Courses in the Department of Mathematics

MA409: Continuous-Time Optimisation 2011/12

Graduate Handbook entry for this course

Lecture Notes

Homework Assignments

Additional Material

Assessment and Mock and Past Exams

Associated M.Sc. Dissertation Projects

Lecture Notes

A full set of Lecture Notes will be distributed by way of this web-site. The current files are last year's unless otherwise stated. I expect to update files a few days ahead of the lectures of the 07/08 session, but any updating will tend to be cosmetic, unless otherwise advised. The homework assignments are as currently given. I will announce one week ahead which exercises I intend to concentrate on in the class presentation.

We begin with an Overview Lecture based on a PowerPoint presentation. This is followed by a series of lectures based on Chapter 23 of my textbook entitled "Advanced Mathematical Methods" (the coursebook for MA200 and MA201). A scanned image of the chapter is provided below; please regard this as Chapter 1 of MA409.

Lecture 1 - PowerPoint Presentation

Lectures 2-5

Euler's Presumed Derivation

Lectures 6++ .

Expanded notes on unconstrained control.

Notes on Brownian Motion .

Bellman Equation under certainty .

Examples under certainty .

Bellman Equation under Uncertainty .

Note on Smooth-pasting .

See also the dedicated section on Additional Materials below.

Homework Assignments

Classes begin by working through an assortment of revision exercises on differential equations. You are advised to look through these during Christmas Vacation!

Assignment 1 - Revision Exercises

Assignment 2 - Continue with Revision Exercises

For general reference only (not really needed in this course) I've listed some well-studied second order differential equations here: Some second order differential equations

After that the initial homework assignments will be taken from the Exercises of Section 23.8 at the end of the Lecture 2++ file. Subsequent assignments will be posted below.

Assignment 3 - Questions 2 & 3 from Section 23.8

The next few assignments will come from the following file:

Further Exercises on the Euler-Lagrange Equation

Assignment 4 - Questions 1-4 from the Further Exercises

Assignment 5 - Questions 5, 8 from the Further Exercises, and the following:

Exercises on computing a derivative

Assignment 6 - Question 2 from `Unrestricted Control' file and Question 3 from `Time Optimality' file

The above assignment comes from the following files:

Exercises on unrestricted control

Exercises on time optimality control

You are evidently welcome to attempt more of the exercises!

Assignment 7 - Questions from the following file:

Exercises on Bellman Equation under certainty

Exercises on the Ito Formula

Assignment 8 - Questions to come from the following file:

Exercises on Bellman Equation under Uncertainty

Additional Materials

Some students asked for an additional example of smooth-pasting considerations. Here is one such:

A worked exercise:

Dividend Skimming: An Example on the Bellman Equation and Smooth-pasting

Exchange Rate pegging: Another Example on the Bellman Equation and Smooth-pasting

Below you will find additional material that is not examinable, but which I hope you may find interesting. Browse through it after the exams!

Economics Applications 1: Using Deterministic methods . I will add diagrams later.

Economics Applications 2: Merton Model of Consumption and Investment under uncertainty .

More to come!

Assessment and Mock and Past Exams

This course ran for the first time in 2004/05. Since no past exam papers were then available a mock examination paper, with full solutions, was set. The mock exam is available from this website (without the full solutions). Solutions of the Summer 2011 examination will be discussed during the revision lectures at the start of Summer Term 2012.

Note that solutions to the 2011 Summer Examination will not be posted here till the start of Summer Term 2012.

MA409 Continuous-Time Optimisation - Mock Exam paper         Sketch solutions to Mock Exam

MA409 Continuous-Time Optimisation - Summer 2005 Exam paper (Office Copy)         Solutions to 2005 exam

MA409 Continuous-Time Optimisation - Summer 2006 Exam paper (Office Copy)         Solutions to 2006 exam

MA409 Continuous-Time Optimisation - Summer 2007 Exam paper          Solutions to 2007 exam

MA409 Continuous-Time Optimisation - Summer 2008 Exam paper          Solutions to 2008 exam

MA409 Continuous-Time Optimisation - Summer 2009 Exam paper         Solutions to 2009 exam

MA409 Continuous-Time Optimisation - Summer 2010 Exam paper         Solutions to 2010 exam

MA409 Continuous-Time Optimisation - Summer 2011 Exam paper         Solutions to 2011 exam (to come)

Associated M.Sc. Dissertation Projects

In the Dissertation Seminar I shall talk about one project in some detail and use the following PowerPoint Presentation.

Market to Book versus Earnings to Book Project - PowerPoint

The above refers to the following paper (caution: long download time)

Burgstahler-Dichev paper

In this space I plan to publish a list of various projects which I would be willing to supervise together with useful information. The majority of them require as background the methods developed in MA409. Some of them may require a good knowledge of advanced calculus. Visit the MA200 page.

MA409 Adam's Projects for 11/12 (Slide Format)

Changes from previous years: (i) there is a project on regularly varying functions; (ii) I have withdrawn two projects from earlier sessions -- those based on work by Michael Schroeder (Free University, Amsterdam) -- as being too far from the objectives of the M.Sc. in Applicable Mathematics. You may want to see the source material nevertheless; it may be viewed using the following link.

Schroeder paper

The Schroeder paper requires some of the techniques offered in the MA200 Lectures.