MA8109 Stochastic processes and differential equations
Fall 2019
Welcome to MA8109 !
This PhD course is given every second year, and usually 4th and 5th year (Master) students take it along with PhD students.
What is this course about?
 Differential equations with noisy/uncertain coefficients (stochastic differential equations), and their solutions, continuous time stochastic processes: We give a mathematical background, the main results, and some applications (more details in 'General Information' in the left menu).
Of the multitude of applications in science, engineering and other disciplines, the most famous one is perhaps the BlackScholes model for option pricing in finance.
Who can take this course?
 Interested students at Master or PhD level.
 The level should be suitable for good 4th year students in the industrial mathematics program.
 It can be taken as a regular course (MA8109) or a 'fordypningsemne' (TMA4505).
Tentative office hours  Exam Wed 11.12.2019 (tentative)  Exam Wed 18.12.2019 (tentative)  
Monday Dec 9  11:1512:00  room 1148, SB2  13:0013:45  room 1148, SB2  Francois  8:309:15  room 1148, SB2  Camilla 

Tuesday Dec 10  11:1512:00  room 1148, SB2  13:5014:35  room 1148, SB2  Oscar  9:2010:05  room 1148, SB2  Anders 
Monday Dec 16  11:1512:00  room 1148, SB2  14:4015:25  room 1148, SB2  Oliver  10:1010:55  room 1148, SB2  Philip 
Monday Dec 17  11:1512:00  room 1148, SB2  15:3016:15  room 1148, SB2  11:0011:45  room 1148, SB2  Erik  
13:1514:00  room 1148, SB2  
14:0514:50  room 1148, SB2 
Messages
Date  Title  Message 

15.11.  Exam dates  We have now fixed the two dates for the exam: December 11 (Wednesday) and December 18 (Wednesday) 
Sign up  Send me an email with the date you want and possible time constraints/wishes Based on this feedback I will give you a time slot. See the table above for a tentative schedule (changes are still possible) 

Reference group meeting  We will do the last meeting in the break of the lectures on Thursday November 21. Do you have feedback on course? What is good/what can be improved? 

Project  Do not forget the deadlines  November 29 for the report! See below and in the left menu for more info. 

Office hours  To be decided for the exam period. Se table above for suggestions. 

Final curriculum  See 'General information' in the left menu. 

Solutions  To homework 4 and 5 now available, see the left menu. OBS: Volunteer still needed to write the solution of Problem 11 in Homework set 4. 

08.11.  Tentative oral exam dates  We discussed in class, Friday November 8: December 11 (Wednesday) and December 18 (Wednesday) Not good for you? Send me an email with your wishes and constraints. Let me know before next Friday (15.11.), we aim to finalize dates then. 
Homework  Homework set 5 is now available. You can do the first 3 problems from Friday onwards.  
Read yourselves:  Note on Linear SDEs and Physical Brownian Motion  
18.10.  Notes on existence and uniqueness  The proofs can be found in my handwritten notes from class: Strong solution of SDEs. Uniqueness and Existence 
17.10.  Reference group  Meeting in the break, Friday October 18. 
Project deadlines  October 18: email me your choice of project. November 29: email me the written report. See "The project" in the left menu for more info. 

Next 2 weeks  You should work on the course project. No lectures, office hours instead in my office room 1148 SB2 (note the times): Wednesday 14:1515:00. Thursday 16:1517:00. Friday 14:1515:30. 

Existence and uniqueness of SDEs  We use the proofs as presented in Schilling. My handwritten notes will be posted later. (There seems to be a mistake in the proof of existence in the 6th edition of Øksendal. The proof in previous editions is correct but longer) 

Read yourselves  Øksendal chp. 5.3 (discussion on weak solutions and uniqueness)  
Problem set 3 and 4  Available under 'Homework' in the left menu.  
10.10.  Lectures  Next week: We finish the discussion on strong solutions of SDEs. Week 43 and 44: Project work, no lectures. Office hours instead, in my office room 1148 SB2. 
Project  The project counts for 20% of the final grade. You may study any relevant topic of you own choosing, see 'The Project' in the left menu for suggestions and more information. 

03.10.  Numerics for SDEs  There will be two weeks on numerical approximation on SDEs in the course: MA8404 Numerical solution of time dependent differential equations Week 44 and 45, but check the webpage for the latest update (click above link). Students from our course are welcome to attend these two weeks. 
Read yourselves  About the Stratonovich integral in chapter 3.3 in Øksendal.  
Homework  After this weeks lectures you will be able to do problem 13 in Homework 3. Problem set 3 is available in the left menu. 

24.09.  Lectures  Thursdays 1618 and Fridays 1416. Last Friday we checked again and found that Wednesdays are no longer free. 
12.09.  Office hours  Wednesday 14:1515:00, room 1148 SB2 (my office) 
Homework  Problem set 2 and Solutions 1 available, see 'Homework' in the left menu.  
Reference group  The whole class will be the reference group. First meeting: During break Friday September 20. 

30.08.  Problem set 1  Available under 'Homework' in the left menu. 
19.08.  First Lecture  Friday 23.08. 14:1516:00, Room 734 SB2 
General info and book  See the left menu.  
03.06.  Info meeting  Monday August 19th, 14:1515:00, at room 734 SB2.  Info about the course.  Decide on times for the lectures.  Set up email list for course. Cannot attend, but want to take the course?  Send me an email where you include the times you are not free (have other lectures). 
Master student?  You can take the course either as MA8109 or TMA4505 (fordypningsemne). Grades: pass/fail in MA8109, AF in TMA4505. 4th year MTFYMA students then take a regular course instead of TMA4505 in the 5th year. 
Mon  Tue  Wed  Thu  Fri  

08:1509:00  Lecture, 734 SB2  
09:1510:00  
10:1511:00  
11:1512:00  
12:1513:00  
13:1514:00  
14:1515:00  Lecture, 734 SB2  
15:1516:00  
16:1517:00  Reserve slot  
17:1518:00 