Math 180C

Fall 2020, Lecture A00 (MWF 5:00-5:50pm)

Introduction to Stochastic Processes II

Announcements
Calendar
Syllabus
Piazza
Gradescope
Homeworks

Announcements

December 21:
- The final exam has been graded. Regrades will be active on Gradescope until Tuesday, December 22, 7 PM.
- Solution to the final are available here pdf.
- Concerning Homework 0: students who indicated wrong student ID in their Gradescope accounts will get 2 points; all other students who submitted Homework 0 will get 5 points.
December 16: The final exam is taking place today
- exam problems will become available on Canvas at 7 PM;
- Zoom meeting will start at 8 PM; use this link to join the meeting (no password)
- you can joint the Zoom meeting until 8:50 PM, but not later; when you join the meeting wait in the waiting room;
- IMPORTANT: the deadline for submitting your exam to Gradescope is 8:40 PM;
- read these instructions to make sure that you are familiar with all the procedures.
December 13:
- Solutions to Homework 9 can be downloaded here pdf.
- Solutions to the practice final can be downloaded here pdf.
December 10: Important announcements
- The official final exam time is Thursday, December 17, 7 PM - 10 PM.
- The instructions for the final exam can be found here. It is your responsibility to read and follow these instructions.
- I would like to thank all the students to filled in the CAPE reports. The response rate reached 80%, therefore the second lowest homework score and the lowest quiz score will be dropped when computing the final grade.
December 9:
- The list of what you are expected to know for the final exam pdf.
- The practice final exam has been posted on Canvas. You do not have to submit anything in Canvas. The solutions will be discusses during the office hours.
December 6:
- Lecture 22-23 is ready (see below).
- Homework 9 has been posted below. You DO NOT have to turn in this homework. The solutions will be posted on this website.
- Quiz 5 will take place on Wednesday, December 9, and will cover the material of lectures 18-20.
December 2: Lecture 21 is ready (see below).
December 1:
- Lecture 20 is ready (see below).
- Homework 8 has been posted below. Due: THURSDAY, December 10, 11:59 PM.
November 29:
- Lecture 19 is ready (see below).
- Homework 6 has been graded. Regrades will be active on Gradescope until Wednesday, December 2, 11:00 PM.
November 28:
- New deadline for Homework 7: Thursday, December 3.
- Solutions to the second midterm can be downloaded here.
November 23:
- Lecture 18 is now ready (see below).
- Homework 7 has been posted below. Due: WEDNESDAY, December 2, 11:59 PM.
- Homework 5 has been graded. Regrades will be active on Gradescope until Tuesday, December 1, 11:00 PM.
November 22:
- Instructions for the second midterm can be found here.
- Check the calendar for updated office hours.
November 18:
- Practice midterm 2 has been posted below. The solutions to the practice midterm will be posted on Thursday..
- Instead of the Q&A session on Friday I will have an additional office hour (50 minutes) during which I will answer your questions about the practice midterm.
November15:
- Homework 6 is available below. Due Saturday, November 21. For problem 6 (Problem 7.4.2 from Pinsky-Karlin), you will get a full credit in you can estimate the linear part of M(t) up to an interval of length one, i.e., if you can find A and D such that M(t) = A t + B + o(1) for large t with B in the interval [D,D+1]. If you find the exact value of B, you will get extra 4 points.
- Homework 3 and Homework 4 have been graded. The regrade requests will be active on Gradescope until Tuesday, November 17, 11 PM.
- Lecture 16 and 17 are now ready (see below).
- Quiz 4 will take place on Wednesday, November 18, and will cover lectures 11-15.
- Midterm 2 will take place on Monday, November 23, and will cover lectures 10-17.
November 8:
- Lectures 14 and 15 are now ready (see below). Both lectures are combined in one video and will be discussed on Monday's and Friday's Q&A sessions.
- Midterm 1 has been graded.
November 6: Homework 5 is available below. Due Friday, November 13.
November 5: Lecture 13 is now ready (see below).
November 3: Lecture 12 is now ready (see below).
November 2: Solutions to the first midterm can be downloaded here.
October 31: Lecture 11 and Homework 4 have been posted below.
October 29: Lecture 10 is now ready (see below).
October 27: Instructions for the first midterm can be found here.
October 22:
- Homework 3 has been posted below. Due Saturday, October 31.
- Here is a list pdf of what you should know for Midterm 1.
- Practice midterm 1 and its solutions has been posted below.
- Instead of Q&A sessions on Monday and Wednesday I will have additional office hours (50 minutes) during which I will answer your questions about the practice midterm.
- In order to give you more time to prepare for Midterm 1 I would like to suggest shifting it by one day, from Wednesday, October 28, to Thursday, October 29; this change of date is only possible if it does not create conflicts for any of the students enrolled in this class; therefore if you cannot take the first midterm on Thursday, October 29 inform me via email BEFORE Saturday, October 24, 1 PM; I will then confirm the change of the date of the first midterm.
October 20: Lecture 9 is now ready (see below).
October 19: Please fill in this Google form link about the issues you are experiencing with the remote learning.
October 18:
- Lectures 7 and 8 are now available (see below).
- Homework 1 has been graded. The regrade requests will be active on Gradescope on Thursday, October 22, 8am - 11pm.
- Reminder: Homework 2 is due on Friday, October 23.
- Reminder: Quiz 2 will take plae on Wednesday, October 21 and will cover the material discussed in lectures 4-6.
October 15:
- Lecture 6 is now available (see below).
- The correct answers to the first quiz are available on Canvas.
- Homework 2 can be downloaded below. Due Friday, October 23.
October 13: Lecture 5 is now available (see below).
October 11:
- Lecture 4 is now available (see below).
- Quiz 1 will take place on Wednesday, October 14, on Canvas (Canvas Quizzes) and will cover lectures 1-3.
- There will be no homework due this week
October 8: There was a mistake on p. 9 of Lecture 2: the tilde sign ~ was missing in several places. There was also a small mistake on p. 17 of Lecture 1: sum starting from 1 instead of 0. I updated the slides.
October 7: You may find the following note pdf useful for solving Problem 6.1.5 from Homework 1.
October 5: This week Jiaqi Liu's office hours will be held on Thursday, October 8, 4:00 - 6:00 PM. You can find the Zoom link below.
October 3:
- You can use paper or tablets to write your homework. You can also type it if you prefer (using e.g. LaTeX).
- Note that the midterm and final exams have to be handwritten, either on paper or using a tablet.
- You will not have to upload any documents for quizzes.
October 1: Homework 1 is now available for download (see below). Due Friday, October 9, 11:59pm.
September 30: on Friday, October 2, 5:00 - 5:50pm, there will be a live Q&A session (Zoom link) about the structure of the course and related administrative questions; attendence is not mandatory. There will be no discussion sections on Thursday, October 1.
September 28: Homework 0 is now available for download (see below). Due Wednesday, October 7, 11:59pm.
September 28: Welcome to MATH 180C: Introduction to Stochastic Processes II!

Course Information

Textbooks:
- The required textbook for this course is
  
  An Introduction to Stochastic Modeling, by Mark A. Pinsky and Samuel Karlin; 4th edition
  Reading and homework assignments will refer to this textbook as PK.
- For certain selected topics of this course we will additionally use the following textbook
  
  Essentials of stochastic processes, by Richard Durrett; 3rd edition
  Reading and homework assignments will refer to this textbook as Durrett.
- The links above go to UCSD library website, where the textbooks can be downloaded freely and legally if you are connecting from a UCSD IP address.
Coursework: There will be
- homework assignments due roughly every week, typically on Fridays (starting in Week 1); they are posted below;
- quizzes taking place on Wednesdays of Weeks 2, 3, 5, 7 and 10; details below;
- two midterm exams on Weeks 4 and 8; details below;
- and a final exam; details below.
The list of the important dates can be found here.

Piazza is an online discussion forum. It will allow you to post messages (openly or anonymously) and answer posts made by your fellow students, about course content, homework, exams, etc. The instructors and TAs will also monitor and post to Piazza regularly.
- For all questions related to the course, ask your questions on Piazza before contacting the instructor/TAs directly.
- You can sign up here.
- Note: Piazza has an opt-in "Piazza Careers" section which, if you give permission, will share statistics about your Piazza use with potential future employers. It also has a "social network" component, based on other students who've shared a Piazza-based class with you, that comes with the usual warnings about privacy concerns. Piazza is fully FERPA compliant, and is an allowed resource at UCSD. Nevertheless, you are not required to use Piazza if you do not wish.

Gradescope is an online tool for uploading and grading assignments and exams (it is now under the umbrella of Turnitin). You will turn in your homework through Gradescope, and you will access your graded exams there as well.
- Access the class Gradescope site here.
- Homework 0 below will guide you through the Gradescope sign up process for this class.

Canvas is a learning management system (LMS) used by UCSD. We will use Canvas for Quizzes and for sharing certain course materials. Make sure that you have acces to the MATH 180C course.

Instructional Staff and Office Hours

Name	Role	Office	E-mail	Office hours	Zoom link
Yuriy Nemish	Instructor	-	ynemish@ucsd.edu	Monday 6:00-7:00pm Wednesday 4:00-5:00pm	link
Jiaqi Liu	Teaching Assistant	-	jil1131@ucsd.edu	Monday 4:00 - 5:00pm Thursday 4:00 - 5:00pm	link

Please, check the following calendar for possible reschedulings of the office hours.

Calendar

Zoom Meetings

	Date	Time	Zoom link
Live Q&A (YN)	Monday, Wednesday, Friday	5:00 - 5:30pm	link
Discussion A01 (Liu)	Thursday	1:00 - 1:50pm	link
Discussion A02 (Liu)	Thursday	2:00 - 2:50pm	link

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Important dates

	Week	Date	Time
Quiz 1	2	Wednesday, Oct 14	see Quizzes
Quiz 2	3	Wednesday, Oct 21	see Quizzes
Midterm Exam 1	4	Wednesday, Oct 28	see Midterm Exams
Quiz 3	5	Wednesday, Nov 4	see Quizzes
Quiz 4	7	Wednesday, Nov 18	see Quizzes
Midterm Exam 2	8	Monday, Nov 23	see Midterm Exams
Quiz 5	10	Wednesday, Dec 9	see Quizzes
Final Exam	Finals	Thursday, Dec 17	7pm - 10pm; see Final Exam

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Syllabus

Welcome to Math 180C: a one quarter course introduction to stochastic processes (II). According to the UC San Diego Course Catalog, the topics covered are Markov chains in discrete and continuous time, random walk, recurrent events and other topics.

Here is a more detailed listing of course topics, in the sequence they will be covered, together with the relevant section(s) of the textbook. While each topic corresponds to approximately one lecture, there will be some give and take here. This is a rough schedule that will be updated during the term.

Q&A	Week	Topic	PK	Durrett	Slides	Lecture videos	Additional videos	Empty slides
10/2	0	Administrivia	1.1	-	-	-	-	-
10/5	1	Birth processes	6.1	-	Slides 1	Lecture 1	-	-
10/7	1	Birth processes	6.1	-	Slides 2	Lecture 2	-	Slides 2
10/9	1	Birth and death processes	6.2 - 6.3	-	Slides 3	Lecture 3	-	Slides 3
10/12	2	Strong Markov property. Hitting probabilities	6.5	-	Slides 4	Lecture 4	-	Slides 4
10/14	2	General continuous-time Markov chains. Q-matrices. Matrix exponentials	6.6	4.1	Slides 5	Lecture 5	-	Slides 5
10/16	2	General continuous-time Markov chains. Q-matrices. Matrix exponentials	6.6	4.1	Slides 6	Lecture 6	-	Slides 6
10/19	3	First step analysis for general Markov chains	6.5, 6.6	4.4	Slides 7	Lecture 7	-	Slides 7
10/21	3	Kolmogorov forward and backward equations	6.3, 6.6	4.2	Slides 8	Lecture 8	-	Slides 8
10/23	3	Stationary distributions and long-run behavior	6.4, 6.6	4.3	Slides 9	Lecture 9	-	Slides 9
10/26	4	Review	-	-	Practice Midterm 1	Solutions	-
10/28	4	Midterm 1
10/30	4	Conditioning on a continuous random variable	2.4	-	Slides 10	Lecture 10	-	Slides 10
11/2	5	Introduction to renewal processes	7.1	3.1	Slides 11	Lecture 11	-	Slides 11
11/4	5	Poisson process as a renewal processes	7.3	3.1	Slides 12	Lecture 12	-	Slides 12
11/6	5	Examples of renewal processes	7.2 - 7.3	3.1	Slides 13	Lecture 13	-	Slides 13
11/9	6	Asymptotic results for renewal processes	7.4	3.1, 3.3	Slides 14-15	Lectures 14-15	-	Slides 14-15
11/11	6	Veterans Day
11/13	6	Asymptotic results for renewal processes	7.4	3.1, 3.3	Slides 14-15	Lectures 14-15	-	Slides 14-15
11/16	7	Generalizations of renewal processes	7.5	3.1, 3.3	Slides 16	Lecture 16	-	Slides 16
11/18	7	Generalizations of renewal processes	7.5	3.1, 3.3	Slides 17	Lecture 17	-	Slides 17
11/20	7	Review	-	-	Practice Midterm 2	Solutions	-
11/23	8	Midterm 2
11/25	8	Martingales	2.5	5.1 - 5.2	Slides 18	Lecture 18	-	Slides 18
11/27	8	Thanksgiving
11/30	9	Definition of Brownian motion	8.1	-	Slides 19	Lecture 19	-	Slides 19
12/2	9	Basic properties of Brownian motion	8.1	-	Slides 20	Lecture 20	-	Slides 20
12/4	9	The reflection principle	8.2	-	Slides 21	Lecture 21	-	Slides 21
12/7	10	Processes derived from Brownian motion	8.3-8.4	-	Slides 22-23	Lecture 22-23	-	Slides 22-23
12/9	10	Processes derived from Brownian motion	8.3-8.4	-	-	-	-
12/11	10	Review	-	-	-	-	-

Prerequisite: MATH 180B or concent of instructor.

Lecture: You are responsible for material presented in the lecture whether or not it is discussed in the textbook. You should expect questions on the exams that will test your understanding of concepts discussed in the lecture.

Homework: Homework assignments are posted below, and will be due at 11:59pm on the indicated due date. You must turn in your homework through Gradescope; if you have produced it on paper, you can scan it or simply take clear photos of it to upload. Your lowest homework score will be dropped. It is allowed and even encouraged to discuss homework problems with your classmates and your instructor and TA, but your final write up of your homework solutions must be your own work.

Quizzes: The quizzes will take place on the dates listed above.

You will able to access the quiz on the course Canvas page starting at 12am (PST) on the day of the quiz. The quiz will be accessible for 24 hours; however, once you start the quiz, you will only have 30 minutes to complete it (unless you start it later than 11:30pm (PST), in which case the 24 hour deadline is enforced since it comes before the end of your 30 minutes). For example, if you start the exam at 11:40pm (PST), you will need to submit it within 20 minutes to receive credit since the 24 hour deadline would come before the end of your 30 minutes.
There will be no makeup quizzes. See below for other policies.

Midterm Exams: The midterm exams will take place on October 28 and November 23 as listed above.

The exam will be available for 24 hours on Gradescope starting at 12am (PST) on the scheduled date: however, once you start the exam, you will have 60 minutes to finish and submit your exam. The exam is intended to take 50 minutes: the remaining 10 minutes are intended for the scanning and submission of the exam on Gradescope. In particular, we will not accept submissions after the 60 minute deadline: it is your responsibility to submit the exam on time. For example, if you choose to write the exam for 60 minutes and do not leave yourself enough time to scan the exam, then you will not receive credit for the exam.
As with the quizzes, the 24 hour deadline supersedes your allotted 60 minutes for the exam. For example, if you start the exam at 11:40pm (PST), you will need to submit it within 20 minutes to receive credit since the 24 hour deadline would come before the end of your 60 minutes.
There will be no makeup midterm exams. See below for other policies.

Final Exam: The final examination will be held at the officially scheduled time: 7pm - 10pm (PST) on December 17.

It is your responsibility to ensure that you do not have a schedule conflict involving the final examination; you should not enroll in this class if you cannot take the final examination at its scheduled time.
The written portion of the exam will be available on Gradescope at 7pm (PST) on December 17. Submissions are due at 8:30pm (PST) on Gradescope.
After the written portion, students will join a Zoom room from 8:30pm - 10pm (PST). During this time, the instructional staff will randomly select students for a follow-up oral exam to ensure academic integrity. A determination will be made based on the oral exam whether or not to file a case with the Office of Academic Integrity. If a student fails to report to the Zoom room, then they will automatically receive no points for the final exam.
There will be no makeup final exam. See below for other policies.
Zoom links and more detailed instructions will be announced later.

Exam policies

Quizzes and exams are open textbook, and you can rewatch lectures and review the lecture slides.
Keep in mind that any computations on exams must be supported by clear explanations. Simply writing down an answer without any justification or work will receive 0 points, even if the answer is correct (unless the problem specifically says that no justification is required).
You are not allowed to use calculators, computer algebra systems, or similar software. Of course, this also means that you will not be asked to solve problems on exams that require the aid of an electronic computing device.
You are not allowed to consult other people or resources on the internet. In particular, contract cheating and other instances of academic dishonesty can result in suspension or dismissmal. According to the Academic Integrity Office,

About 15% of students reported for cheating are suspended or dismissed from UCSD.

Administrative Links: Here are two links regarding UC San Diego policies on exams:

Exam Responsibilities An outline of the responsibilities of faculty and students with regard to final exams

Policies on Examinations The Academic Senate policy regarding final examinations (These are the rules!)

Regrade Policy:

Your exams and homeworks will be graded using Gradescope. You will be able to request regrades directly from your TA through Gradescope for a specified window of time. Be sure to make your request within the specified window of time; no regrade requests will be accepted after the deadline. Note: Your grader will consider your regrade request only if you have explained clearly, thoroughly, and politely why you think an error in grading was made.

Grading: Your cumulative average will be computed as the best of the following weighted averages:

20% Homework, 20% Quizzes, 10% Midterm 1, 10% Midterm 2, 40% Final Exam
20% Homework, 20% Quizzes, 10% Best midterm, 50% Final Exam

Your course grade will be determined by your cumulative average at the end of the quarter, and will be based on the following scale:

A+	A	A-	B+	B	B-	C+	C	C-
97	93	90	87	83	80	77	73	70

The above scale is guaranteed: for example, if your cumulative average is 80, your final grade will be at least B-. However, your instructor may adjust the above scale to be more generous.

Academic Integrity: UC San Diego code of academic integrity outlines the expected academic honesty of all studentd and faculty, and details the consequences for academic dishonesty. The main issues are cheating and plagiarism, of course, for which we have a zero-tolerance policy. (Penalties for these offenses always include assignment of a failing grade in the course, and usually involve an administrative penalty, such as suspension or expulsion, as well.) However, academic integrity also includes things like giving credit where credit is due (listing your collaborators on homework assignments, noting books or papers containing information you used in solutions, etc.), and treating your peers respectfully in class. In addition, here are a few of our expectations for etiquette in and out of class.

Noise and common courtesy: When class/section begins, please stop your conversations. Do not disturb others by engaging in disruptive behavior. Disruption interferes with the learning environment and impairs the ability of others to focus, participate, and engage.
Electronic devices: No visual or audio recording of the class meetings is allowed without prior permission of the instructor (whether by camera, cell phone, or other means).
E-mail etiquette: You are expected to write as you would in any professional correspondence. E-mail communication should be courteous and respectful in manner and tone. Please do not send e-mails that are curt or demanding.

Accommodations:

Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (OSD) which is located in University Center 202 behind Center Hall. The AFA letter may be issued by the OSD electronically or in hard-copy; in either case, please make arrangements to discuss your accommodations with me in advance (by the end of Week 2, if possible). We will make every effort to arrange for whatever accommodations are stipulated by the OSD. For more information, see here.

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Homework

Weekly homework assignments are posted here. Homework is due by 11:59pm on the posted date, through Gradescope. Late homework will not be accepted.

Homework 0, due Wednesday, October 7.

Homework 1, due Friday, October 9.

Homework 2, due Friday, October 23.

Homework 3, due Saturday, October 31.

Homework 4, due Friday, November 6.

Homework 5, due Friday, November 13.

Homework 6, due Saturday, November 21.

Homework 7, due THURSDAY, December 3.

Homework 8, due THURSDAY, December 10.

Homework 9, NOT TO BE TURNED IN.