1. First Lecture Introductory Lecture on Modeling
2. Second Lecture Examples of Markov chains and their properties.
3. Third Lecture Computing Probabilities Associated with Markov Chains.
4. Fourth Lecture Recurrence and Transience
5. Fifth Lecture Class Properties, Limiting Probabilities
6. Sixth Lecture Limiting Probabilities, Gambler's Ruin.
7. Moment Generating Functions and other transforms.
8. The Central Limit Theorem for unequal components.
9. Random Sums .
10. Supplement: Poisson, Binomial and Negative Binomial Distributions
11. Supplement: Solving difference equations
12. Poisson Processes I
13. Continuous Time Markov Chain Basics
14. Supplement: Basic facts about exponential functions including matrix exponential functions.
15. Mean and Variance for Birth and Death Processes
16. Supplement: Fair Games and Martingales
17. Brownian Motion
18. Geometric Brownian Motion and Option Pricing
19. Arbitrage and Probability

This course is a natural extension to any first course in probability or statistics for students interested in modeling physical phenomena by probabilistic rather than deterministic methods. We will study Markov chains in general in both continuous and discrete time. Particular models include queues, branching processes, birth and death models, Poisson processes, and Brownian motion. The focus is on applications rather than theory. For example, we will derive the Black-Scholes pricing formula for stock options and use it to price simple European calls.

As you might expect, the course supposes familiarity with basic ideas in probability. These are summarized in the first three chapters of the text, Introduction to Probability Models, by Sheldon Ross. Students are expected to be fluent with this material. In addition, we will suppose that you have fundamental computational skills in matrix algebra, including the computation of eigenvectors and eigenvalues. We also suppose familiarity with the solution of simple differential equations. This material is ordinarily covered in Math 234.

There will be random quizzes in on the reading (about 5% of your grade), weekly homework assignments (about 30% of your grade), midterm exams (two or three, about 35% of your grade), and a final exam (about 30% of your grade). You may be required to re-submit homework problems until you have solved them correctly if you are to get credit for them. For a list of the reading, homework problems and their initial due dates, go here .

If you have any doubts about your preparation, or any other questions, contact me .