Finance and insurance

GABRIELE STABILE

The course provides students with the essential foundations of linear algebra, multivariable functions, unconstrained and constrained optimization, and solution methods for differential equations. These topics constitute the mathematical background necessary for implementing mathematical models in economics and finance. Students who successfully pass the exam will be able to understand and apply economic and financial modelling techniques. They will be able to use fundamental mathematical tools for analyzing financial problems (such as matrix diagonalization, examination of properties of multivariable functions, maximization/minimization of functions with or without constraints, and solving differential equations and systems) consistently with economic and financial theories. They will acquire knowledge of mathematical models and the ability to apply these methodologies to real problems, identifying the most appropriate model and correctly interpreting the results. The teaching approach, based on theoretical lectures and guided exercises, supports the development of independent judgment in the selection of mathematical tools, communication skills in presenting quantitative results, and autonomous learning abilities in pursuing advanced topics. Therefore, the course represents a fundamental pillar of the study program, closely related to the course Probability and Stochastic Processes for Insurance and Finance, and essential for subsequent courses such as Risk Theory, Quantitative Finance, Methods and Models for Finance, Time Series Analysis, and Actuarial Mathematics for Private Insurance.

The course provides students with the essential foundations of linear algebra, multivariable functions, unconstrained and constrained optimization, and solution methods for differential equations.

Specific Objectives (Dublin Descriptors):

Knowledge and understanding: Upon completion of the course, students will be able to identify and formulate new procedures for assessing economic and financial problems and to understand the outcomes derived from the application of mathematical models used to solve them.

Applying knowledge and understanding: Students who successfully pass the exam will be able to identify the most appropriate mathematical modelling approach to describe a given economic or financial context and to determine the most efficient methodologies to solve the related problem.

Making judgments: Thanks to the competences acquired during the course, students will be able to independently analyse problems in the economic and financial domain, assess the mathematical tools to be employed, and interpret the obtained results even in new contexts.

Communication skills: After passing the exam (consisting of a written test involving exercises and analysis of the obtained results), students who achieve a positive evaluation will be able to effectively describe the topics learned during the course, both orally and in written form, and communicate them to specialists and non-specialists alike.

Learning skills: After successfully completing the exam, students will have a full command of advanced mathematical concepts and related tools for handling economic and financial models. This competence will enable them to develop a method for autonomously acquiring new knowledge and skills, both at the theoretical and practical levels.

FUNDAMENTALS
Real numbers: operations and properties. Sets and intervals. Absolute value
and distance. Exponents and radicals.
Algebraic expressions: monomial and polynomial, operations, factorization,
special formulas. Rational expressions: operations, least common
denominator of polynomials, rationalizing the denominator.

EQUATIONS AND INEQUALITIES
Linear equations. Quadratic equations: discriminant, quadratic formula.
Equations with higher powers, radicals, absolute values.
Linear inequalities. Quadratic inequalities.
Inequalities involving quotients, absolute values. System of inequalities.

EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Definition of exponential function. Exponential equations and inequalities.
Definition and properties of logarithms.
Equations and inequalities involving exponential and logarithmic functions.

ANALYTIC GEOMETRY
Cartesian coordinates in the plane, distance between two points, equation of a
line, parallel and perpendicular lines, distance from a point to a line, parabola,
line–parabola intersections and conditions for tangency.
DIFFERENTIAL CALCULUS
Definition of derivative. Geometric interpretation of the derivative – Differentiation rules: related theorems. Mean value theorems: Rolle’s, Cauchy’s, Lagrange’s – Increase and decrease and related theorems – Indeterminate forms. de L'Hôpital's Theorem – Differential – Derivative of a composite function – Second and higher-order derivatives – Concave and convex functions at a point – Inflection points. Convexity and concavity. Taylor’s formula. Asymptotes – Function analysis.

INTEGRAL CALCULUS
Definition of integral – Integral: geometric meaning. Properties – Mean value theorem – Definite integral. Integral function – Fundamental theorem of calculus – Calculation of definite integrals using antiderivatives – Indefinite integrals – Integration methods: by decomposition, by transformation, by substitution, by parts.

Introduction
Set Theory: Relations and operations between sets and their properties; Cartesian product; power set; illustrative examples. Quantifiers. Absolute value of a real number.
Metric spaces: definition of distance; Euclidean distance and other examples of distance functions.

Linear algebra
Vectors and scalar quantities. Vector operations. Inner (dot) product of vectors in the plane or in space. Properties of the inner product. Vector norm (length). Characterization of orthogonality using the inner product. Linear combinations of vectors. Notion of subspace, generated subspace, and spanning sets. Linearly independent vectors. Basis and dimension of a vector space.
Definition of linear transformation. Examples of linear transformations. Correspondence between linear maps and matrices. Kernel and image of a linear transformation. The dimension theorem (rank-nullity theorem).
Eigenvectors and eigenvalues of an endomorphism. Eigenspaces. Characteristic polynomial. Existence of a basis of eigenvectors. Matrix diagonalization.
Quadratic Forms
Definition. Matrix associated with a quadratic form. Sign of a quadratic form and criteria for positive definiteness, semi-definiteness, and negative definiteness.
Real-Valued Functions of Several Variables
Sequences in Euclidean spaces. Neighborhoods, open and closed sets. Accumulation points. Functions of two or more variables. Domain of definition. Level curves. Continuity. Partial derivatives. Differentiable functions. Directional derivatives. Gradient vector. The gradient as the direction of maximum increase. The total differential theorem. Continuity and differentiability of differentiable functions. Examples of non-differentiable functions. Higher-order derivatives. Taylor’s formula.
Unconstrained optimization: Necessary condition for a point to be a local extremum. Hessian test for determining the nature of stationary points. Computation of absolute maxima and minima on compact domains.
Constrained optimization: The method of Lagrange multipliers. Economic interpretation of the multipliers. Bordered Hessian matrix. Examples of applications in economics. Introduction to constrained optimization problems with inequality constraints.
Introduction to Ordinary Differential Equations
Examples of differential models. Definitions: n-th order differential equation, normal form, solution, general integral, initial conditions, Cauchy problem.
First-order ODEs: Separable equations, linear equations, Bernoulli equations.
Second-order ODEs: Second-order linear equations with constant coefficients. Structure of the general solution: relation between the general solution of the complete equation and the corresponding homogeneous equation. Finding a particular solution of the complete equation using the method of undetermined coefficients (method of resemblance): discussion of selected cases (*): polynomial, exponential, and trigonometric functions.

C. P. Simon, L. Blume, Mathematics for Economists, Norton, 1994.
C. Mattaglia, F. Privileggi, Matematica per le scienze economiche e sociali. Algebra lineare, funzioni di più variabili e ottimizzazione
statica. Apogeo
The supplementary teaching material (lecture slides, exam prototypes) is available on the Classroom site of the course

Rangarajan K. Sundaram
A First Course in Optimization Theory
Cambridge University Press, 1996.

Angelo Guerraggio, Sandro Salsa
Metodi matematici per l'economia e le scienze sociali
Giappichelli, 1997.

The course is issued in a traditional way, by classroom lessons mainly dedicated to illustrating and explaining the formal theoretical concepts and the quantitative tools used to represent and solve problems. The course is structured so that theoretical concepts are presented with mathematical rigor and are accompanied by a discussion of their practical aspects, with particular reference to the economic and financial fields. In addition to the theoretical lessons there are also classes focused on the solutions of practical exercises and self-assessment test.

Frequency is not compulsory but recommended

The exam consists of a written test, lasting 2 hours, made up of exercises covering the entire course program.
Each question may in turn include several sub-questions. The purpose of the exam is to assess the student’s ability to use theoretical results to solve applied problems. To this end, the student is required to show all steps and provide references to the theoretical results used in the solution.
In assessing the exam, the final grade takes into account the following elements:
1. the logic followed by the student in solving the question %30
2. the ability to solve exercises by choosing the appropriate procedures %40
3. correctness of the calculation procedures %20
4. understanding and use of specific and symbolic language %10

- Study problems of unconstrained and constrained optimization
- Study the linead dependence/independence between vectors
- Matrix diagonalization
- Solve first or second order ordinary differential equations

Sample exams and exam questions from previous sessions are available at the Classroom site of the course
https://classroom.google.com/c/MjM1MzI3OTYxMzda?cjc=lfmdxmxa

In assessing the exam, the final grade takes into account the following elements:
1. the logic followed by the student in solving the question %30
2. the ability to solve exercises by choosing the appropriate procedures %40
3. correctness of the calculation procedures %20
4. understanding and use of specific and symbolic language %10

• Weekly Course Schedule
  Week 1
  Course introduction. Sets: relations and operations between sets and their properties; Cartesian product; power set; examples. Number sets. Quantifiers. Absolute value of a real number. Metric spaces: concept and definitions of distance. Euclidean distance and other examples of distance functions.  
  • Books: Chapter 1 - C. Mattaglia, F. Privileggi, Matematica per le scienze economiche e sociali. Algebra lineare, funzioni di più variabili e ottimizzazione.


• 2 Introduction to vector spaces. Physical quantities and vectors. Vector operations. Inner (dot) product of two vectors in the plane or space. Properties of the inner product. Vector norm (length). Characterization of orthogonality via the inner product. Linear combinations of vectors. Notion of subspace, generated subspace, and spanning sets. 
  • Books: Chapter 2 - C. Mattaglia, F. Privileggi, Matematica per le scienze economiche e sociali. Algebra lineare, funzioni di più variabili e ottimizzazione.


• 3 Linearly independent vectors. Basis and dimension of a vector space. 
  • Books:  Chapter 11 - C. P. Simon, L. Blume, Mathematics for Economists, Norton, 1994.


• 4 Definition of linear map. Examples of linear transformations. Correspondence between linear maps and matrices. Kernel and image of a linear map. The dimension theorem (rank-nullity theorem). 
  • Books: Chapter 3 - C. Mattaglia, F. Privileggi, Matematica per le scienze economiche e sociali. Algebra lineare, funzioni di più variabili e ottimizzazione.


• 5 Eigenvectors and eigenvalues of an endomorphism. Eigenspaces. Characteristic polynomial. Existence of a basis of eigenvectors. Matrix diagonalization. 
  • Books: Chapter 23 - C. P. Simon, L. Blume, Mathematics for Economists, Norton, 1994.


• 6 Quadratic forms. Definition. Matrix associated with a quadratic form. Sign of a quadratic form and criteria for positive definiteness, semi-definiteness, and negative definiteness. 
  • Books: Chapter 16 - C. P. Simon, L. Blume, Mathematics for Economists, Norton, 1994  


• 7 Sequences in Euclidean spaces. Neighborhoods, open and closed sets. Accumulation points. 
  • Books: Chapter 12 - C. P. Simon, L. Blume, Mathematics for Economists, Norton, 1994. 


• 8 Functions of two or more variables. Domain of multivariable functions. Level curves. Continuity. Partial derivatives. Differentiable functions. 
  • Books: Chapter 4 - C. Mattaglia, F. Privileggi, Matematica per le scienze economiche e sociali. Algebra lineare, funzioni di più variabili e ottimizzazione.  


• 9 Directional derivatives. Gradient vector. The gradient as the direction of maximal increase. Total differential theorem. Continuity and differentiability of differentiable functions. Examples of non-differentiable functions. Higher-order derivatives. Taylor’s formula. 
  • Books: Chapter 4 - C. Mattaglia, F. Privileggi, Matematica per le scienze economiche e sociali. Algebra lineare, funzioni di più variabili e ottimizzazione.


• 10 Unconstrained optimization: necessary condition for a point to be a local extremum. Hessian test to determine whether a stationary point is a local extremum. Computation of absolute maxima and minima over compact domains. 
  • Books: Chapter 17 - C. P. Simon, L. Blume, Mathematics for Economists, Norton, 1994.


• 11 Constrained optimization. The method of Lagrange multipliers. Economic interpretation of the multipliers. Bordered Hessian matrix. Examples of applications in economics. Introduction to constrained optimization problems with inequality constraints. 
  • Books: Chapter 6 - C. Mattaglia, F. Privileggi, Matematica per le scienze economiche e sociali. Algebra lineare, funzioni di più variabili e ottimizzazione 


• 12 Introduction to ordinary differential equations: examples of differential models. Definitions: n-th order differential equation, normal form, solution, general integral, initial conditions, Cauchy problem. First-order equations: separable equations, linear equations, Bernoulli equations. 
  • Books: Chapter 24 - C. P. Simon, L. Blume, Mathematics for Economists, Norton, 1994.


• 13 Second-order ordinary differential equations. Second-order linear equations with constant coefficients. Structure of the general solution: relationship between the general solution of the complete equation and that of the homogeneous equation. Method of undetermined coefficients: discussion of selected cases (*): polynomials, exponentials, trigonometric functions. 
  • Books: Chapter 24 - C. P. Simon, L. Blume, Mathematics for Economists, Norton, 1994.