MAP 6468 – Stochastic Differential Equations and High-Dimensional Probability

The course is on Canvas

Homework

Homework #1
Homework #2
Homework #3
Homework #4

 

Lecture Notes

Simulation of random variables

Syllabus

Syllabus

Time and Location

M-W-F Period 5 (11:45 AM – 12:35 PM), TURL 2328

Office Hours

Monday 10am – 10:50am, Wednesday 10am – 10:50am, or by appointment

Textbook

There is no required text, but the following textbooks are suggested:

  • G. F. Lawler, Stochastic Calculus: An Introduction with Applications (PDF available on Prof. Lawler’s website).
  • P. E. Protter, Stochastic Integration and Differential Equations, Springer.
  • K. Ball, An elementary introduction to modern convex geometry, Cambridge University Press (PDF available on Prof. Ball’s website).
  • R. Vershynin, High-Dimensional Probability, An Introduction with Applications in Data Science, Cambridge University Press (PDF available on Prof. Vershynin’s website).

Scope of the course

The aim of the course is to provide students with strong foundations in the area of probability theory. At the end of the course, students will be acquainted with the language of probability and will gain sufficient experience to successfully apply probabilistic tools to most areas of pure and applied sciences.

The course is intended for graduate students as part of their PhD requirement, and for students considering studying probability theory at a research level.

Prerequisite

MAP 6467 — Stochastic Differential Equations

Topics Covered

Topics include: Stochastic differential equations, Stochastic integral with respect to martingales, Geometric objects in high-dimension, Sphere packing and covering, Dimension reduction: the Johnson-Lindenstrauss lemma, Concentration inequalities.

Weekly Schedule

W1: Stochastic differential equations, Existence and uniqueness of solution.
W2: Simulation.
W3: Construction of stochastic integral with respect to a martingale.
W4: Change of measure, Girsanov’s theorem.
W5: Introduction to high-dimension: distribution of the volume of the Euclidean ball.
W6: John’s theorem, Sections of the cube and Dvoretsky’s theorem.
W7: Dimension reduction: the Johnson-Lindenstrauss lemma.
W8: Geometry of log-concave distributions.
W9: Bourgain’s hyperplane conjecture.
W10: Concentration inequalities.
W11: Sub-Gaussian and sub-exponential distributions.
W12: Concentration for Gaussian and log-concave distributions in Rn.
W13: Shannon entropy.
W14: Asymptotic equipartition property and typicality.