List of topics chosen so far.

### Possible projects for Stat 251/551

Here are some suggestions for projects, showing the amount of work I expect from groups of size one, two, or three.
• Card shuffling
one person:
Write out a complete explanation of the argument for the upper bound on total variation distance from uniformity for the riffle shuffle.
Reference: Aldous & Diaconis (1986), "Shuffling cards and stopping times", American Mathematical Monthly 93, 333--348.
two person:
Same as for one person, but also explain the role of a-shuffles in finding the exact total variation distance.
Additional reference: Mann, "How many times should you shuffle a deck of cards?", (manuscript available from http://www.dartmouth.edu/~chance/teaching_aids/activities.html )
three person:
Same as for two person, but also explain the duality construction of strong stationary times.
Additional reference: Diaconis & Fill (1990), "Strong stationary times via a new form of duality", Annals of Probability 18, 1483--1522.
• Hidden Markov models
See the description at the end of the handout EM.pdf.
• Image analysis
Write a detailed technical review of:
Besag, "On the statistical analysis of dirty pictures", Journal of the Royal Statistical Society, Series B, vol 48 (1986) pp 259--302.
or of:
Geman and Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images", IEEE-PAMI, 6, 1984, 721-741. (ask DP for paper or download from S. Geman website)
• MCMC in Bayesian statistics
Write a short account of resampling methods for calculating Bayesian posterior distributions, illustrated by some calculations for an example.
References:
Smith and Gelfand, "Bayesian Statistics without tears: a sampling-resampling perspective", American Statistician 46 (1992), pp 84--88.
Smith and Roberts, "Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods", Journal of the Royal Statistical Society, Series B, vol 55 (1993) pp 3--23.
(See also two other papers in JRSSB vol 55 on MCMC topic, followed by fifty pages of discussion.)
• Markov chains on general state spaces

• Optimal stopping (and/or American option)
Explain some form of the optimal stopping problem (finding a stopping to maximize expected reward at that time or to maximize probability of best choice at that time). For a two-person project: also explain the connection with pricing an American option.
References:
Look up optimal stopping in the index to
• the first edition of Billingsley Probability and Measure
• Karlin and Taylor Second course in Stochastic Processes
• Oksendal Stochastic Differential Equations: An Introduction with Applications
• For a more advanced treatment of optimal stopping in dsicrete time, see Neveu Discrete-Parameter Martingales

• Option pricing via arbitrage arguments
Many of you have expressed an interest in this topic. I am slightly fearful, because I have never heard of some of the options mentioned as possible topics. I don't know how much math is needed to run the arbitrage arguments. Nevertheless, I am prepared to let you try. Some of the following might be helpful.
References:
• Wilmot, Howison, and Dewynne The Mathematics of Financial Derivatives: A student introduction. I am finding this book quite helpful. It reduces many problems to the partial differential equations (not one of my strengths). Nothing about Girsanov or martingale measures.
• Duffie Dynamic Asset Pricing Theory. Written slightly above the level of Stat 251.
• There are many web sites that contain notes for courses on stochastic calculus or finance. See, for example, Per Myklund's pages for two courses, Stat 390 and Stat 391. The former has a link through to some nice lecture notes by Steve Lalley.

Per recommended to me several papers, which are are mentioned on the Stat 391 page:

• The paper by Miyani talks about American options, but it gets into heavy math pretty quickly.
• The paper by Longstaff and Schwartz describes a method for approximating the value of American options.
• Foellmer and Leukert
• Volume 21 issues 8-9 of Journal of Economic Dynamics and Control , in particular the paper by Phelim Boyle, Mark Broadie and Paul Glasserman. You can obtain this paper online, starting from the link to Online Journals and Newspapers on the Research Tools page at the Yale Library.
I have not yet looked at these papers in any detail.

• I also found some interesting lecture notes by Steve Shreve on Stochastic Calculus and Finance.