MATH 286: SDES (FALL 2005)

Copyright c2005, R. J. Williams

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Please send notice of typos to williams@math.ucsd.edu

  • Lecture 1. Introduction to SDEs. Transparencies.

    Click here for some notes related to the topics of lectures 2-3.

  • Lecture 2. Overview of continuous time stochastic processes, Brownian motion.
  • Lecture 3. Brownian motion continued. Overview of continuous time martingales.
  • Lecture 4. Continuous time martingales (continued): Doob's inequalities, stopping times, local martingales.

    Click here for some notes on the definition of the stochastic integral. (Lectures 5-7).

  • Lecture 5. Uniform integrability. Definition of the stochastic integral with respect to Brownian motion - simple integrands.
  • Lecture 6. Definition of the stochastic integral with respect to Brownian motion - integrable integrands.
  • Lecture 7. Local integrands, Ito processes.

    Click here for some notes on Ito processes and Ito's formula.

  • Lecture 8. Ito's formula and applications
  • Lecture 9. Proof of Ito's formula
  • Lecture 10. Proof of Ito's formula (cont.). Stochastic differential equations: weak and strong solutions, Tanaka's formula.
  • Lecture 11. Stochastic differential equations: strong uniqueness.
  • Lecture 12. Stochastic differential equations: strong existence (Picard iteration).
  • Lecture 13. Stochastic stability.
  • Lecture 14. Stochastic stability (cont.). Martingale Representation Theorem.
  • Lecture 15. Martingale representation thm (cont.). Characterization of Brownian motion.

    Click here for some notes on Girsanov's theorem and a martingale representation theorem. Click here for Nate Eldredge's notes on martingale representation.

  • Lecture 16. Girsanov transformation.
  • Lecture 17. Proof of Novikov's condition, diffusions (longer lecture).
  • Lecture 18. Proof of Markov property and strong Markov property (longer lecture).

    Last updated November 30, 2005.