Stochastic Calculus, Fall 2004

Course Home Page

Lecture Problem session
Thursdays from 5:10 to 7pm
Room 713 Silver Center, NYU
Starting September 9, 2004
Mondays from 5:30 to 6:30pm
Room 1302, Warren Weaver Hall
Starting September 13, 2004
Jonathan Goodman
Office hours: Wednesday, to be determined
Office: 617, Warren Weaver Hall, NYU
Teaching assistant
(to be determined)
(to be determined)
Office hours: to be determined
Program in Financial Mathematics
Department of Mathematics
Courant Institute of Mathematical Sciences

Questions or comments

If you have questions, please check the FAQ, then contact me or the teaching assistant.

Before you register, read carefully

To make sure everyone in the class has the prerequisites, the first assignment is due at the first class. See the Prerequisites section below and find the assignment at the bottom of the page.


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


  • 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.


Lecture Notes

I will post some lecture notes. I will revise posted notes as errors and confusions are reported, so check the date to make sure you have the latest version.