Math/Stat 453 (Stochastic Models). Spring 2024


Instructor: Steve Krone

Office: 421 Brink Hall. Office Hours: by appointment.

Class Time: MWF 12:30 - 1:20
Place: TLC 248

Prerequisite: Math/Stat 451 (Probability) and a strong desire to learn. Math 451 is a serious prerequisite, as is the "mathematical maturity" necessary to know that subject at a high level. Just "getting by" in 451 will not put you in a good position to be successful unless you are willing and able to put in the extra work needed to address any defficiencies.

Text: R. Durrett, Essentials of Stochastic Processes, 3rd ed. (but earlier editions OK), Springer

Grading

Homework .... 25%
Exam 1 .......... 25%
Exam 2 .......... 25%
Final Exam .... 25%

Exam 1:

Exam 2:

Final Exam:

This is a first course in stochastic processes, the mathematics behind time-dependent random phenomena. We will introduce the main ideas and techniques from the subject and relate them to nontrivial models in science. In order to facilitate a deeper understanding, we will spend less time on the artificial examples one often finds in textbooks, and more time developing a family of related models from population genetics (no previous knowledge necessary). This will allow us to become more intimately aware of what is going on in the models, and so make the theory more transparent with the intuition we will develop.

A rough outline of the topics is as follows:

1. Introduction

2. Discrete-Time Markov Chains and the Wright--Fisher Model

  • Transition probabilities
  • Classification of states
  • Stationary distributions and other limits
  • Wright--Fisher model and other applications

3. Martingales

  • Examples of martingales and connections between martingales and Markov processes
  • Stopping times and the Optional Stopping Theorem
  • Other calculations via martingales
4. Continuous-Time Markov Chains and the Moran Model
  • Poisson processes
  • Birth-death processes
  • Transition rates
  • Kolmogorov backward and forward equations
  • Limiting probabilities
  • Time reversiblity
  • Moran model and other applications

5. Brownian Motion and Diffusion Processes

  • Diffusion processes and stochastic differential equations
  • Approximating scaled Markov chains with diffusion processes
  • Calculations with diffusions
  • Applications to Wright--Fisher diffusion in population genetics

Additional References:

  • L. Allen, An Introduction to Stochastic Processes with Applications to Biology
  • S. Ross, Stochastic Processes.
  • S. Karlin and H. Taylor, A Second Course in Stochastic Processes.

Note: This course can also be taken for graduate credit as Math 538 or Stat 544. In addition to the exams and homework, students taking the graduate-level option will be expected to complete an extra reading assignment and present that material to me.