomorrow... we'll see.
Version 6.3 of the Final ( 12:30 PM Nov 24). Constant in 11c really fixed this time. ( PDF and Source)
Version 6.2 Modified at 6:30 PM Nov 23. Constant in 11c fixed. ( PDF and Source)
Version 6.2 ( PDF and Source) has been posted (1:30 Nov 23). One sign fixed. Problem 12 simplified.
Version 6.0 ( PDF and Source) has been posted (10:30 AM Nov 23). You now have 13 problems. The last two problems (numbers 14 and 15) will be up by December 2. If you see a bug or have a question, drop me an email. Also check in here occasionally to see if there are any bug fixes.
Version 5.2 has been posted (9:30 M Nov 23). You now have 12 problems. I also added a further part to Problem 8. Some small clarifications made since 5.0 Also the links have been corrected  PDF and Source. You can anticipate 15 problems.
Please also note that you are required to use the cover sheet. The exam is due Monday December 16 at NOON. Please read all instructions carefully. NOTE: To use the Source you may need the file mydefs.sty
PSA for a former student
"Please join representatives from Citi’s Sales and Trading team on Thursday, November 7th, to participate in a Sales and Trading Workshop that will cover Sales and Trading basics, current topics in the market, common pitfalls on the road to a summer internship, and tips for how to succeed in the Sales and Trading Analyst program. Citi traders and salespeople will be available following the event for networking. Details as well as upcoming key dates are listed below.
To RSVP, please email your resume to julie.blair.samuel@citi.com prior to the session. We look forward to meeting you!
S&T 101 Workshop:
Date: 11/7/2013
Time: 8:00  9:00 pm
Location: Jon M. Huntsman Hall in the Wharton School Building, Room G55
*This session will be most useful to Sophomores and Juniors interested in Citi’s Summer Analyst program but Ph.d. and Masters students are also welcome.
Is this course for you?

This course should be useful for wellprepared 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 (Lone, Ltwo, 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.

"Measuretheoretic 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 BorelCantelli lemmas, the definition of convergence with probability one, and the Dominated Convergence Theorem.

Students who have had Statistics 530531 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 530531.

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 uptodate 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 takehome 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 laptops, 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 FeynmanKac connection