Topics from 2001

• Conditioning
  • elementary case
  • conditional distributions (postpone until discussion of BM)
  • fair price interpretation
  • the abstract Kolmogorov conditional expectation; sigma-fields as ``information"; projection interpretation
• Stochastic processes
• filtrations
• optional times (a.k.a. stopping times)
• sample path properties
• versions of processes and the "usual conditions"
• Martingales
  • a quick overview of theory for discrete-time martingales, as treated in Statistics 600
  • uniform integrability
  • sample path properties of martingales in continuous time: existence of cadlag versions
  • preservation of martingale properties at stopping times
  • the martingale central limit theorem, as an introduction to the role of quadratic variation
• Brownian motion
  • construction of a Brownian motion with continuous sample paths
  • Lévy's martingale characterization of Brownian motion
  • Itô integral as the prime example of a stochastic integral with respect to a (locally) square integrable martingale
• Predictability (omit some parts if short of time)
  • discussion of the subtle connections between prediction of processes and measurability with respect to the predictable sigma-field
  • statement and explanation of the section and projection theorems (maybe)
  • foretelling of predictable stopping times
  • characterization of predictable processes; predictable projections
  • predictable measures and predictable increasing processes
  • compensators and the Doob-Meyer decomposition of a submartingale
  • maybe something on point processes
• Stochastic integral with respect to a square integrable martingale
  • the Doléans measure for a submartingale
  • the stochastic integral via the L²(Doléans) isometry
• Localization and semimartingales
  • local martingales etc
  • stochastic integral with respect to a semimartingale
  • quadratic variation [square-brackets process]
  • Itô's formula for semimartingales
• Change of measure
  • Black-Scholes formula via change of measure
• Diffusions and stochastic differential equations (maybe)