30063  MATEMATICA  MODULO 2 (APPLICATA) / MATHEMATICS  MODULE 2 (APPLIED)
Dipartimento di Scienze delle Decisioni / Department of Decision Sciences
Per la lingua del corso verificare le informazioni sulle classi/
For the instruction language of the course see class group/s below
Docente responsabile dell'insegnamento / Course Director:
SIMONE CERREIA VIOGLIO
SIMONE CERREIA VIOGLIO
Class group/s taught in English
Suggested background knowledge
A basic knowledge is recommended on:
a) proof techniques;
b) onevariable differential calculus;
c) linear algebra.
Mission & Content Summary
MISSION
Following the ideas of module 1, the aim of this course is to give students the basic mathematical knowledge and instruments that are necessary to cope with the quantitative study of problems in Economics, Finance and Management. In order to reach this goal, it is necessary that students understand the internal structures and the essential procedures of Mathematics, and that they get to comprehend the nature of Mathematics as an axiomaticdeductive system. To complete the course, some basic knowledge in probability calculus and mathematical finance is given.
CONTENT SUMMARY
 Linear algebra. Eigenvalues and eigenvectors of a symmetric matrix, spectral theorem. Quadratic forms and their classification with respect to their sign, SylvesterJacobi's theorem.
 Nvariable differential calculus. Hessian matrix, secondorder conditions for unconstrained local maximizers /minimizers, case of the concave / convex differentiable functions, unconstrained optimization problems. Implicitly defined real functions of one real variable, Dini's theorem. Constrained optimization problems, Lagrangean function, Lagrange's theorem.
 Integral calculus. Riemann integral for a bounded function, integrability conditions, classes of integrable functions, properties of the Riemann integral, integral mean value. Indefinite integral, first fundamental theorem of calculus. Integral function, second fundamental theorem of calculus. Properties of the indefinite integral, integration by parts, integration by substitution. Improper integral, integrability criteria. Stieltjes integral for a bounded function f with respect to an increasing function g.
 Probability calculus. Axiomatic approach, probability measures. Random variables: distribution function, probability function, probability density function, notable examples. Expected value and variance of a random variable, moments of a random variable.
 Mathematical finance. Accumulation, discount. Compound accumulation. Financial operations. Financial markets. Portfolios, payoffs, contingent claims. Law of one price. Arbitrages, no arbitrage conditions. Fundamental theorem of finance.
Intended Learning Outcomes (ILO)
KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
 Know the fundamental notions of linear algebra, nvariable differential calculus, integral calculus, probability calculus, and mathematical finance.
 Articulate these notions in a conceptually and formally correct way, adequately using definitions, theorems, and proofs.
APPLYING KNOWLEDGE AND UNDERSTANDING
At the end of the course student will be able to...
 Apply the fundamental theoretic results in linear algebra, nvariable differential calculus, integral calculus, probability calculus, and mathematical finance to the resolution of problems and exercises.
 Actively search for the most adequate ideas and deductive chains, in order to prove possible links between the properties of mathematical notions and to solve assigned problems.
 Interpret the fundamental theoretical results in the framework of the mathematical modeling processes that are necessary for the analysis of problems in Economics, Finance, and Management.
Teaching methods
 Facetoface lectures
 Online lectures
 Exercises (exercises, database, software etc.)
DETAILS
Online lectures will be included or not according to the external constraints and will be analogous to the usual facetoface lectures.
Exercise sessions are dedicated to the application of the main theoretical results obtained to problems and exercises of various nature.
Assessment methods
Continuous assessment  Partial exams  General exam  


x  x 
ATTENDING AND NOT ATTENDING STUDENTS
Students are evaluated on the basis of a written exam. This exam may be taken in one of the two following ways.
 It can be split in two oncampus partial exams. Each of the two partial exams contains both openanswer questions and closedanswer questions; each one covers one half of the course syllabus; each one weighs for approximately one half of the final grade. Each type of questions contributes in a specific way to the assessment of the students' acquired knowledge. In particular, closedanswer questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises. While openanswer questions mainly aim at evaluating:
 The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
 The ability to actively search for deductive ideas and chains that are fit to prove possible links between the properties of mathematical objects.
 The ability to apply mathematical notions to the solution of more complex problems and exercises.
 It can be taken as a single oncampus general exam. This exam contains both openanswer questions and closedanswer questions, covers the entire course syllabus, and can be taken in one of the four general sessions which are scheduled in the academic year. This way is mainly meant for students who have withdrawn from the partials procedure or could not follow it. Each type of questions contributes in a specific way to the assessment of the students' acquired knowledge. In particular, closedanswer questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises. While openanswer questions mainly aim at evaluating:
 The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
 The ability to actively search for deductive ideas and chains that are fit to prove possible links between the properties of mathematical objects.
 The ability to apply mathematical notions to the solution of more complex problems and exercises.
We take a special care to adjust the raw grades assigned in each exam, to obtain final grades whose distribution follows as closely as possible the normal distribution of grades that is recommended by Bocconi University.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
Texts:
 S. CERREIA VIOGLIO, M. MARINACCI, E. VIGNA, Principles of Mathematics and Economics, Milano (draft version, available as a pdf file).
 Integrative teaching materials.
Last change 08/12/2023 10:12