## Statistics 603a, Fall 2004 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) to be announced Friday 1:30 -- 3:30 plus weekly meetings with DP (see email.3sept04.txt for details) 24 Hillhouse Stephan Winkler

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

As with all my graduate courses, there will be no exams. The final grade will be based completely on the homework assignments and work discussed with DP during weekly meetings. I am also open to creative suggestions for other ways to assign a grade. See email.3sept04.txt for details.

### 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 and the outline for the weekly projects. See also the notes from 2001 and some simplified notes from an introductory stochastic processes course, giving a nonrigorous account of some material relevant to Stochastic Calculus.
• extracts from UGMTP, with access only for Yale students

### Topics for 2004 (tentative)

I am still making changes to topics covered in 2001.
• Conditioning
• elementary case
• conditional distributions
• fair price interpretation
• the abstract Kolmogorov conditional expectation; sigma-fields as "information"
• Stochastic processes
• filtrations
• optional times (a.k.a. stopping times)
• sample path properties
• versions of processes and the "usual conditions"
• stochastic processes as functions on a product space
• progressive measurability and stopped processes
• 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
• strong Markov property of 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