Discrete Stochastic Processes

Overview

Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance.

Syllabus

1. Introduction and Probability Review.
2. More Review; The Bernoulli Process.
3. Law of Large Numbers, Convergence.
4. Poisson (the Perfect Arrival Process).
5. Poisson Combining and Splitting.
6. From Poisson to Markov.
7. Finite-state Markov Chains; The Matrix Approach.
8. Markov Eigenvalues and Eigenvectors.
9. Markov Rewards and Dynamic Programming.
10. Renewals and the Strong Law of Large Numbers.
11. Renewals: Strong Law and Rewards.
12. Renewal Rewards, Stopping Trials, and Wald's Inequality.
13. Little, M/G/1, Ensemble Averages.
14. Review.
15. The Last Renewal.
16. Renewals and Countable-state Markov.
17. Countable-state Markov Chains.
18. Countable-state Markov Chains and Processes.
19. Countable-state Markov Processes.
20. Markov Processes and Random Walks.
21. Hypothesis Testing and Random Walks.
22. Random Walks and Thresholds.
23. Martingales (Plain, Sub, and Super).
24. Martingales: Stopping and Converging.
25. Putting It All Together.