Textbook: "Probability and Random Processes" by Grimmett and Stirzaker, Third edition 2009, Oxford Univ Press, ISBN 978-019-857222-0.

Pre-requsite: Undergraduate probability (ACMS 30530 or MATH 30530, preferably ACMS 50850) and the equivalent of courses such as Multi-variable calculus, Linear algebra, ODE, Fourier and Laplace transforms (ACMS 20550 and 20750 and 20620, or MATH 20550 and MATH 20750 and 20610) will be assumed.

Qualifying exams: This course is a course for qualifying exam for ACMS Ph. D. students. First year students who choose this course (must be approved by DGS and advisor) as part of the qualifying exam must pass the qualifying exam by the end of the first year (the exam will be given in June). Anyone who fails the exam will have a second (final) chance immediately after summer (after an entire summer effort).

Syllabus: We will cover selected materials:

1. Basic setup of probability theory (including sample spaces, conditional probability, independence). Random variables (including the elements of measure and integration theory).
2. Discrete random variables (including random walks).
3. Continuous random variables, the basic distributions, sums of random variables.
4. Generating functions, branching processes, basic theory of characteristic functions, central limit theorems.
5. Markov chains (embedding, birth and death processes, Poisson processes)
6. Monte Carlo simulations
7. Laws of large numbers, Martingales.
8. Various stochastic processes, including Brownian motion, and applications.
9. Martingales, including stopping times and optimal stopping.
10. The rudiments of stochastic integration (including Ito's formula and the Black-Scholes differential equation).

Office Hours: You are welcome to stop by my office at any time. You can always make an appointment with me by email (b1hu@nd.edu).

Homework: Homework problems will be assigned during class meetings and will be collected once a week on each Wednesday. The main purpose of the homework is to help you learn the material. Experience shows that students who take their homework seriously do very well in the course because they have a better understanding of the material.

Exams: There will be a midterm exam (in class) as well as a final exam (take home). Your final grade is based on a total of 350 points listed below. The grade might be curved if it becomes necessary.

Recording Keeping: All records will be kept on Sakai