IQM WS 2008

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This is the homepage of the IQM-course in 2008 of the VGSF-program. The page serves as a basis of communication between students and lecturer. It is updated regularly. You may check whether there are any changes of the page by viewing the date of the most recent update below.

Last update: 14.10.2008

Contents

Organisation

The course is given by Prof. Dr. K. Pötzelberger and by Prof. Dr. H. Strasser.

Part I: Mathematics - Prof. Strasser

Class 1: October 1st, 13:00-17:00

Class 2: October 2nd, 13:00-17:00

Class 3: October 6th, 13:00-17:00

Class 4: October 7th, 13:00-17:00

Part II: Probability - Prof. Pötzelberger

Class 5: October 8th, 13:00-17:00

Class 6: October 9th, 13:00-17:00

Class 7: October 13th, 13:00-17:00

Class 8: October 14th, 13:00-17:00

Goals

This course reviews the basic mathematical and statistical knowledge that is expected to be known from previous studies (master programs). The selection of subjects is strongly biased towards those parts of mathematics which are prerequisites of the VGSF-program.

The course is a classroom lecture which is supported by blackboard, sheets and handouts.

The course is not based on a particular book. As further readings we recommend:

Sydsaeter K. and Hammond P.: Essential Mathematics for Economic Analysis. Prentice-Hall, 2002
Sydsaeter K., Hammond P., Seierstad A. and Stroem A.: Further Mathematics for Economic Analysis. Prentice-Hall, 2005
Rice J. A.: Mathematical Statistics and Data Analysis. 2nd Ed., Duxbury Press, 1995
Jacod J. and Protter P.: Probability Essentials. Springer 2000

Handouts

The handouts contain a collection of some sheets which are presented during the lecture. They provide partial information about the subjects of the lecture.

The last year version can be downloaded from:

Version 02: part 1, part 2

Prerequisites

For beginners who have problems with undergraduate mathematics (and who are familiar with the German language) we recommend the e-learning page of the WU (Vienna University of Economics and Business Administration). You have to register as a student. Then you can get an admission to the computer system of the university (powernet login).

  • Goto -> Quick Links and click at -> Learn@WU.
  • Login with your powernet account.
  • You arrive at your own startpage of the WU learn-server.
  • Goto the tab called -> Eigene Startseite.
  • Take -> Group:LV-Quicklinks and choose -> Grundlagen Teil B Mathematik
  • Choose -> Zur Vorlesung Mathematik

Here you find extensive materials of the basic mathematics course of the WU for business and economics studies. In particular the IQM-course assumes knowledge of

  • matrices (WU script 197-230, IQM handouts 1-30)
  • vector geometry (WU script 231-234)
  • linear equations (WU script 180-196)
  • differentiation (WU script 73-115, IQM handouts 123-166)
  • Integration (WU script 117-135, IQM handouts 173-209)
  • partial derivatives (WU script 235-245, IQM handouts 276-280)
  • probability (WU script 136-179, IQM handout Part II 11-22, 60-104)


Contents of Part I

  • Orthogonality: elementary notions (handouts 31-44), subspaces (handouts 57-76), projections (handouts 92-105), application to multiple regression (blackboard)
  • Eigenvalues: concept of eigenvalues (handouts 106-114), symmetric matrices (handouts 115-122), quadratic functions (handouts 259-275)
  • Convexity: separation (handouts 45-49, 53,54), linear inequalities (handouts 50-52, 55, 56), applications to finance (blackboard)
  • Calculus: univariate Taylor expansions (handouts 167-172), multivariable differentiation (handouts 281-289), multiple integration (blackboard)
  • Optimization: unconstrained optimization (handouts 290-298), constrained optimization (handouts 299-307)
  • Dynamical systems: 1st order differential equations (handouts 210-239), trigonometric functions (blackboard), complex numbers (blackboard), 2nd order linear differential equations (blackboard), 1st order systems of linear differential equations (blackboard)

Contents of Part II

  • Preliminaries: sets, functions, real numbers (handouts(blackboard) chapter 1)
  • Axioms of probability: definitions and axioms, independence and conditional probability, examples (handouts(blackboard) chapter 2, 3, Jacod and Protter chapter 2, 3)
  • Discrete probabilities: measurable faunction, distribution of random variable, examples (handouts(blackboard) chapter 4, Jacod and Protter chapter 4, 5)
  • Probability on R: distribution functions, densities, random variables, examples (handouts(blackboard) chapter 5, 6, Jacod and Protter chapter 6, 7, 8)
  • Integration: definition, properties, computation, Lebesgue measure, examples (handouts(blackboard) chapter 7, Jacod and Protter chapter 9, 11)
  • Independent random variables: Fubini's theorem, properties of sequences of independent random variables, examples (handouts(blackboard) chapter 8, Jacod and Protter chapter 10)
  • Probability on R^n: multivatiate distributions (handouts(blackboard) chapter 9, Jacod and Protter chapter 12)
  • Characteristic functions, moment generating functions: properties, computation, c.f. of sums, examples (handouts(blackboard) chapter 10, Jacod and Protter chapter 13, 14)

Exam

There are two written exams to be taken, exam 1 about part 1 of IQM and exam 2 about part 2 of IQM.

Exam 1: Monday, 20.10.2008, 14:00-17:00, H 4.37 (A)
Exam 2: Thursday, 23.10.2008, 8:30-11:30, H0.2(B/C)

The exams are closed books exams ! Each test consists of 8 problems to be solved and 8 questions to be answered.

For the tests you may not use other tools than simple calculators, and those tables and formula collections which are authorized by the course leader as part of the published problem collection. The formula collection contains important mathematical formulas without any description and comments.

Exam Part 1

Problems and questions can be found in collection: Sections 1.2 (all), section 1.3 (RevQu 1,2,4,5, Prob 1-4), section 1.4 (all), section 2.3 (all), section 2.4 (all).

Further problems and questions:

  • Explain how to use the Gram-matrix for finding multiple regression coefficients.
  • Explain the concepts of no arbotrage, price measure and eqaunivalent price measure in a single period setup.
  • State the funcdamental theorem of asset pricing in single period setup.
  • Verify the fundamental theorem of asset pricing in the single period binomial case.
  • Prove for a single period setup that the existence of an equivalent price measure implies the NA property.
  • Explain how the NA-lemma for matrices can be used for proving the fundamantal theorem of asset pricing.

All examples of multiple integration presented in the lecture are exam problems.

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