MATH 606, Summer 2020.
Topics in Stochastic Processes (054–39482R)
Understanding the Bichteler-Dellacherie Theorem
Class meetings: MW, 9:30am-12:30pm, On line.
Information on this and related pages changes frequently.
Instructor: Sergey Lototsky.
Office: KAP 248D.
Phone: 213 740 2389
E-mail: lototsky usc edu.
URL: https://dornsife.usc.edu/sergey-lototsky/
Office Hours: MW 12:30-1:30pm, on line [May 20-July 6]
Appointments at other time are welcome.
Course objective To understand why a consistent stochastic integration is only possible with respect to semi-martingales
Course work: Class participation, homework assignments, final presentation.
Official grading scheme: 20% class participation, 40% homework assignments, 40% final presentation.
Main reference: Stochastic Integration and Differential Equations, by Philip Protter; Any Edition, Springer, from 1990 on, 300 pp.
Other references
- Karatzas and Shreve, Browinian Motion and Stochastic Calculus, Springer, from 1988 on, 500 pp.
- Liptser and Shiryaev, Theory of Martingales, Kluwer/Springer, from 1986 on, 800 pp.
- An alternative proof of the theorem [pre-print; published as Beiglböck, M. and Siorpaes, P. Riemann-integration and a new proof of the Bichteler-Dellacherie theorem. Stochastic Process. Appl. 124 (2014), no. 3, pages 1226–1235.]
Closely related fun reads
- Burk, A Garden of Integrals, MAA, 2007.
- Gelbaum and Olmsted, Counterexamples in Analysis, from 1964 on [e.g. Dover, 2003].
- Stoyanov, Counterexamples in Probability, Wiley, from 1987 on.
- Szekely, Paradoxes in Probability Theory and Mathematical Statistics, Kluwer, from 1986 on.
Highlights of the course, including homework problems, will be updated on a regular basis and appear at the top of the content section of Blackboard
Lecture notes: at least two versions will be uploaded to the content section of Blackboard after every lecture.
The plan
Introduction and outline; Deterministic integration; Summary of Functional Analysis; Review of Basic Probability; Stochastic Analysis I: sigma-algebras and stopping times; Stochastic Analysis II: different types of processes; Stochastic Analysis III: martingales and related concepts; Properties of the Stochastic Integral, Cameron-Martin-Girsanov-Meyer; The Proof; Some examples and counterexamples.
Our Progress
May 20. Understanding the task: stochastic integration as a bounded linear operator.
May 25. Memorial Day: no class.
May 27. An overview of integration: from Cauchy and Riemann to Henstock-Kuzweil and Ito.
June 1. Deterministic integration: importance of bounded variation.
June 3. Functional analysis and probability: topology vs sigma-algebra.
June 8. Functional analysis: the main theorems.
June 10. Stochastic analysis: the foundations.
June 15. Martingales and the idea of the proof.
June 17. Quasimartingales and the proof.
June 22. What is behind: Theorems of Cameron-Martin-Girsanov-Meyer, Doob-Meyer, and Rao.
June 24. Semimartingales and stochastic integrals: general properties
June 9. Martingales: inequalities, convergence, and change of measure.
July 1. A summary: examples and counterexamples.
July 6. A problem-solving session.