MTH U581 -- Spring 2009

Possible Projects

• The Arbitrage Theorem and Option Pricing: a simple model analysing options pricing. See here for a quick introduction. Project should explain the Arbitrage Theorem and apply it to some interesting case. Here are some further notes.
• Shannon's Entropy and Typical Sequences from an Information Source here.
• Markov Chain models of the English language here.
• Random Walk on a Graph -- how a Knight moves on the chessboard. Set up and solve a Markov chain describing how a knight moves on the chessboard. These notes on random walks on graphs are useful. Consider extensions to other chess pieces, and maybe two knights moving simultaneously.
• Random Coin Tossing and the Arc-Sine Law governing fraction of time Heads leads Tails. See here and here.
• Parrondo's Paradox. An interesting example from Game Theory, which is a branch of mathematics used in Economics, Biology, Political Science and many other areas. Here is a good introduction. Here are some further notes.
• Some applications of probability in genetics.
• Hidden Markov models. A useful variation on the basic Markov chain idea, where there are several possible transition matrices producing the observed sequence of jumps, and you need to figure out what is happening behind the scenes. This paper by Rabiner gives an excellent introduction.
• Simulation of continuous random variables (inverse and rejection method). See here
• Monte Carlo method for generating Markov chain -- Hastings--Metropolis algorithm.
• Google's Page Rank. This is a fun topic -- see how mathematics can turn two graduate students into billionaires! Here are two introductions: some notes by Jim Carlson and wikipedia.