Stochastic Calculus, Spring 2007

Communication

There is an nyuhome page for this class that has a message board for the class. If you are registered for the class, you will be able to get to the class page from your nyuhome account. Please post any questions related to the homework on that message board rather than emailing them to me or the TA. Contact me directly for personal questions.

Course Description

As of December 27, 2006, the course description and outline are exactly as they were for the Fall 2004 version of the class. I plan to revise them to include more about time series.

Discrete dynamical models (covered quickly): 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 December 27, 2006, due January 18, 2007, the first day of class.
• Assignment 2, due February 1, 2007.
• Assignment 3, Revised February 2, due February 8, 2007.
• Assignment 4, due February 15, 2007.
• Assignment 5, due February 22, 2007. Corrected version posted February 21.
• Assignment 6 is cancelled -- don't do it.
• Assignment 7, due March 8, 2007. Corrected version posted March 5.
• Assignment 8, due March 22, 2007.
• Assignment 9, due March 29, 2007, Problem 3 corrected March 25, Problem 1d corrected March 27.
• Assignment 10, Due April 5.
• Assignment 11, Due April 12, revised April 9, April 10 (both for #2c).
• Assignment 12, Due April 19, corrected version posted April 18.
• Assignment 13, Due April 26.
• Practice for the final

Lecture Notes

• Part 1 Last revised Jan. 2.
• Part 2 Last revised Jan. 22.
• Part 3 Last revised Jan. 22.
• Part 4 Last revised March 2. These are very preliminary & in the middle of a revision. Look for a slightly better version in a few days.
• Part 5 Last revised March 7.
• Part 6 Last revised March 7.
• Part 7 Last revised March 29 (barely).
• Part 8 Last revised April 5 (barely).