Mathematics I
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This is the homepage of my course Mathematics I for the Master Program in Quantitative Finance.
Last update: 28.9.2009
Contents |
[edit] Handouts
There is a preliminary version of handouts ready for download:
[edit] References
The contents of the course are covered by the following references:
SH1: K. Sydsaeter and P. Hammond (2008): Essential Mathematics for Economic Analysis. Prentice Hall, Third Edition. ISBN 978-0-273-71324-1.
SH2: K. Sydsaeter, P. Hammond, A. Seierstad and A. Stroem (2005): Further Mathematics for Economic Analysis. Prentice Hall, First edition. ISBN 0-273-65576-0.
LL: S. Lipschutz and M. L. Lipson (2009): Linear Algebra. Schaum's Outline Series. Mc Graw Hill, Fourth Edition. ISBN 978-0-07-154352-1.
[edit] Prerequisites
The following topics are not taught in the course but are assumed to be known.
[edit] Basic Business Mathematics
The following notions are subject of every basic course in business mathematics. In particular, they are part of the introductory mathematics course at the WU.
- Introductory Algebra (SH1, chapter 1)
- Introductory Equations (SH1, chapter 2)
- Functions of One Variable (SH1, chapter 4)
- Differentiation (SH1, chapter 6, sections 7.1-7.4, 7.7)
- Single-Variable Optimization (SH1, chapter 8)
- Integration (SH1, sections 9.1-9.6)
- Interest Rates (SH1, sections 10.1-10.5)
- Functions of Many Variables (SH1, sections 11.1-11.7, 12.1-12.3)
- Multivariable Optimization (SH1, sections 13.1-13.4)
- Constrained Optimization (SH1, sections 14.1-14.4)
- Matrix and Vector Algebra (SH1, chapter 15, sections 16.1 and 16.9)
[edit] Undergraduate Business Mathematics
The following notions are standard content of undergraduate business mathematics. In particular, they can be obtained from elective courses of undergraduate studies at the WU.
- Introduction to mathematical foundations (SH1, chapter 3)
- Properties to functions (SH1, chapter 5)
- Advanced differentiation (SH1, sections 7.6-7.12)
- Advanced integration (SH1, section 9.7)
- Trigonometric functions and complex numbers (SH2, Appendix B)
- Sets, completeness and convergence (SH2, Appendix A)
[edit] Course Outline
The course is devoted to Advanced Mathematics for Business and Finance. All notions from Basic Mathematics and Undergraduate Mathematics are supposed to be well-known and familiar.
- Part 1: Linear Algebra
- Class 1: Vector Spaces and linear mappings
- Class 2: Inner products, orthogonality and diagonalization
- Class 3: Orthogonal projections and multiple regression
- Class 4: Convexity and separation
- Midterm Test
- Part 2: Analysis
- Class 5: Multivariable differentiation
- Class 6: Multiple integration and first order differential equations
- Class 7: Differential equations
- Endterm Test
[edit] Course Details
[edit] Vector Spaces and linear mappings
definition of a vector space - linear combinations - spanning sets - subspaces - linear spans, row space, column space - linear dependence and linear independence - basis and dimension - rank of a matrix - coordinates
linear mappings - kernel and image - singular and nonsingular linear mappings, isomorphisms - matrix representation - change of basis
References: LL chapters 4,5,6.
[edit] Inner products, orthogonality and diagonalization
inner product - norm - Cauchy Schwarz inequality - orthogonality - orthogonal sets and bases - orthogonal projection - Gram Schmidt orthogonalization process - orthogonal matrix
quadratic functions - positive definite matrix - diagonalization, eigenvalues and eigenvectors
[edit] Orthogonal projections and multiple regression
minimal distance problem - least squares principle - forcing linear restrictions - uniqueness - orthogonality criterion - normal equations - linear orthogonal projection - generalized inverse
descriptive multiple regression - Cholesky decomposition - inverse matrix representation - residual variance - partial correlations
[edit] Convexity and separation
open and closed sets - continuous functions - maximum theorems
convex sets - separating hyperplane theorem - nullation lemma - Farkas lemma - strong separating hyperplane theorem - no arbitrage theorem - application to finance
Frobenius theorem - application to productive economies
References: SH2 chapter 13
[edit] Multivariable differentation
gradients and directional derivatives - convex and concave functions - Taylor's formula - inverse function theorem - implicit function theorem
partial elasticities - homogeneous functions - Euler's theorem - chain rule - implicit differentiation along a level curve - elasticity of substitution
References: SH1 section 11.8, chapter 12, SH2 chapter 2
[edit] Multiple integration and first order differential equations
multiple integrals - section theorem - Fubini's theorem - transformation theorem
separable equations - first order linear equations
References: SH2 chapter 4, SH1 sections 9.8 and 9.9
[edit] Differential equations
second order linear differential equations
linear systems of equations in the plane
partial differential equations
References: SH2 chapters 5-7 buy essays online
