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## Syllabus

EN IT

### Learning Objectives

Calculus and Optimization

LEARNING OUTCOMES:
Integrals and optimization in several variables.

KNOWLEDGE AND UNDERSTANDING:
Change of variables in integrals and use of polar coordinates.
Lagrangian multipliers to study optimization under costraints.
Simple cases of Kuhn-Tucker theorem.

APPLYING KNOWLEDGE AND UNDERSTANDING:
To be able to evaluate integrals in several variables (by means of Fubini Theorem, …).
Evaluation of integral by the derivation under the integral sign technique.
To solve simple differential equations.
To calculate the Hessian matrix and its eigenvalues.
To calculate local minimum and maximum for a several variable function.

MAKING JUDGEMENTS:
Translation in mathematical terms of optimizations problems from real life.

COMMUNICATION SKILLS:
To be able to present quantitative aspects of economic and financial models.

Linear Algebra and Probability

LEARNING OUTCOMES:
Basic properties of abstract vector spaces and linear transformations.
How the main discrete and (absolutely) continuous distribution arise from real life problems and their properties.

KNOWLEDGE AND UNDERSTANDING:
To be able to determine eigenvalues and eigenvectors of a matrix.
Symmetric matrices.
Notions of projections and idempotent matrices.
Meaning of the basic limit theorems: (weak) law of large numbers and central limit theorem.

APPLYING KNOWLEDGE AND UNDERSTANDING:
To know how to apply basic properties of matrix algebra with special emphasis on block matrices.
To know how to diagonalize a matrix (under suitable conditions).
Geometric meaning of conditional expectation and its applications to multivariate gaussian.

MAKING JUDGEMENTS:
How to model economic and financial situations using stochastic models.

COMMUNICATION SKILLS
To be able to present quantitative aspects of economic and financial models.

### Prerequisites

It is taken for granted that students have a basic knowledge of calculus and linear algebra. In particular they know: how to study a function in one variable, the fundamental theorem of calculus, how to evaluate a definite integral, how to study a system of linear equations, the basic geometry of three-dimensional space.
As a reference one can use the Appendices A1, A2, A3, A4 and Part I – Part II of the book by Simon-Blume.

### Program

Linear Algebra
Systems of linear equations. Matrix Algebra. Algebra of square ma- trices. Transpose and its properties. Determinant. Groups, fields, vector spaces. Linear independence and basis. Dimension of vector spaces. Linear transformations. Kernels. Scalar products. Cauchy-Schwartz inequality. Eigenvalues, eigenvectors and the characteristic polynomial of a square matrix. Basic properties of eigenspaces. Symmetric, and orthogonal matrices. Positive definite matrices. Projection operators. Cholesky decomposition. Diagonalizable matrices. The spectral theorem.
Calculus
Series. The complex numbers. Complex series and the complex exponential. The Euler formula. Differentiability for functions of several variables: examples and counterexamples. The gradient. The Jacobian matrix. The chain rule for differentials. Mixed partial derivative. The Schwartz (Young) theorem. Integration in n dimension. The Fubini theorem. The change of variable formula. Integration using polar coordinates. Differentiation un- der the integral sign. Introduction to differential equations. The Cauchy problem. The L^^2 scalar product on R^2, on C[0, 1] and for random variables. Quasiconcave functions. Implicit functions. The contraction mapping principle.
Optimization
The Taylor polynomial in n-dimensions. The Hessian matrix. Uncon- strained optimization: necessary and sufficient conditions for maxima and minima. Constrained optimization. Lagrangian function and Lagrange mul- tiplier. Introduction to Kuhn-Tucker. The Envelope Theorem.
Probability
Elements of a probability space. Algebras of events and information about random experiments. Introduction to combinatorial calculus. Finite probability spaces, probability measures, introduction to Kolmogorov theory. Conditional probability, total probability formula, Bayes formula. Independent events. Random variables and their properties. Probability distribution, distribution function and densities function of a random vari- able. Expectation and variance of a random variable and their properties. Expectation and variance for the main kinds of random variables. Covariance and scale-invariance of the correlation coefficient. Random vectors and their properties. Probability distribution, distribution functions and densities functions of a random vector. Independent random variables, covariance and correlation. Conditional expectation of a random variable and its prop- erties. Conditional expectation as best estimator. Geometric approach to the conditional expectation. Sequences of random variables. Convergence in probability and in law. The (weak) law of large numbers. The characteristic function. Central limit theorem. Multivariate Gaussian distribution. Conditional expectation for the bivariate gaussian.

### Books

C. P. Simon and L. Blume. Mathematics for Economists. Norton & Company

G. Casella and R.L. berger. Statistical Inference. Duxbury

A. Mas-Colell, M. D. Winston and J.R. Green. Microeconomic Theory

### Bibliography

T.M. Apostol. Calculus, Vol. I and Vol. II, Wiley & Sons.

D. C. Lay, Linear Algebra and Its Applications, Pearson

P. Biilingsley, Probability and Measure (Wiley Series in Probability and Statistics)

S. Ross, A First course in Probability

J.P. Romano and A.F. Siegel, Counterexamples in Probability And
Statistics (Wadsworth and Brooks/Cole Statistics/Probability Series)

P. Lockhart. A Mathematician Lament.

### Teaching methods

The course has lectures and practice sessions.

Also during the lectures, examples, problems and exercises play a central role in order to immediately apply the theoretical knowledge proposed.
This is done keeping in mind the areas (statistics, economics, finance) where the students will apply the mathematical contents of the course.

### Exam Rules

The students evaluation is done through a written examination.

To guarantee the achievement of the learning outcomes the structure of the examination has three parts. One part is devoted to the proof of theorems in order to check the theoretical understanding of the program (Learning Outcomes). In an intermediate situation is a second part made by quizzes (true-false alternative): the questions involved test the ability of the students to apply the theoretical comprehension to new situations (Applying Knowledge and Understanding). A third part is made by more standard exercises (Knowledge and Understanding).

Students who withdraw or fail an exam may take the exam again in the same exam session.