Stochastic Processes (MATH136/STAT219, Winter 2019)

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 alone is NOT adequate preparation 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 (syllabus). 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''. Kevin Ross short notes on continuity of processes, the martingale property, and Markov processes may help you in mastering these topics.

Supplementary material: (texts on reserve at science library)

Meeting: 200-305, Mo/We 1:30-2:50, replacement lectures: 380-380F, Fr 1:30-2:50, 1/25, 2/22 and 3/8.

Instructor: Amir Dembo, office hours: Seqouia 129, Mo 2:55-3:45 p.m. or e-mail adembo at stanford.edu (please include MATH136 in your email title).

CA: Panagiotis Lolas, office hours: 380-380T, Tu 3:00-5:00 p.m., We 10:00-11:00 a.m. or e-mail panagd at stanford.edu (please include MATH136 in your email title).

Grading : Judgement based on Final (50%) and Midterm (25%) exam marks and on consistent Homework efforts (25%). At least 60% required for CR grade.

Midterm: Tuesday 2/12, 6:00-7:30 p.m., Room 380-380C. Three pages of notes (2 sides each) allowed, handwritten or computer generated, at any font readable without artificial magnification. Material: Sections 1.1-3.3 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 and its solution.

Final: Wednesday, 3/20, 3:30-6:30, Room 380-380X. Six pages of notes (2 sides each) allowed, handwritten or computer generated, at any font readable without artificial magnification.

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 Section 6.3; everything marked as ``omit at first reading'' and all ``proofs'' unless done during lectures (80% of exam shall be from Sections 4.1--6.2).

Study tools: List of key items and Final-2019: Exam Format. Exercises 4.3.20, 4.4.6, 4.5.4, 4.6.7, 5.1.8, 5.2.6, 5.3.9, 6.1.19 and 6.2.12 are from previous finals. See also practice exam and its solution, or Final-2017: with solution.

Homework of 2019: Problems from the text as listed on HW1--HW9, are due each Wednesday 1:30 p.m. on a weekly basis ( Solutions: see Canvas page) Collaboration allowed in solving the problems, but you are to provide your own independently written solution. Please deliver your assignment to class on due date (late homework solutions will not be graded).

Schedule (Read corresponding sections of notes before class):

1/7         Mo(1.1/1.2.1)             We(---) 
1/14        Mo(1.2.1/1.2.2/1.2.3)     We(1.3.1/1.4.1)      
1/21        Mo(---)                   We(1.3.2/1.4.2)     Fr(1.4.3/2.1/2.3)
1/28        Mo(2.3/2.4)               We(3.1) 
2/4         Mo(3.2.1/3.2.2)           We(3.2.3/3.3/5.1)       
2/11        Mo(Review:1-3)            We(5.1/4.1.1/4.1.3)                       
2/18        Mo(---)                   We(4.1.3/4.2)       Fr(4.3.1/4.3.2)
2/25        Mo(5.2/4.4.1)             We(4.4.2/4.5/4.6)
3/4         Mo(4.6/5.3)               We(6.1)             Fr(6.1/6.2)
3/11        Mo(6.2)                   We(Review:4-6)          

Approximately equivalent material (outdated):

List of key items: