Online Class starting Monday, March 16
Lecture notes for Friday, March 13: Lecture-notes-3-13-2020 (By Joel Mathias)
Lecture notes for Monday, March 16: Lecture-notes-3-16-2020
Lecture notes for Wednesday, March 18: Practice exercises from Homework #3
Lecture notes for Friday, March 20: Solution of Homework #3
Lecture notes for the week of March 23 (March 23 – March 27): Lecture-notes-3-23-27-2020
Lecture notes for the week of March 30 (March 30 – April 3): Lecture-notes-3-30–4-3-2020
Lecture notes for the week of April 6 (April 6 – April 10): Lecture-notes-4-6-10-2020
Lecture notes for the week of April 13 (April 13 – April 17): Lecture-notes-4-13-17-2020
Lecture notes for Monday, April 20: Practice exercises from Homework #4
Lecture notes for Wednesday, April 22: Solution of Homework #4
Homework
Homework #4
Homework #3
Homework #2
Homework #1
Syllabus
Time and Location
M-W-F Period 5 (11:45 AM – 12:35 PM), LIT 221
Office Hours
Monday 1:55 PM – 2:45 PM, Wednesday 1:55 PM – 2:45 PM, or by appointment
Textbook
There is no required text, but the following textbooks are suggested:
- R. Durrett, Probability: Theory and Examples, 5th edition (PDF available on Prof. Durrett’s website)
- P. Morters and Y. Peres, Brownian Motion, Cambridge University Press (PDF available on Prof. Morters’ website)
- D. Khoshnevisan, Probability, Graduate studies in mathematics vol. 80, 2007
Final Exam Date
Thursday, April 30 (4/30/2020) at 10:00 AM – 12:00 PM in LIT 221
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 6472 – Probability and Potential Theory I
Topics Covered
Topics include Conditional Expectation, Martingale, Stopping time, Uniform Integrability, Continuous time stochastic processes (Poisson process, Gaussian process, Brownian motion), Potentials and excessive functions. Below is the weekly schedule:
W1: Review of probability (random enumeration of random variables, sigma-algebra generated by random variables, Lebesgue integral, Lebesgue theorems, etc.).
W2: Conditional expectation, construction in L^2, properties of conditional expectation.
W3: Conditional expectation in L^1, independence.
W4: Conditional expectation with respect to random variables, special case: discrete/continuous.
W5: Martingale, Doob-Meyer decomposition.
W6: Stopping time, Doob’s inequality.
W7: Stopped martingale, stopping theorems.
W8: Convergence of martingales, uniform integrability.
W9: Poisson process.
W10: Brownian motion, Levy’s construction of the Brownian motion.
W11: Blumenthal’s 0-1 law, law of large numbers for Brownian motion, long-term behavior, nowhere differentiability of the trajectories.
W12: The Brownian motion as a Gaussian process, as a martingale, as a Markov process.
W13: The reflexion principle.
W14: Potentials and excessive functions.
Process for online Exam:
On Thursday, April 30 at 10:00am, the final exam will be posted here (course website).
You will have 2 hours to complete the exam.
Then, you will need to scan or take a picture of your test, and send it to me via email at a.marsiglietti@ufl.edu (pdf or jpeg file).
The deadline to submit your test is 12:30pm on Thursday, April 30.
I will acknowledge receipt of your email. If you do not receive an acknowledgement from me by 12:45pm, you need to contact me (and resend your file).
I trust that you will not seek external help.
Please keep in mind the Honor Code at the University of Florida, the following pledge is either required or implied:
“On my honor, I have neither given nor received unauthorized aid in doing this assignment.”