Stochastic Calculus, Fall 2004

Communication

To help people communicate with each other, there is a class bboard. You will get a popup asking for a userid and password. Use your NYU netid (mine is jg10) and corresponding password. Do not use your userid and password for the CIMS network. Please check this regularly since I will also post announcements there. If you have questions or problems with the homework or notes, please post them rather than emailing them to me. This way everyone can see them.

Course Description

Discrete dynamical models (covered quietly): Markov chains, one dimensional and multidimensional trees, forward and backward difference equations, transition probabilities and conditional expectations, algebras of sets of paths representing partial information, martingales and stopping times. Continuous processes in continuous time: Brownian motion, Ito integral and Ito's lemma, forward and backward partial differential equations for transition probabilities and conditional expectations, meaning and solution of Ito differential equations. Changes of measure on paths: Feynman--Kac formula, Cameroon--Martin formula and Girsanov's theorem. The relation between continuous and discrete models: convergence theorems and discrete approximations. Measure theory is treated intuitively, not with full mathematical rigor.

The course requires a working knowledge of basic probability, multivariate calculus, and linear algebra. The first homework assignment is a review of basic probability. It is due on the first day of class to ensure that all students start the class with the tools to succeed. The FAQ. has references and hints on how to review and fill in any missing background

Outline

• Week 1: Discrete tree models and Markov chains: transition probabilities, the forward and backward equations and their duality relations. Application to simple random walk.
• Week 2: Increasing algebras of sets to represent increasing information, conditional expectation as projection, nonanticipating functions and stopping times.
• Week 3: Martingales, the martingale property for conditional expectations, martingales and stopping times (Doob's stopping time theorem).
• Week 4: Multivariate normal random variables and the associated linear algebra for sampling and marginal and conditional probability densities. The central limit theorem for iid random variables.
• Week 5: Brownian motion as a multivariate normal (not entirely rigorous). The Brownian bridge construction. The independent increments and Markov properties of Brownian motion. Definition of conditional expectations and conditional probabilities.
• Week 6: The relationship between Brownian motion and partial differential equations. Evolution (forward) of transition probabilities, and (backward) of conditional expectation. Hitting probabilities and the reflection principle.
• Week 7: Sets of paths, partial information, and conditional expectation as projections in continuous time(not entirely rigorous). Martingales and the martingale property of conditional expectations. Progressively measurable functions.
• Week 8: The Ito integral with respect to Brownian motion. Convergence of approximations for Lipschitz progressively measurable functions under the Brownian bridge construction. Examples.
• Week 9: Ito's lemma and Dynkin's theorem as tools for solving Ito differential equations and Ito integrals. Geometric Brownian motion and other examples.
• Week 10: Partial differential equations for transition probabilities and conditional expectations for general Ito differential equations. Applications to hitting times and stopping times.
• Week 11: Change of measure, Feynman Kac, and Girsanov's theorem.
• Week 12: Convergence of random walks and tree models to Ito processes (Donsker's theorem, stated, not proved). Applications to approximations of hitting times in tree models and stopping times in sequential statistics.
• Week 13: Approximation of Ito processes by trees. Applications to approximate solution of forward and backward partial differential equations and to simulating Ito processes.

Assignments

• Assignment 1, given summer, due September 9, first day of class. The PDF version, the Postscript version, the LaTeX source file. Last revised May 26.
• Assignment 2, given September 9, due September 23. The PDF version, the Postscript version, the LaTeX source file. Last revised September 10. Note: Assignment 2 was not changed, but it will be due in two weeks rather than in one week. Please start working on the first two questions right away.
• Assignment 3, given September 16, due September 30. The PDF version, the Postscript version, the LaTeX source file. Last revised September 16.
• Assignment 4, given October 1, due October 14. The PDF version, the Postscript version, the LaTeX source file. Last revised October 1.
• Assignment 5, given October 7, due October 21. The PDF version, the Postscript version, the LaTeX source file. Last revised October 7.
• Assignment 6, given October 21, due October 28. The PDF version, the Postscript version, the LaTeX source file. Last revised October 23.
• Assignment 7, given November 4, due November 11. The PDF version, the Postscript version, the LaTeX source file. Last revised November 9.
• Assignment 8, given November 12, due November 18. The PDF version, the Postscript version, the LaTeX source file. Last revised November 12.
• Assignment 9, given December 14, due December 23. The PDF version, the Postscript version, the LaTeX source file. Please put the completed final exam in my mailbox in Warren Weaver Hall by the end of the day on December 23 at the latest. Please do not consult other people or books for this assignment. Last revised December 22 to fix a very important formula.

Lecture Notes

• Lecture 1, last revised September 12, the PDF version, the Postscript version, and the LaTeX source.
• Lecture 2, last revised September 16, the PDF version, the Postscript version, and the LaTeX source.
• Lecture 3, last revised September 23, the PDF version, the Postscript version, and the LaTeX source.
• Lecture 4, last revised October 1the PDF version, the Postscript version, and the LaTeX source. This is a very rough draft and will be revised hopefully over the next few days.
• Lecture 5, last revised October 21, the PDF version, the Postscript version, and the LaTeX source.
• Lecture 6, last revised October 23, the PDF version, the Postscript version, and the LaTeX source.
• Lecture 7, last revised November 9, the PDF version, the Postscript version, and the LaTeX source.
• Lecture 8, last revised Dec 9, the PDF version, the Postscript version, and the LaTeX source.