MA409: Continuous-Time Optimisation 2009/10
General information about MA409: Continuous-Time
Optimisation
Lectures and classes
This is a half-unit course, with lectures in the Michaelmas Term.
There will also be revision lectures in the Summer Term.
Classes start in Week 2 of the Michaelmas Term.
Please check the
Timetables
website for further information.
Exercises
Homework announcements will be made in the first lecture each week. The
homework has to be handed in during the
class to which they relate. No late work will be accepted. If you don't
finish the assignment, hand in whatever you have.
Label all work with your name and the course number (MA409).
Corrected work will be handed back and discussed in the classes.
Course-material
A full set of lecture notes will be provided. Reference will be made to the following books:
E.R. Pinch, Optimal Control and the Calculus of Variations, Oxford University
Press, ISBN: 0198532172.
W.H. Fleming and R.W. Rishel, Deterministic and Stochastic Control Springer, ISBN:3540901558.
G. Leitmann, Calculus of Variations and Optimal Control, Plenum, ISBN: 0306407078.
A.K. Dixit and R.S. Pindyck, Investment under Uncertainty, Princeton
University Press, ISBN: 0691034109.
D. Duffie, Security Markets, Academic Press, ISBN: 0691034109.
Further sources of material are:
W.H. Fleming and H.M. Rishel, Controlled Markov Proccesses and Viscosity Solutions, Springer, ISBN:3540979271.
D.J. Bell and D.H. Jacobsen, Singular Optimal Control, Academic Press, ISBN: 0120850605.
G. Hadley and M.G. Kemp, Variational Methods in Economics, North Holland, ISBN: 0444100970.
All course material distributed, including lecture notes, exercise sheets,
model answers, etc., will also be made available on this website, via
http://www.maths.lse.ac.uk/Courses/MA409/.
Course description of MA409: Continuous-Time
Optimisation
Overview
This a high level methods course centred on the establishment of a calculus appropriate to
optimisation problems in which the variable quantity is a function or curve. Such a curve might describe the evolution over continuous time of the state of a dynamical system. This is typical of models of consumption or production in economics and financial mathematics (and for models in many other disciplines such as engineering and physics). We will be studying two contexts. In the first the future state of the modelled environment is predetermined by the equations describing the model. In the second context we introduce and model uncertainty by making use of a mathematical device known as Brownian Motion as the generator of uncertainty. The emphasis of the course is on calculations, but there will also be some theory.
Course Content
The main topics that the course covers are the following:
- Brief review of ordinary-differential equations;
- Calculus of variations and the Euler-Lagrange Equations;
- Pontryagin Maximum Principle;
- Switching functions, Bang-bang control and Singular control;
- Dynamical programming and Value functions;
- Brownian Motion and Itô's Lemma;
- Hamilton-Jacobi-Bellman equation and the Verification lemma.
- Applications to Economics and Finance: Economic Growth models, Consumption and investment, Optimal Abandonment. If time allows: Black-Scholes model.
Lectures
During the lectures, the theory will be developed and
explained, proofs given, and many examples demonstrated. Students are expected
to make their own notes during the lecture. Additional notes/summaries will be
distributed from time to time.
Classes and Exercises
In this course, as in other courses in Mathematics, it is very important
that all homework questions are attempted and handed in for grading. There
is a big difference between watching other people presenting a proof or
performing some calculations and being able to do these things yourself. It
is vital to get practice in the various techniques covered in the
course.
It is also important to hand in homework in, so that feedback on it can be
given. Corrected work will be handed back and discussed in the class.
The lecturer of this course pays great attention to the presentation of the
homework. It is not enough to have the correct answer, but also the reasoning
used to obtain that answer should be correct and understandable. This involves
that minimum standards of the use of the English language and mathematical
argumentation will be kept.
Office Hours
The office hours are meant for any questions and problems with the course
material that have not or cannot be covered in the normal lectures and
classes. You are strongly recommended to make use of them.
The times of the office hours for lectures and class teachers can be found
on the departmental
office hours page.
Connections to Other Courses
As prerequisite you will need the kind of skills in calculus that are developed in MA200 Further Mathematical Methods (Calculus).
From the outside options, a course
which has close relations with this one is
ST409 Stochastic Processes,
and there are some connections with
FM402 Financial Risk Analysis,
FM441 Derivatives and
FM442 Quantitative Methods for Finance and Risk Analysis.In regard to courses in the Mathematics Department the
following two courses (though not required) could be useful in building up an
appreciation for the rigorous and detailed development of some aspect of the
methods employed in MA409. These are MA412:
Functional Analysis and its Applications, useful for the theoretical basis
of the Calculus of Variations, and MA411:
Probability and Measure, useful for a rigorous development of Brownian
Motion which we treat informally.
Past exam papers for this course
Please note: students are advised not to rely too heavily on past exam
papers when revising for their exams, as they can only offer a limited
indication of what might be covered in a future exam. For further information,
please see the guidance here:
http://www.maths.lse.ac.uk/examinations_in_mathematics.html#past_papers
Exam paper of
2006
Exam paper of
2007
Exam paper of
2008
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of Economics & Political Science 2009
Last change: 14th October 2009
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