MATH 777: Stochastic Processes: Characterization and Convergence (Fall 2007)

Instructor: Rick Durrett

Meeting Time & Room

Rather than a trip to the frontiers of knowledge, this is a course on a topic that every probabilist should know. In many situations one would like to show that a sequence of stochastic processes converges to a limit. Examples are the convergence of rescaled random walks or martingales to Brownian motion, and Markov chains to limiting diffusions.

The most commonly used approach is to show that the sequence is relatively compact, and that every subsequential limit has properties that uniquely characterize the limit. For example, Brownian motion is the only martingale with quadratic variation t, or more generally it solves a well-posed martingale problem.

A rough outline is:

  • generalities about weak convergence in metric spaces
  • path spaces C and D, their topologies, conditions for relative compactness
  • convergence of random walks to Brownian motion and stable processes
  • generators of Markov processes, and martingale problems
  • convergence to diffusions.

For more details consult our main sources:

  • Billingsley (1968), Weak Convergence of Probability Measures (Chapters 1-3) with improvements from his 1971 CBMS lecture notes
  • Ethier and Kurtz (1986), Markov Processes: Characterization and Convergence (parts of Chapters 3-11)
  • Jacod and Shiryaev (1987), Limit Theorems for Stochastic Processes (Chapters VI, VIII, IX)