Probability Theory II
Spring 2011

Instructor: Ben Steinhurst
Email: steinhurst at math dot cornell dot edu. (When you send an email message, please identify yourself at the end.)
Office hours: W 1:30-3:30 and by appointment
TA
Pengsheng Ji
Email: pj54 `at' cornell.edu
Office Hours: Th 1:30-2:30 Malott 115
Course info:
Lecture MW 10:10 - 11:25 in Malott 205
Course description: The goal of the course is to cover stochastic integration in as much dept as time allows. This will cover along the way Brownian motion and continuous time martingales. We will explicitly be using the topics covered in Math6710 Fall 2010 at the starting point for this course.
Course evaluation:
Will be by written homework assigned every two or three weeks and by a final presentation. Written homework will not be accepted late without prior approval. Feel free to discuss and work on the homework with your classmates but you must deliver your own typed version to recieve credit.
Course documents:
  • R. Durrett, Probability: Theory and Examples, 4th edition. Electronic version PDF. The hardcover version is now available and the campus bookstore may or may not have it but the other usual places most likely do.
  • Richard Bass's notes on probability theory (Spring 2008 version and `Stochastic calculus with applications to finanace PDF and potential theory') here will also be used.
  • Revuz and Yor pages about Feller processes here.

Lecture List with assignments

This list will mostly be updated after the fact as a reference for what we have discussed and to post homework assignments.

Lecture Date Topics Discussed See
1 24 January Administrative details, a doomed proof of the existence of Brownian motion. Durrett 8.1
2 26 January Kolmogorov's extension theorem, a successful proof of the existence of Brownian motion. Durrett A.3, 8.1
3 31 January Some path properties of Brownian motion and Markov property. Durrett 8.1, 8.2.
Homework Due 7 February
4 2 February Snow day The sky
5 7 February Blumenthal's 0-1 Law and more consequences of the Markov property. Durrett 8.2.
6 9 February Strong Markov Property, Zeroes of Brownian motion. Durrett 8.2 and 8.3
7 14 February Properties of hitting times of Brownian motion. Durrett 8.3
Homework Due 21 February
8 16 February Modulus of continuity of Brownian motion. Continuous time martingales. Durrett 8.4 and 8.5, Bass Sections 2 and 5.
9 21 February Martingales and path hitting properties of Brownian motion. Durrett 8.5
10 23 February Law of the Iterated Logarithm Bass Sections 3 and 6
11 28 Feubrary Skorokhod Representation Theorem and weak convergence. Durrett 8.6
Homework Due 7 March
12 2 March Square Integrable martingales. Bass Section 7
13 7 March Snow day II The sky
14 9 March Quadratic variation of square integrable martingales. Bass Section 7
15 14 March Doob-Meyer Decomposition by Nate 2003 version of Bass Section 10.A plus corrections provided by Nate
16 16 March Doob-Meyer Decomposition by Nate completed and Introduction to Stochastic Integrals by Martingales 2003 version of Bass Section 10.A plus corrections and Bass Section 8.
17 28 March Stochastic integrals and semi-martingales Bass Section 8
Solution to problem 5 on HW3.
Homework Due 4 April
18 30 March Ito's Formula Bass Section 9
19 4 April Applications of Ito's Formula, Intro to Girsonov Transformation Bass Section 10, 11
20 6 April Proof of Girsonov Transformation, Intro to SDEs Bass Section 11, 12
21 11 April Existence and Uniqueness of Solutions to SDE Bass Section 12
22 13 April BGS Inequalities and intro to Financial models. Bass Sections 13, 15 and 16
23 18 April Block-Scholes Theorem, Fundamental Theorem of Finance, Feller semi-groups. Bass Sections 16 and 18, Revuz and Yor pages.
24 20 April Generators of Feller processes, Semigroups and resolvents, and Martingale problem. Revuz and Yor pages
25 25 April Martingale problem and PDE, Poisson Equation Bass PDE notes Sections 7, 14, 20, 21, and 22.
26 27 April Dirichlet, Cauchy, and Schrodinger Problems Bass PDE Notes 8, 9 and 10.
27 2 May First Day of Presentations Gabriel, Alex
28 4 May Second Day of Presentations Felipe Mathav
Special 13 May Third Day of Presentations 9am-12pm Malott 205 Cyrus, Siva, Yue, Zhuo, Cristina

Back to my page.
Back to the Math department.
Lasted Updated: 13 May 2011.