Stochastic Processes (MATH136/STAT219, Winter 2021)

This course prepares students to a rigorous study of Stochastic Differential Equations, as done in Math236. Towards this goal, we cover -- at a very fast pace -- elements from the material of the (Ph.D. level) Stat310/Math230 sequence, emphasizing the applications to stochastic processes, instead of detailing proofs of theorems. A critical component of Math136/Stat219 is the use of measure theory.

The Stat217-218 sequence is an extension of undergraduate probability (e.g. Stat116), which covers many of the same ideas and concepts as Math136/Stat219 but from a different perspective (specifically, without measure theory). Thus, it is possible, and in fact recommended to take both Stat217-218 and Math136/Stat219 for credit. However, be aware that Stat217-218 can not replace Math136/Stat219 as preparation for a study of Stochastic Differential Equations (i.e. for Math236).

Main topics of Math136/Stat219 include: introduction to measurable, Lp and Hilbert spaces, random variables, expectation, conditional expectation, uniform integrability, modes of convergence, stationarity and sample path continuity of stochastic processes, examples such as Markov chains, Branching, Gaussian and Poisson Processes, Martingales and basic properties of Brownian motion.

Prerequisites: Students should be comfortable with probability at the level of Stat116/Math151 (summary of material) and with real analysis at the level of Math115. Past exposure to stochastic processes is highly recommended.

Text: Download the course lecture notes and read each section of the notes prior to corresponding lecture (see schedule). When doing so, you may skip items excluded from the material for exams (see below) or marked as ``omit at first reading'' and all ``proofs''. Alternatively, view prior to each lecture the relevant pre-recorded annotated reading from the notes, or go over the slides for each lecture (as posted on Canvas). Kevin Ross short notes on continuity of processes, the martingale property, and Markov processes may help you in mastering these topics.

Supplementary material: (online, or on reserve at science library).

Meeting: Tu/Th 8:30-9:50pm (except: Tu 3/16 5:30-7:00pm). Synchronous recorded discussions, with breakout, anonymous polling and a TA monitored chat-line, serving also as instructor's (public) office hours.

Instructor: Amir Dembo. For questions on material, use our Piazza page, or TA office hours, or e-mail adembo at (with MATH136/STAT219 as subject), for setting a (confidential) private office-hours meeting.

CA1 (HW1/HW3/HW5/HW7/HW9; chat Tue meetings): Sky Cao, office hours Mo 12:00-1:30pm, Tu 12:00-1:30pm (to 3/17) or e-mail skycao at (with MATH136/STAT219 as your email subject).

CA2 (HW2/HW4/HW6/HW8/HW9; chat Thu meetings): Youngtak Sohn, office hours Fr 5:30-7:00pm, Mo 5:30-7:00pm, (to 3/16) or e-mail youngtak at (with MATH136/STAT219 as your email subject).

Grading : Judgement based on two Midterm exam marks (36% each) and on consistent Homework (22%) and Participation (6%) efforts (see Panopto recorded introduction on Canvas). At least 60% required for CR grade.

Midterm 1: Open books, timed 1.5h exam, taken via Gradescope within a 16h frame starting 6:00am PST on Th 2/18 (upload frame ends 10:00pm PST, Th 2/18).

Material: Sections 1.1-3.3 and 5.1 of lecture notes, except: all of Section 2.2; from Section 2.4: up to 2.4.3; from Section 3.1: the cylindrical sigma-field; from Section 3.3: Fubini's theorem (practice exam+solution posted on Canvas).

Midterm 2: Open books, timed 1.5h exam, taken via Gradescope within a 16 frame starting 6:00am PST on Th 3/18 (upload frame ends 10:00pm PST, Th 3/18).

Material: Everything in lecture notes, except: all of Section 2.2; from Section 2.4: up to 2.4.3; Section 4.1.2; all of Sections 6.2-6.3; everything marked as ``omit at first reading'' and all ``proofs'' unless done during lectures (over 90% of exam shall be from Sections 4.1--6.2). Practice Exam Posted (Canvas), solutions provided 3/16 (via Gradescope).

Study tools: List of key items, Exercises 4.3.20, 4.4.6, 4.5.4, 4.6.7, 5.1.8, 5.2.6, 5.3.9 and 6.1.19 are from previous Midterm2 exams.

Homework of 2021: Problems from the text as listed on HW1--HW9 ( Posted! ), are to be submitted through Gradescope each Tuesday at 6:30pm (no grading of late submissions). Collaboration allowed in solving the problems, but you are to provide your own independently written solution. Your assignment will typically be graded and returned on Gradescope the following week. Solutions are posted (on the course Canvas page), within 48h of due date.

Schedule (Read corresponding sections of notes before class):

1/11       Tu(1.1/1.2.1/1.2.2)       Th(1.2.3/1.3.1) 
1/18       Tu(1.3.2/1.4.1)           Th(1.4.2/1.4.3/2.1)   
1/25       Tu(2.1/2.3)               Th(2.4/3.1)
2/1        Tu(3.2.1/3.2.2)           Th(3.2.3/3.3)
2/8        Tu(5.1/4.1.1)             Th(4.1.1/4.1.3)
2/15       Tu(Review:1-3+5.1)        Th(Asynchronous:4.2/4.3.1)               
2/22       Tu(4.3.1/4.3.2)           Th(4.4.1/5.2)
3/1        Tu(5.3/4.4.2)             Th(4.5/4.6)
3/8        Tu(6.1)                   Th(6.1/6.2)
3/15       Tu(Review:4-6)            Th(--)          

Approximately equivalent material (outdated):

List of key items: