In Spring 2007 I taught MATH 778.
This is an advanced graduate course on stochastic processes. My focus was on continuous time stochastic processes and a major objective was to display the power of stochastic calculus.

Below, I describe an outline of what I taught.

  1. Week: Review of stochastic integration and Ito's rule.
    • martingales, local martingales
    • quadratic and mutual variations
    • progressive measurability
    • random times and filtrations

  2. Week: Representation of martingales and Girsanov's theorem
    • Levy's characterization
    • integral representations
    • predictable representation property
    • Brownian motion with drifts

  3. Week: Markov Processes and the martingale problem
    • SDE
    • fundamental martingales
    • concept of weak and strong solutions

  4. Week: BM and differential equations - Feynman-Kac's theorem
    • Heat equation
    • Occupation times and F-K

  5. Week: Local time for Brownian motion
    • Definition, properties
    • Levy's theorem

  6. Week: Bessel and bessel square processes

  7. Week: Ray-Knight and related theorems

  8. Week: Dvoretzy-Erdos-Kakutani and related theorems

  9. Week: Cieselski-Taylor and related theorems

  10. Week: Planar BM, conformal invariance, winding number

  11. Week: Schramm-Loewner evolutions

  12. Week: SLE contd.

  13. Week: SLE contd.

  14. Week: SLE contd.