Copyright c2011, R. J. Williams


Please send notice of typos to williams at math dot ucsd dot edu

  • Lecture 1. Doob's inequalities, L^p convergence theorem for martingales.
  • Lecture 2. Convergence of conditional expectations, backwards submartingale convergence theorem.
  • Lecture 3. Proof of backwards submartingale convergence theorem, convergence of more conditional expectations, start of continuous time submartingale theory.
  • Lecture 4. Path regularity for submartingales.
  • Lecture 5. Usual conditions, right continuous modifications of submartingales with right continuous expectations, Doob's inequalities and submartingale convergence theorem for right continuous submartingales.
  • Lecture 6. Stopping times, properties of stopping times, right continuous, adapted processes evaluated at stopping times.
  • Lecture 7. Stopping theorem, Doob-Meyer decomposition theorem, Brownian motion - definition, some properties.
  • Lecture 8. Gaussian processes, time inversion of Brownian motion.
  • Lecture 9. Existence of Brownian motion via Kolmogorov extension theorem and Kolmogorov's continuity criterion.
  • Lecture 10. Completion of proof of K's continuity criterion. Path properties of Brownian motion: not Holder continuous of exponent 1/2, nowhere differentiability.
  • Lecture 11. Introduction to Levy processes. Markov property of Brownian motion.
  • Lecture 12. Proof of the Markov property for Brownian motion.
  • Lecture 13. Blumenthal's zero-one law and applications; strong Markov property of Brownian motion.
  • Lecture 14. A process that is Markov but not strong Markov; reflection principle; multidimensional Brownian motion.
  • Lecture 15. Poisson random measures, definition, Laplace functional, example, existence for Sigma finite mean.
  • Lecture 16. Examples of Poisson random measures: Poisson process, pure jump Levy process with finite Levy measure.
  • Lecture 17. Construction of Levy process with general Levy measure.

    Last updated March 30, 2011.