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.
Prerequisites
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
Each assignment is due on the date given. I will accept homeworks up
to one week late. Late homework will have a 10% penelty, i.e. a maximum
of 18 points instead of 20.
- 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
I will revise lecture notes from the 2004 class and post them here.
The old versions are full of mistakes and omissions. The revised versions
may be a little better. See the
page from 2004
for the old versions.
- 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).