Room 1302, Warren Weaver Hall, NYU

Starting September 5, 2002

Department of Mathematics

Courant Institute of Mathematical Sciences

If you have questions, please check the
FAQ,
then contact me or the teaching assistant (see the contact information above).

To help people communicate with each other, there is a class bboard. 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.

**Prerequisites: **Basic Probability and Stochastic Processes
(GS63.2901)
or equivalent, multivariate calculus and linear algebra.

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. One way to get the basic probability is to take the Courant Institute course Basic Probability and Stochastic Processes (GS63.2901). Many people will have studied probability elsewhere. The first homework assignment (posted below) exercizes some of the skills you will need for the course. Try it. For more information, see the FAQ.

- Week 1: Quick review of discrete probability, conditioning, and independence. Finite state space Markov chains, transition probabilities, and the space of paths,
- Week 2: Tree models in one or more dimensions. Backward equations for conditional expectation for Markov chains and trees. Sets of paths corresponding to information available at time T. Conditional expectation as a projection, ``progressively measurable'' functions, and stopping times.
- Week 3: Martingales, the martingale property for conditional expectations, stopping times and conditional expectation with respect to a stopping time.
- Week 4: Review of continuous finite dimensional probability: densities, expectations, and conditioning. Multivariate normal random variables and the accompanying linear algebra. 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 (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.

- Assignment 1, given September 5, due September 12. The PDF version, the Postscript version, the source file in LaTeX. Last revised September 4.
- Assignment 2, given September 12, due September 19. The PDF version, the Postscript version, the source file in LaTeX. Last revised September 15.
- Assignment 3, given September 20, due September 26. The PDF version, the Postscript version, the source file in LaTeX. Last revised September 23.
- Assignment 4, given September 26, due
**October 10**. The PDF version, the Postscript version, the source file in LaTeX. Last revised October 7 - Assignment 5, given October 19, due
**October 24**. The PDF version, the Postscript version, the source file in LaTeX. Last revised October 23. - Assignment 6, given October 25, due
**October 31**. The PDF version, the Postscript version, the source file in LaTeX. Last revised October 30 (minor typo in question 1). - Assignment 7, given October 31, due
**OcNovember 7**. The PDF version, the Postscript version, the source file in LaTeX. Last revised October 31. - Assignment 8, given November 7, due
**November 14**. The PDF version, the Postscript version, the source file in LaTeX. Last revised November 8. Although we did not cover Ito's lemma for Xt in class, it is in the notes. This should allow you to do questions 3 and 4. - Assignment 9, given December 5, due
**December 19**. The PDF version, the Postscript version, the source file in LaTeX. Last revised December 11. - The practice final exam, given December 11. The PDF version, the Postscript version, the source file in LaTeX. Note, this is much much longer than the final will be. Think about having five out of these ten questions and fewer true false. Last revised December 11.

- Assignment 2 solutions in Postscript format only.

- Lecture 1, last revised Sept 9 PDF version, the Postscript version, the source file in LaTeX.
- Lecture 4, last revised Sept 17 PDF version, the Postscript version, the source file in LaTeX. These are not finished but may be helpful now. I'll post revisions in a few days.
- Lecture 5, last revised October 5 PDF version, the Postscript version, the source file in LaTeX.
- Lecture 6, last revised October 17 PDF version, the Postscript version, the source file in LaTeX. The brownian motion figure file also exists in eps format or in pdf format.
- Lecture 7, last revised October 23 PDF version, the Postscript version, the source file in LaTeX.
- Lecture 8, last revised October 31 PDF version, the Postscript version, the source file in LaTeX.
- Lecture 9, last revised November 7 PDF version, the Postscript version, the source file in LaTeX.