# MA8109 Stochastic processes and differential equations

## Fall term 2015

Messages
• (2016-01-05) Grading nearly done, finally! I found some minor misprints in the solution, and uploaded a new version (now at 1.1).
• (2015-12-07) Here are today's exam and solution. Please let me know if you find any misprints or mistakes in the solution. I am aware that the solution is a bit brief in places; also let me know if it is too brief.
• (2015-12-05) I hope everybody has realized by now that the exam is in Trondheim Spektrum, known to old timers as Nidarøhallen. Regarding the project work, in order to preserve students' anonymity as best I can, I will not look at your reports until after I have graded the main exam.
• (2015-12-02) On allowed aids on the exam: The code is C, which means I could allow some specified set of aids. However, I have not chosen to specify any. You may bring a dictionary if you think you need it, though. English or between English and your favourite language. (I don't think you'll need it, but at least you have the option.)
• (2015-11-26) The curriculum is finalized. See the Info page.
The “synopsis” is not part of the curriculum, but you may find it useful anyhow. It is unfinished, in part because of a muscular problem that made working on the computer painful. (It's much better now.)
• (2015-11-11) Lectures resume this week. Have a look at Harald Krogstad's note on the Kolmogorov and Fokker–Planck equations (listed below).
• (2015-10-28) There are no lectures this week and next week. Instead, I will be available in my office for questions during the regular lecture hours. (Both weeks. I thought I might be away the second week, but I had misread my calendar.) Regular lectures resume on Thursday, 12 November.
• (2015-10-21) The lecture last Friday ended with this expression, that we needed to show vanishes in the limit as $k\to\infty$: $\int_k^\infty E([t<\tau]Z_t^2)\,dt.$ It had been assumed that $|Z_t|\le M$, so the factor $Z_t^2$ can be removed. For the rest, we just interchange the order of integration, to get $E\Bigl(\int_k^\infty[t<\tau)\,dt\Bigr)=E\bigl((\tau-k)^+\bigr)\to0,$ because $(\tau-k)^+\to0$ pointwise and $E(\tau)<\infty$ by assumption, so we can use DCT.
• The project page is up; see the menu on the left
• I have been informed that the exam (for MA8109) will be on December 7. I was not aware that this would be decided without my input; let's hope the date will work for all.

The synopsis is not quite a set of lecture notes, as it will contain few proofs, if any. But it is intended to list important definitions and concepts, as well as any unusual notation and terminology I might employ.

The synopsis will keep growing, hopefully on a weekly basis, so keep coming back for more. Also, for that reason, you probably don't want to print the whole thing every time it's updated. Your print dialog should allow you to select a page range; use it.

Unfortunately, the synopsis stopped growing too soon. I was going to give it a facelift towards the end of the term, but a muscular problem made it too painful to sit at the computer for more than a few minutes at a time, so that fell by the wayside. (It is almost healed now.)

• Updated 2016-12-15 (after I was reviewing and found some misprints):
• Synopsis (yellow background, variable page length)
• A5 format (more conventional, can also be used for screen reading)
• A4 format (with two A5 pages per sheet, for printing)

For the first few weeks, I am lecturing mainly from my own head than any given notes. But some notes from earlier versions of the course, by Harald Krogstad, are quite relevant:

• Lebesgueintegralet (in Norwegian, unfortunately)
• Additionally, I wrote an extremely brief note giving a super quick derivation of the abstract Lebesgue integral. It comes in three flavours:
• Note on the Lebesgue–Radon–Nikodym theorem (screen/A5/A4)
• I don't plan to depart radically from the way the course was taught last year. In particular, we'll use the same textbook (Øksendal's Stochastic differential equations) and a collection of notes and other resources.

The following is shamelessly copied from the 2013 course page:

• 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. Of the multitude of applications in science, engineering and other disciplines, the most famous one is perhaps the Black–Scholes 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 or a 'fordypningsemne'.