## Stochastic Processes

• Time and place: MWF at 11 in MS 5117
• Instructor: Thomas M. Liggett (tml@math.ucla.edu)
• Office hours: MWF 1-2 in MS 7919 (Students should feel free to send questions by email as well, though we can usually accomplish more by speaking in person.)
• Teaching Assistant: Yao Yao. Her office hours are Tu 11-12 and Wed 2-3 in MS 6160.
• Text: "Essentials of Stochastic Processes", by Rick Durrett. There is a review chapter at the beginning of the text, which is a good summary of the probability that is assumed in the course. You should read this to make sure you are up to speed. The main subject of the course is Markov chains, which is one of the most important classes of stochastic processes. An errata for the book can be found at http://www.math.duke.edu/~rtd/books.html
• A stochastic process is simply a collection of random variables indexed by time. These come up in many contexts: position of a particle that is influenced by some random phenomenon, price of an option or stock, winnings of a gambler, size of a queue (a queue is a waiting line, such as at a checkout counter, in the skies above LAX, or in a buffer at a node in a communications network), and many others. The main issue we will discuss is the behavior of the random system after it has evolved for a long time -- is it close to being in equilibrium, does it grow without bound,...?
• Prerequisites: Officially, the prerequiste is Mathematics 170A or equivalent. Mathematics 170B (taken earlier or concurrently) is definitely helpful. If you are not on top of the topic of convergence of sequences and series, you should review that before the class begins.
• Discussion sections: (Thursdays at 11 in MS 5117) These provide an opportunity for you to ask questions about and discuss more fully issues that are covered in the lectures. The TA will discuss the homework that is due that day, but all questions related to the course are fair game. In the first discussion section (March 31), the TA will do some reviewing of probability theory, based on the material in "Review of Probability" at the beginning of the text. In preparation for this session, you should look at (and try to do) problems 1.15, 1.17, 1.21, 1.25, 1.28, 2.9, 2.17, 2.19, 3.11, 3.15, 3.17, 3.20, 3.22 of this pre-chapter. They are not to be turned in, however.
• Homework will be assigned each Friday (and listed on this web page), and will be due at the beginning of the discussion section the following Thursday. It is essentially impossible to learn mathematics without doing a lot of problems. I urge you to do the homework with care and on time. You should also look through and attempt as many of the unassigned problems as you have time for. The basic rule is: The more problems you do, the more thoroughly you will learn the material. Note that there are answers to many of the problems at the end of the book, so you can check your work that way. The lowest HW score will be dropped before grades are assigned.
• Midterms: There will be two midterms, on Monday, April 25 and Monday, May 16. Both will be in Haines A25.
• The first midterm: It will be on Monday, April 25, and will cover up to page 81. I will take questions about this material at the beginning of the lecture on Friday, April 22. The exam will be in Haines A25.
• Test 1 Solutions
• The second midterm It will be on Monday, May 16, in Haines A25. It will cover Section 1.7, Chapter 2, and Sections 3.1,3.2,3.4.
• Test 2 Solutions
• Final exam: The final exam will be Wednesday, June 8, 3-6 in MS 5117. It will cover the entire quarter -- Chapters 1-4. The emphasis is on Chapter 4. I won't do much on the last two sections of Chapter 4 in class -- just an example or two. You should think of this material as being further examples of the theory that we developed in the earlier sections, so that you will benefit from reading the latter sections.
• Final Solutions
• Finals week OH: Monday, June 6, 2pm.
• Grades will be based on homework (20%), midterms (20% each) and final exam (40%).