Statistics 603a, Fall 2001
Stochastic Calculus

Martingales in discrete and continuous time, Brownian Motion, Sample path properties, predictable processes, stochastic integrals with respect to Brownian motion and semimartingales, stochastic differential equations. Applications mostly to counting processes and finance. Knowledge of measure-theoretic probability at the level of Statistics 600 is a prerequisite for the course, although some key concepts, such as conditioning, are reviewed. After Statistics 600.

Instructor:	David Pollard (email: david.pollard@yale.edu)
Office hours:	Wednesday 4:00 - 5:00, or by appointment
Time:	Tuesday, Thursday 10:30-11:50
Place:	24 Hillhouse
TA:	Gheorghe Doros (office hours: Friday 5:00)

My aim is to explain enough theory to give students an understanding of the calculus of stochastic integration with respect to semimartingales.

Grading

References

• There is no single text. I have borrowed material from many sources, some of which are included in the annotated bibliography.
• detailed notes on some special topics
• extracts from UGMTP, with access only for Yale students

Topics

• 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)