Index of /~pollard/Courses/251.spring09/Handouts
Index of /~pollard/Courses/251.spring09/Handouts
Stochastic Calculus
The page Steele2001p142.pdf is taken from the very good book
Stochastic Calculus and Financial Applications by J. Michael Steele. The book is written at a slightly higher level than
Stat 251/551, but it contains much wisdom, wit, and insight. If you ever decide to get really serious about
Stochastic Calculus, this book should be high on your list of essential references.
Projects
Students should discuss their planned projects with DP in the first
week after the break.
Some possible topics
 Riffle shuffles

Give a complete account of the analysis of the riffle shuffle. Do not use the language of convolutions for walks on finite groups; use the methods developed during the course.
 Simulated annealing


S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi,
Optimization by Simulated Annealing,
Science 220 (1983), 671680.
[JSTOR]

Stuart Geman and Donald Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images,
IEEE Transactions on Pattern Analysis and Machine Intelligence,
Volume PAMI6, Issue 6, Nov. 1984 Page(s):721  741,
Digital Object Identifier 10.1109/TPAMI.1984.4767596 .
[IEEE]
 Image analysis
 Do something clever with Markov chains
 The decoding example described by Diaconis, The Markov Chain
Monte Carlo Revolution, Bulletin of the American Mathematical
Society, 46 (2009), 179205 & [AMS]
looks promising.
 Random walks and electrical networks

Write a complete probabilistic account of Rayleigh's Monotonicity Law (see section 1.4 of Doyle and Snell).
 Optimal stopping

Explain the solution to the classical Secretary Problem.
Then explain how the solution can be derived using the Snell envelope. Maybe apply the same method to derive pricing for an American option over a finite set of trading times.
 Notes on optimal stopping by
Tom Ferguson
 Section 8 (Example 8.5 and pp111116) of: P. Billingsley
(1979), Probability and Measure (first edition). Wiley.
 Algorithms for HMM and Graphical models
 First read Chapter 3 of the Chang notes. Maybe move on to
the Jordan article. Explain how the messagepassing schemes
work (on a tree, at least). Implement. Of course I expect
you to go beyond what is in the Chang notes.

Michael I. Jordan, Graphical Models,
Statistical Science, Vol. 19, No. 1 (Feb., 2004), pp. 140155.
[JSTOR]

See also other papers and notes by Jordan and by Wainwright at the YPNG
web site.
 Engel algorithm?
 Maybe too easy. Not quite enough for a whole project just to
understand material in engel.pdf.