Stochastic Processes (MATH136/STAT219, Autumn 2013)

This course prepares students to a rigorous study of Stochastic Differential Equations, as done in Math236. Towards this goal, we cover elements from the material of Stat310/Math230 sequence, emphasizing the applications to stochastic processes, instead of detailing proofs of theorems (see also comparison with Stat217/218 and Stat310/Math230).

Main topics are introduction to measurable, Lp and Hilbert spaces, random variables and (conditional) expectation, uniform integrability and 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/Math105/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 math. library)

Meeting: 380-380Y, MWF 12:15-1:05.

Instructor 1: Amir Dembo, Sequoia 129, F 1:15-2:30, lecture/office-hours, on Oct 19 - Nov 24; or e-mail amir at stat.stanford.edu (please include MATH136 in your email title).

Instructor 2: Tianyi Zheng, 380-382H, F 3:10-5:10, lecture/office-hours relevant on Sep 23 - Oct 18, Dec 1 - Dec 9. or e-mail tzheng2 at math.stanford.edu (please include MATH136 in your email title).

CA1 (grading HW1/HW3/HW5/HW7/HW9) : Li-Cheng Tsai, 380-380H, M 2:00-4:00 and W 11:00-12:00, Sep 30 - Dec 9, or e-mail lctsai at math.stanford.edu (please include MATH136 in your email title).

CA2 (grading HW2/HW4/HW6/HW8/HW9) : Ruojun Huang, Sequoia 241, Tu 10:30-12:00 and Tu 10:30-12:00. or email hruojun 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: Friday, 11/1, 12:10-1:10, alternate seating in room 370-370, exam's solution . 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: from Section 1.4: uniform integrability; all of Section 2.2; from Section 2.4: up to 2.4.3; from Section 3.1: the cylindrical sigma-field. practice exam and its solution.

Final: Monday, 12/9, 8:30-11:30,alternate seating in McCullough 115, exam's solution . 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, 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.

Homework of 2013: Problems from the text as listed on HW1--HW9, are due 12:15 p.m. on a weekly basis (see dates below). 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). Homework solutions posted on Fridays and graded assignments returned in class on the following Wednesday (HW9 returned in class 12/6). HW1 due Wednesday 10/2, solution; HW2 due Wednesday 10/9, solution; HW3 due Wednesday 10/16, solution; HW4 due Wednesday 10/23, solution; HW5 due Wednesday 10/30, solution; HW6 due Wednesday 11/6, solution; HW7 due Wednesday 11/13, solution; HW8 due Wednesday 11/20, solution; HW9 due Wednesday 12/4, solution. All solutions posted!,

Schedule (Read corresponding sections of notes before class):

9/23       M(1.1)           W(1.2.1/1.2.2)    F(1.2.2/1.2.3)        
9/30       M(1.3.1/1.4.1)   W(1.4.1/1.3.2)    F(1.4.2)
10/7       M(1.4.2/1.4.3)   W(2.1.1/2.1.2)    F(2.3)
10/14      M(3.1)           W(3.1/3.2.1)      F(3.2.2)
10/21      M(3.2.3/3.3)     W(5.1)            F(4.1.1)   
10/28      M(4.1.3/2.4)     W(2.4/Review:1-3) F(Midterm)      
11/4       M(4.2)           W(4.3.1)          F(4.3.1/4.3.2)     
11/11      M(5.2/4.4.1)     W(4.4.1/4.4.2)    F(4.5/4.6)           
11/18      M(5.3)           W(6.1)            F(6.1)               
11/25      M(---)           W(---)            F(---)
12/2       M(6.2)           W(6.2;Review:4-6) F(Review+Q/A)

Approximately equivalent material (outdated):

List of key items: