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Regular version of the site
Master 2018/2019

## Mathematics for Economists

Type: Compulsory course (Financial Economics)
Area of studies: Economics
When: 1 year, 1 semester
Mode of studies: Full time
Master’s programme: Financial Economics
Language: English
ECTS credits: 5

### Course Syllabus

#### Abstract

The objective of the course is to equip the students with some of the theoretical foundations of the modern mathematics and what is more important with the analytical methods of solving problems posed by the micro and macro analysis Prerequisites include undergraduate level mathematics: Calculus (both single and multi-dimensional), Linear Algebra, Probability theory and Mathematical Statistics, Ordinary Differential Equations. #### Learning Objectives

• The course has been designed to convey to the students how mathematics can be used in the modern micro and macro economic analysis
• Upon completion an individual will:  have the ability to solve differential equations and systems of differential equations,
•  have acquired the knowledge of the methods of the optimal control theory and dynamic programming and its applicability for solving problems in economics,
•  have developed skills in working with the Brownian and Wiener stochastic processes and have the idea how Ito’s integral is applied. #### Expected Learning Outcomes

• Apply multidimensional calculus, optimization to economic problems
• Apply methods of linear algebra to economic issues
• Apply the theorems that provide sufficiency conditions in the problems of optimization
• Apply statistical methods to economic tasks
• Handle first-order DE and linear DE of higher order with the constant coefficients
• Solve problems of calculus of variations as well as optimal control theory
• Apply Lagrange method as well as analysis of Bellman’s equations to macroeconomics #### Course Contents

• Multidimensional calculus, basics of optimization
1. Euclidean spaces: basic notions and definitions  vector  distance  open and closed sets  neighborhood of a point, limiting points, boundary points  bounded sets, compact sets 2. Functions and their generalizations  vector-functions  limit of a function  continuity 3. Multidimensional calculus  Total differential and more - partial derivative - relation between partial and total derivatives - implicit function theorem - higher-order derivatives and differentials - Young’s theorem, Hessian 4. Optimization in many variables. Unconstrained optimization at first followed by constrained optimization  concept of extrema  Necessary conditions of extrema  Bordered Hessians
• Linear Algebra
Basic notions, definitions and propositions  operations on matrices  linear spaces and subspaces, their properties  Gauss method of solving linear systems  eigenvalues and eigenvectors (definition, relation to the matrix rank, case of a symmetric matrix)  quadratic forms: sign-definiteness of for  kernel and image of a linear operator  Eucleadean spaces  orthogonalization by Gramm-Schmidt’s method  quadratic form sign-definiteness criterion (by eigenvalues)  Sylvester’s criterion  reduction of a matrix to a diagonal form  Projectors as operators in vector spaces  Normal Jordan form of a matrix and its applications  Notion of a pseudoinverse matrix
• Convex analysis and Kuhn-Tucker theorem
1. Convexity (convexity of a set, convex/concave functions, their properties)  separability theorem, separating hyperplane  saddle point  necessary and sufficient conditions of concave functions  strict convexity of a function 2. Unconstrained optimization in many variables  Taylor’s expansion in a multivariable case  Jacobi’s matrix  sufficient conditions for extrema 3. Constrained optimization  Lagrange’s classic problem 4. Constrained optimization with inequality constraints  necessary and sufficient conditions for extrema  problem modification for the nonnegative variables  differential characteristics of Kuhn-Tucker conditions  the meaning of Lagrange multiplier
• Theory of probability and statistics
1. random variable, sample space 2. cumulative distribution function and its density 3. uniform distribution 4. normal distribution, reduction of the Gaussian variable to variable 5. standard expectation E[ X], E[f (X)] 6. initial and central moments 7. joint distributions of the random variables 8. conditional distributions 9. iterated expectations formula 10. limiting densities 11. covariation and correlation 12. standard normal vector and its properties 13. marginal and conditional normal distributions 14. quadratic forms in a standard normal vector 15. X2 distribution and its properties 16. Student’s distribution and its properties 17. Fisher’s distribution and its properties 18. point estimation of parameters 19. unbiasness and efficiency of estimators 20. elements of large-sample distribution theory 21. convergence in probability and convergence in distribution 22. asymptotic distribution 23. interval estimation 24. hypothesis testing 25. errors of the first and second type 26. critical region of the test, decision rule
• Differential Equations
1. First-Order and higher-order Ordinary Differential Equations (a) Types of equations (b) Linear, first-order differential equations with constant and variable coefficients (c) Higher-order differential equations, the case of linear with the constant coefficients 2. Systems of Linear Ordinary Differential Equations (a) Phase Diagrams. (b) Analytical Solutions of Linear, Homogeneous Systems. (c) The Relation between the Graphical and Analytical Solutions. (d) Stability. (e) Analytical Solutions of Linear, Nonhomogeneous Systems. (f) Linearization of Nonlinear Systems
• Dynamic Optimization in Continuous Time
1. The Typical Problem. 2. Derivation of the First-Order Conditions. 3. Transversality Conditions. 4. The Behavior of the Hamiltonian over Time. 5. Sufficient Conditions. 6. Infinite Horizons. Example: The Neoclassical Growth Model. 7. Transversality Conditions in Infinite-Horizon Problems. 8. Summary of the Procedure to Find the First-Order Conditions. 9. Present-Value and Current-Value Hamiltonians. Multiple Variables.
• Finite-Horizon Dynamic Programming
1. Examples of the Dynamic Programming Problems 2. Histories, Strategies and the Value function 3. Existence of an Optimal Strategy 4. The Bellman Equation 5. Stationary Strategies 6. Example: the Optimal Growth Strategy
• Uncertainty, information, and stochastic calculus
1. Probability essentials  Sigma-algebras  Basic properties of sigma-algebras  Borel sigma algebras  Measurable functions  Probability as measure  Expectation 2. Conditional expectation  Definition  Calculation of conditional expectation  Properties of conditional expectations 3. Discrete-time stochastic processes  Filtration  Adapted process  Predictable process  Markov process  Markov chains  Examples 4. Martingales  Definitions of martingales  Properties  Examples  Random walk 5. Continuous-time stochastic process  Arithmetic and geometric Brownian motion  martingales in continuous time  multi-dimensional processes 6. Ito calculus  Stochastic integral  Ito’s lemma  SDE 7. Change of measure  Girsanov theorem  solution of Black-Scholes model via Girsanov theorem 8. Introduction to Matlab  Basic matrix operations  functions  scripts  graphs  flow control #### Assessment Elements

• home work
• math refresher
Students who miss the examination due to a valid reason are assigned an additional date to sit it.
• midterm exam
• final exam #### Interim Assessment

• Interim assessment (1 module)
0.2 * home work + 0.4 * math refresher + 0.4 * midterm exam
• Interim assessment (2 module)
0.35 * final exam + 0.15 * home work + 0.5 * Interim assessment (1 module) #### Recommended Core Bibliography

• Mathematics for economists, Simon C. P., Blume L., 1994