Statistics 251b/551b: Stochastic Processes. Time: Mon., Wed., 2:30-3:45 Prerequisite: a first course in probability, like Stat 241a. Instructor: Joseph Chang. Phone: 2-0642. Teaching assistant: Ronald Fricker. Text: I will distribute my own typed notes. For an alternative treatment, a recommended (not required) text is "Probability and Random Processes," by Grimmett and Stirzaker. The coop has ordered this book. Grading: Grades are based on weekly homework assignments and a "project"; no exams. The project should fun, not a painful ordeal; trust me... ********************************************************************* This is a first course in random processes. Both undergraduate and graduate students are welcome. My hope is that the course will be interesting and useful to students in a variety of fields and with a variety of backgrounds: from physics to finance, economics to engineering, mathematics to molecular biology. Here are some typical questions addressed by this subject. I hope to discuss a good selection of these sorts of questions in class. (1) How many times do you have to shuffle a deck of cards to mix them up well? (2) "Simulated annealing": How can we use randomness to solve large, complicated optimization problems like the "traveling salesman problem"? (3) Evolutionary trees: There are a number of possible trees that describe the relationship among the species human, chimpanzee, and gorilla. How can we use DNA samples to deduce which is the correct tree? (4) Quality control: How can we detect quickly when a noisy process has shifted from one "good" probability distribution to another "bad" one? (5) Finance: With stock prices fluctuating randomly, what is the "fair" price for a "call" option--the right to buy a certain stock at a specified price by a specified time? (6) What is the probabilistic basis of the statistical methods used in image processing and speech recognition? (7) Genetic mapping: how can probability calculations help find the location of a gene responsible for some trait? *************** LIST OF TOPICS *************** (order of presentation subject to change) Markov chains. Random walks. Convergence to stationary distribution, coupling. Applications: Generating random objects, optimzation, branching processes, Markov random fields. Hammersley-Clifford Theorem, hidden Markov models. Applications: Evolutionary trees, image reconstruction, speech recognition. Brownian motion and Diffusions. Construction, reflection principle, escape time calculations. Introduction to stochastic calculus. Applications: Black-Scholes formula, diffusion approximations. Other topics. Poisson processes, renewal theory. Martingales. Extremes, large deviations, Poisson clumping heuristic. Some statistical inference problems for stochastic processes. (I hope.) This year I will make an attempt to touch on all of the major topics rather early in the semester, so that people will have at least been exposed to them early enough to make them feasible as project topics. Later on in the semester we can come back to fill in some of the topics in more depth. *********************** WHAT IS A PROJECT? ************************** I want to be quite flexible about this, subject to the natural constraints that it should involve stochastic processes and have some sort of connection with the class. To give you an idea of the scope I have in mind, I'd say that the amount of work involved should be roughly like doing a few good homework assignments. I hope you will choose to investigate something that you are interested in. For example, think about why you want to take this class in the first place. Many of you want to learn about stochastic processes, naturally enough, in order to apply that knowledge to a particular field of application. Maybe you know of some important work in your field that requires some understanding of probability and stochastic processes. By all means, then, use this project as an opportunity to get started on your study of that work. On the other hand, you may not be coming into the class with a ready-made subject area. That's OK; here's how I hope things will work out for you. We will discuss a lot of ideas and topics in the class, and you'll see more in the homework problems. I'll be mentioning further ideas and references in class. I hope and believe that some of this will spark your interest. When that happens, it's a good sign that you might do some further reading or thinking or problem-solving or computation in that area. Your project could be ``theoretical,'' or ``applied,'' or both. On the theoretical side, a good way to start is to choose an interesting paper to read. The project could involve almost a ``book-report'' like activity, where you would explain what the idea of the paper is, explain the main result or results, and hopefully do some other thinking about the paper besides this (although understanding and explaining the main ideas of a paper may already be a significant achievement in many cases). For example, you might work out in detail an interesting illustrative example of some general theorem that provides some insight into what is going on. Or you could take a result in the paper and explain it carefully, filling in the sort of details that are often missing in journal articles. On the ``applied'' side, you could try something out on a computer. For example you could simulate some process that you are interested in. Or read a paper that proposes an algorithm or method, and you could program the method and try it out on some examples. Especially after getting some ``hands-on experience'' implementing a method and seeing how it works, you might have an idea of your own for some variation, which you could then implement and see if it seems to provide some improvement. That could be a lot of fun. In general, we can talk after class or you can stop by during office hours; I'd be glad to discuss ideas with you and suggest references or whatever toward formulating a feasible and enjoyable project. ******************** If you have questions or would like to discuss the course, you can send email to chang@stat.yale.edu or call me at 432-0642.