Stochastic Processes (MATH136/STAT219, Autumn 2017)

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 science library)

Meeting: 380-380D, TuTh 9:00-10:20.

Instructor: Amir Dembo, Sequoia 129, Mo Nov 27, 2:00-3:00 p.m.; Th Dec 7, 4:30-5:30 p.m. or e-mail adembo at stanford.edu (please include MATH136 in your email title).

CA : Yue Hui 380-380L, Tu Nov 28, Dec 5, 2:00-3:00; W Nov 29, Dec 6, 3:30-5:30; Mo Dec 11, 2:00-4:00; Tu Dec 12, 2:00-4:00 or e-mail yueh 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: (solution) Tuesday, 10/31, 6:00-7:30 p.m., alternate seating in 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: from Section 1.4: uniform integrability; all of Section 2.2, all of Section 2.4; from Section 3.1: the cylindrical sigma-field. practice exam and its solution.

Final: Wednesday, 12/13, 8:30-11:30, alternate seating in 380-380D. 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 2017: Problems from the text as listed on HW1--HW9, are due each Thursday 10:20 a.m. on a weekly basis. 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). Graded assignments returned in class the following Thursday. For solutions, see SOL1, SOL2, SOL3, SOL4, SOL5, SOL6, SOL7 Posted! SOL8, SOL9.

Schedule (Read corresponding sections of notes before class):

9/25         Tu(1.1/1.2.1)           Th(1.2.1/1.2.2/1.2.3)        
10/2         Tu(1.3.1/1.4.1)         Th(1.3.2/1.4.2)
10/9         Tu(1.4.2/1.4.3/2.1.1)   Th(2.1.2/2.3)
10/16        Tu(3.1)                 Th(3.2.1/3.2.2)
10/23        Tu(3.2.3/3.3/5.1)       Th(5.1/4.1.1)   
10/30        Tu(Review:1-3)          Th(4.1.3/2.4)           
11/6         Tu(4.2/4.3.1)           Th(4.3.1/4.3.2)     
11/13        Tu(5.2/4.4.1)           Th(4.4.2/4.5/4.6)           
11/20        Tu(---)                 Th(---)            
11/27        Tu(5.3/6.1)             Th(6.1)               
12/4         Tu(6.2)                 Th(Review:4-6) 

Approximately equivalent material (outdated):

List of key items: