MAT 486 Random Processes (Fall 2019)
Instructor Allan
Sly (asly@princeton.edu)
Syllabus On blackboard or here.
Class time 11:00 - 12:30 on Mondays and Wednesdays at 401 Fine Hall.
Office hours
Instructor: Monday 12:30-2:00 or by arrangement. Room 405 Fine Hall.
Special refresher session: Friday September 13, 11:00 - 12:30. Room 405 Fine Hall.
Course Description
The course will begin with a short introduction to concentration of measure and large deviations, essential topics in the analysis of random processes. The largest topic will be on diffusions which will include the following:
- Gaussian processes and construction of Brownian motion.
- Properties of Brownian motion.
- Stochastic Calculus and Ito's formula.
- Stochastic differential equations.
- Brownian excursions and the continuum random tree.
- Applications of Brownian motion to partial differential equations.
For the remainder of the course we will cover additional topics to be selected based on a poll of student interest in the first class.
Grading
The grade for the course will be made up of problem sets (50%) and a final take home exam (50%) after scaling.
Probability is a subject where actively working on problems is essential to your understanding of the material, it is not enough simply to listen in class or read the text. There will be a weekly problem set which will be posted each Wednesday (starting September 18) on blackboard and will be due the following Wednesday in class. These can be done in pairs (or with permission groups of 3 if there is an odd number of students in the class). You should list who you worked with on the problem set. Each of you should write your own solutions separately - this is important to make sure you understand the solutions yourself. I also strongly encourage you to try the problems yourself first before working with your partner.
There will be a take home final exam covering all the material of the course. This must be done individually between January 15 and January 21.
Text Books
We will use two texts, both of which have free online copies:
- Probability: Theory and Examples by Rick Durrett.
- Theory of Probability and Random Processes by Koralov and Sinai. You can access a free to download copy through the library website.
These will be supplemented by other readings.
Probability refresher
I have put together a sheet of definitions, theorems and facts from probability theory that I will be assuming that you know from a previous course. Please read over it and come and talk to me if any of the concepts are unfamiliar or if there is anything you'd like me to clarify. [Notes]
Schedule
Sep 11: Probabalistic Inequalities. Durrett 1.6, 2.7 [Notes]
Sep 16: Large Deviation Theory. Durrett 2.7 [Notes]
Sep 18: Azuma-Hoeffding Inequality.
See text of Roch Chapter 3.2 [Notes]
Sep 23: Gaussian vectors and processes. KS 9.3 Durrett 3.10 [Notes]
Sep 25: Properties of Brownian Motion. Durrett 7.3, 7.4 [Notes]
Sep 30: Donsker's Theorem. Durrett 8.1 [Notes]
Oct 2: More properties of Brownian Motion. Durrett 7.3, 7.4 [Notes]
Oct 6: Stochastic Integration. Durrett 7.6 [Notes]
Oct 8: Ito's Formula. Durrett 7.6 [Notes]
Oct 14: General stochastic integration. Durrett 7.6, KS 20.1, 20.2 [Notes]
Oct 16: Quadratic variation and general stochastic integration. KS 20.1, 20.2 [Notes]
Oct 20: Motivations for SDEs [Notes]
Nov 4 : Strong solutions for SDEs (Karatzas and Shreve 5.2) [Notes]
Nov 6 : Strong solutions for SDEs (Karatzas and Shreve 5.2) [Notes]
Nov 11 : Local Times and Tanaka's Formula (Karatzas and Shreve 3.7) [Notes]
Nov 13 : Weak solutions for SDEs and Girsanov's Formula (Karatzas and Shreve 3.3, 5.3) [Notes]
Nov 18 : Calculations for one dimensional diffusions [Notes]
Nov 23 : Brownian motion and the solution to the Dirichlet Problem (Durrett 9.5) [Notes]
Dec 2 : Hausdorff Dimension of Brownian motion [Notes]
Dec 4 : Random Graphs and diffusions [Notes]
Dec 9 : Random Graphs and diffusions [Notes]
Dec 11 : Random Graphs and diffusions [Notes]