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¤sigh¤ ...

Yet another error in the 5155_20100326_note.pdf : Order of multiplication was adapted to the other possible way to organise eigenvectors. New version uploaded again, see corrections in red on last page.

There was unfortunately a substantial error in the 5155_20100326_note.pdf . New version uploaded. Be sure to check the differences (indicated in colour; for your reference, the old one is still available (renamed) in the Undervisningsmateriale folder).

Sorry for the inconvenience, - Nils

Syllabus final, with reservation for errors; please report any such to me a.s.a.p. Available via the 'Syllabus/achievement requirements' document ; the document itself is in the 'Undervisningsmateriale' folder

As noted in the May 03 message, the Math5_bookpages.pdf file needs the pw ECON5155

The seminars Mondays May 3rd and May 10th are moved to room 504 Eilert Sundts Hus. Time the same: 1015--12.

(Corrected version uploadet 17:38)

Problems for next seminar: Due to time constraints, I have as of today

• only two problems
• of which one I have not calculated myself, but "should be doable" (and if not, I guess time will not be wasted anyway).
• and not typeset, only scanned. Available here .

I hope to put out more soon, but I might not have time until next Wednesday -- I will be absent the rest of this week.

Jeg klarte å få med meg en genser som ikke var min. Har lagt den på studentenes pauserom 12. etasje, oppå bokhyllene inn mot trappegangen. Beklager.

(Corrected!)

Seminar the 19th -- problems:

• Optimal control: Consider the portfolio optimization problem on the blackboard, with infinite horizon and running utility f(c)=ln c. Show that for some constants a, b, then a ln (x+h)+b is superoptimal for all h>0, and optimal for h=0.

• Optimal stopping: Seierstad exercise 4.16--4.18.

• Optimal stopping: Consider section 3.3 of this article from the Undervisningsmateriale folder . Test the proposed solution (eqs. (28)--(33)) against the Bellman equation and the smooth fit condition.

Updates (just before Easter, just in case you want to keep up):

Teaching:

• Monday's seminar: Not very well visited. I take it as a sign that I assign far too easy problems ;>
• On Wednesday, we covered continuous-time optimal stochastic control
• Next Wednesday (after Easter), we will complete this and cover optimal stopping.
• Then the following Wednesday: Filtering.

Material:

• The note under "Various material" is updated.
• For reference on language on information, you might want to read this note I wrote for ECON5160 last year . Optional reading (pun intended, will be revealed if you read the note ;) )

Sorry for the late call here:

• I did not have time to cover the martingale concept on Wednesday's lecture.
• So in the problem set for Monday: In the note, problem 1.3. (b) and (c), replace "is a martingale" by "has the property that expected change from time 0 to time T is zero". (This is not the martingale property, merely a consequence of it.)

Problems for the seminar Monday 22nd:

• From Seierstad: Problems 4.2 through 4.4

• From this note : I intend to cover material enabling you to do problems 1.1 through 1.3. I might have to amend this by Wednesday though.

Past, present, future:

Seminar: No seminar March 15th. Next March 22nd.

Lectures:

The Feb 3rd lecture: as announced in the Feb 24th message below, including the "-- probably --" part. Yet to do: Update the note with "what do you need to know".

The Feb 10th lecture covered

• Brownian motion (and the Poisson process);
• Basic properties of Brownian motion;
• The Itô integral (wrt. Brownian motion);
• The formulae d(B^2(t)) = 2 B(t) dB(t) + dt and more generally d g(B(t)) = g'(B(t)) dB(t) + g''(B(t)) dt / 2.

Next lecture:

• The Itô formula in full generality (no proof)
• Stochastic differential equations: concept
• Geometric Brownian motion and the Ornstein--Uhlenbeck process: the SDEs, the solutions and their properties
• Dynkin's formula and, if time: martingales.

- Nils

• Today, we went through my note and its applications to numerical solution of the Bellman equation.

• My note has broken links to the Darling and Arouba et al. articles. The note will probably be updated (for other reasons too), but the references are available in the undervisningsmateriale subfolder to these course pages.

• For next time: We will consider numerical procedures for optimal stopping in discrete time, based on the Darling article. We will -- probably -- also discuss an algorithm for continuous time; although the concepts are not introduced yet, it involves reduction to discrete time. A few book pages (of which the last one will include the algorithm itself) will be available in the "undervisningsmateriale" folder very soon.

A note to read on iterative methods is available. This principle will be useful for computer implementation. On Wednesday, I will start more or less from scratch (basic principles and Newton's method), and let's see how far we get.

- Nils

Rough lecture plan until Easter:

• Atle completes his series on discrete-time stochastic optimization on the 18th.
• Feb 24th [corrected from 25th, - Nils] I will start on numerical methods. I expect this to take two weeks -- we'll see what we need. I will find and post some reading material in due course.
• March 10th I will start on continuous-time stochastic processes and stochastic calculus (integration and differential equations). I will also cover some linear algebra for solving linear differential equations (the Ornstein--Uhlenbeck process), and a bit of general martingale theory. Reading material here will be from Seierstad chapter 4, but I will probably find supplementary material.

Seminars:

• Next seminar: 22nd.
• Then: To be decided.

Teaching-free week:

• Suggestion: Not until Easter, unless it turns out to fit a revised schedule.

-- Nils

Seminars:

There will be (approximately!) six seminars.

They will be on Mondays 10:15 to 12, room 1220 (where the lectures are; this room will become unavailable late April though).

First seminar: Februar 1st.

On the choice of topics:

Content will be discussed on first lecture this Wednesday.

This course has usually covered topics in stochastic systems, and that will be at least the main part of the menu for this semester's version as well. There will have to be some optimization (dynamic programming/optimal control theory), with stochastic systems.

We might also cover some of the following (related) topics:

• The mathematics of finance (arbitrage-pricing)
• Filtering (i.e. predictions from a noisy signal)
• Some principles for numerical solutions
• Other suggestions from you?

We intend to start with stochastic optimization, leaving some time to fix the schedule in detail. However, the broad content should be settled on first lecture.

It will be relevant to know whether you have credits in ECON5150 (Mathematics 4) or ECON5160.

-- NCF