Math 504 (ST: Stochastic Differential Equations). Fall 2005

Instructor: Steve Krone

Office: 416 Brink Hall. Phone: 885-6317
Office Hours:

Class Time: TTh 10:30 - 11:45
Place: McClure 315

Prerequisite: Graduate Probability (Math 536), or Measure Theory (Math 535) and Stochastic Models (Math 453).

Text:F. Klebaner, Introduction to Stochastic Calculus with Applications, Imperial College Press. NOTE CHANGE: This book is apparently out of print, so the recommended text for the course in the bookstore will be Oksendal (see below).

This is a special topics course on stochastic differential equations and diffusion processes. We will begin with a development of the Ito stochastic integral. This leads to a ``stochastic calculus" that is genuinely and beautifully different from ordinary calculus. For example, there is an extra (and very important) term in the usual integration by parts formula. With these tools, we then explore stochastic differential equations and their solutions, called diffusion processes. We will learn where stochastic differential equations come from (i.e., why they are so important) and how to do calculations of various quantities of interest. We also consider some real examples from population genetics.

Grades will be based on homework. No exams.

Homework Solutions:

Preliminary Course Outline: (subject to some change in content and order)

  1. Brief introduction to diffusion processes and how they arise as natural approximations to certain discrete processes. This will provide motivation and give some intuition.
  2. Review of some basic facts about Brownian motion and martingales.
  3. Ito stochastic integrals
    • construction
    • martingale properties
    • Ito's formula, integration by parts
    • comparison with Stieltjes integrals and ordinary calculus
  4. Stochastic differential equations
    • existence and uniqueness of solutions
    • Markov property
    • generators
    • martingale problem characterization of weak solutions
  5. Diffusion processes
    • Dynkin's formula
    • calculations of expectations and probabilities
    • converting to the natural scale
    • stationary distributions
  6. Examples from biology
    • Wright-Fisher diffusions
    • branching diffusions

Additional References:

  • B. Oksendal, Stochastic Differential Equations. Springer.
  • S. Karlin and H. Taylor, A Second Course in Stochastic Processes. Academic Press.