# 30063 - MATEMATICA - MODULO 2 (APPLICATA) / MATHEMATICS - MODULE 2 (APPLIED)

For the instruction language of the course see class group/s below

Class group/s taught in English

Class-group lessons delivered on campus

A basic knowledge is recommended on: a) demonstration techniques; b) one-variable differential calculus; c) linear algebra.

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 aim, 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 axiomatic-deductive system. To complete the course, some basic knowledge in probability calculus and mathematical finance is given.

- Linear algebra. Eigenvalues and eigenvectors of a symmetric matrix, spectral theorem. Quadratic forms and their classification with respect to their sign, Sylvester-Jacobi's theorem.
- N-variable differential calculus. Hessian matrix, second-order 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, Lipschitz property, 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.
- Introduction to differential equations. Cauchy problem, existence and uniqueness theorem, notable examples of differential equations.
- 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. Axiomatic approach, compound accumulation. Financial operations, annuities, investments, loans. DCF, NPV, internal rates of a financial operation. Financial markets. Portfolios, payoffs, contingent claims. Law of one price. Arbitrages, no arbitrage conditions. Fundamental theorem of finance.

For the entire academic year 2022/2023, students not belonging to the first year and who have not yet passed the exam will be responsible of a specific exam syllabus. Those students will receive detailed info.

- Know the fundamental notions of linear algebra, n-variable differential calculus, integral calculus, differential equations, probability calculus, and mathematical finance.
- Articulate those notions in a conceptually and formally correct way, adequately using definitions, theorems, and proofs.

- Apply the fundamental theoretic results in linear algebra, n-variable differential calculus, integral calculus, differential equations, 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.

- Face-to-face lectures
- Online lectures
- Exercises (exercises, database, software etc.)

Online lectures will be included or not according to the external constraints and will be analogous to the usual face-to-face lectures.

Exercise sessions are dedicated to the application of the main theoretical results obtained to problems and exercises of various nature.

Continuous assessment | Partial exams | General exam | |
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x | x |

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 on-campus partial exams. Each of the two partial exams contains both open-answer questions and closed-answer 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, closed-answer 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 open-answer 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 on-campus general exam. This exam contains both open-answer questions and closed-answer 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, closed-answer 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 open-answer 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 Università Bocconi.

For the entire academic year 2022/2023, although they are responsible of a specific exam syllabus based on using specific teaching materials, all students not belonging to the first year and who have not yet passed the exam will have the same exam rules as above. Those students will receive detailed info.

Texts:

- S. CERREIA VIOGLIO, M. MARINACCI, E. VIGNA, Principles of Mathematics and Economics, Milano (draft version, available as a pdf file).
- Integrative teaching materials.

For the entire academic year 2022/2023, all students not belonging to the first year and who have not yet passed the exam will use specific teaching materials. Those students will receive detailed info.