Math/Stat 453 (Stochastic Models). Spring 2004


Instructor: Steve Krone

Office: 416 Brink Hall. Phone: 885-6317
Office Hours: Tu 3:30, Th 2:30, and by appointment.

Class Time: MWF 12:30 - 1:20
Place: Morrill 302

Prerequisite: Math/Stat 451 (Probability) and a strong desire to learn.

Text: L. Allen, An Introduction to Stochastic Processes with Applications to Biology, Prentice-Hall, 2003.

Exam #2, Fri, April 9.

Grading

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

Solutions (HW and Exams): http://www.uidaho.edu/newton.

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. 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

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:

  • R. Durrett, Essentials of Stochastic Processes, Springer, 1999
  • 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. Graduate students will be expected to do a little extra reading to get graduate credit.