Theory of Probability (MATH230C/STAT310C, Spring 2019)

The third quarter in a yearly sequence of probability theory serves as an introduction to the theory of continuous time stochastic proceses. Covering continuity and modification, Gaussian and Markov processes, continuous time martingales, Brownian motion and its properties, invariance principles with applications to CLT and LIL.

Prerequisites: Students should have mastered a graduate probability course covering conditional expectation, discrete time martingales and Markov chains. Specifically, you may take this class for credit if you had at least grade B in Stat310B/Math230B. Otherwise, you'll need instructor's permission for doing so.

Posted lecture notes: Chapters 8-10 of STAT310 notes; See also list of updates

Supplementary texts:

• Durrett, Probability: Theory and Examples, 4th edition (Ch. 8).
• Karatzas and Shreve, Brownian Motion and Stochastic Calculus, 2nd edition (Ch. 1-2).
• Billingsley, Probability and Measure, 3rd edition (Ch. 7).
• Dudley, Real Analysis and Probability (Ch. 12).
• Breiman, Probability (Ch. 12--15).
• Morters and Peres, Brownian motion (Ch. 1,2,5).
• Kallenberg, Foundations of Modern Probability (Ch. 12--14 and part of Ch. 17--19).

Meeting: Sequoia 200, Tu/Th, 9:00-10:20 a.m.

Instructor: Amir Dembo, Office hours: Thu 4:30-5:30 p.m. at Sequoia 129 or e-mail

TA1 (grading HW1, HW2, HW5, HW7, HW9): Theo Misiakiewicz, Office hours: Mon 5:00-7:00 p.m. at Sequoia 220, Wed 4:30-5:30 p.m, at 420-245, or e-mail .

TA2 (grading HW3, HW4, HW6, HW8, HW9): Andy Tsao, Office hours: Tue 10:30-11:30 at Sequoia 220, Fri 10:30-12:30, at Sequoia 105, or e-mail .

Grading : Judgement based on final exam mark (73%) and on consistent Homework efforts (27%). Typically, total score above 63% needed for passing.

Final exam: Monday, June 10, 8:00-12:00 p.m., Sequoia 200; Bring only lecture notes and HW solutions; Material: Chapters 8-10 of lecture notes. To prepare, see practice final , with solution, and list of key-items.

Homework: Four HW problems out of the five assigned are due Tue at 10:30 a.m. on a weekly basis. See HW1--HW9 and their solution. Please deliver your assignment to the lecture or to instructor's mailbox in Sequoia, by the due time. Late homework will not be accepted. Solutions posted on the course Canvas page (within 36h). Graded homework returned in class (within a week).

Syllabus (per lecture notes):

        4/1   Tu(8.1)           Th(8.2)        
	4/8   Tu(8.2/8.3)       Th(9.1)        
	4/15  Tu(9.2.1/9.2.2)   Th(9.2.2/9.2.3)         
	4/22  Th(9.2.3/9.3.1)   Th(9.3.1/9.3.2)
        4/29  Tu(9.3.2/10.1)    Th(10.1)     
	5/6   Tu(10.2)          Th(10.2/10.2.1)        
        5/13  Th(10.2.1/10.2.2) Th(10.2.2/10.3)              
	5/20  Tu(10.3/9.2.4)    Th(9.2.4)           
	5/27  Tu(9.2.4/9.3.3)   Th(9.3.3)            
	6/3   Tu(Review:TA1)

Material covered, including HWs and self-reading:

• Durrett: Ch. 8 and Karatzas and Shreve 1-2; Supplements from Breiman 12-15, Billingsley 36-38 and Dudley 12.

See also seminar for current activity in related areas.