- Vector spaces. Operations between vectors and matrices. Determinant. Linear transformations and matrices. Evigenvalues and eigenvectors.
- Functions of several variables: Taylor Series, Double Integrals, Differential, Partial Derivatives. Maximum and minimum. Linear differential equations.
- Random variables, distributions and laws, computation of probability laws. Central limit theorems.
- Poisson processes, Markov chains. Random walk
-Financial transactions and markets. Interest rates
-I.Khuri-Advanced Calculus. Second Edition Wiley Series in Probability and Statistics.
-R.Magnus e H. Neudecker. "Matrix differential Calculus with Applications in Statistics". Third Edition. John Wiley and Sons.
- M.Bramanti, C.D. Pagani e S.Salsa. Calcolo infinitesimale e algebra lineare. Zanichelli.
-S.Abeasis. Elementi di algebra lineare e geometria. Zanichelli.
- P. Baldi, Calcolo delle probabilita`.
- S.M. Ross, Calcolo delle probabilita`.
-G. Scandolo, Matematica finanziaria.
The course aims to provide students with the knowledge and the ability to understand and apply the basic concepts and results of linear algebra and differential and integral calculus in two or more dimensions. The course also intends to develop the ability to use matrix and differential calculus, especially in the case of two real variable functions, in solving the exercises.
The course aims to provide to students the knowledge and understanding of concepts and results on discrete and continuous random variables to calculate probabilities required by concrete situations and to calculate the laws (or distributions) of one-dimensional and multidimensional random variables models for known models and random variables that cannot be studied using known models. The course aims to provide to students the knowledge and understanding skills of the limit theorems (with the related proof), Poisson processes and the Markov chains (based on the Monte Carlo method) with attention to capacity development to apply these results to problems that require the modeling to a concrete situation.
Particular attention is given to developing the communicative skills needed to expose the main arguments of the lessons and to solve the problems by justifying the correct statements and deductions with correct mathematical language.
Differential and integral calculation in one variable for real functions. Basic knowledge of algebra and geometry.
Lectures and discussion and correction of homework
Type of Assessment
The exam consists of two written examinations per session with open questions of two types. The first type in which the student should state and prove results explained during the lessons, with the aim to verifying the knowledge, the understanding and the quality of the exposition. A second type in which the questions are conceived to assess the ability of the students to apply their skills to problem modelling and solving, and to give the rigorous justification using formule and the appropriate scientific language. Moreover, the questions will be formulated to highlight whether the student is able to choose the best probabilistic models to solve the concrete situation described in the exercise.
In addition, concerning the part related to: "Financial transactions and financial markets, and interest rates", the student must solve a concrete mathematical problem related to a portfolio situation using appropriate scientific language.
Additionally, the students will have the opportunity to perform two partial tests that, if both successful, will allow them to access directly to the oral examination.
The student may also choose to take an oral test in order to improve the mark of the weigted mean of the written parts.
The course introduces the basic elements of linear algebra and real analysis in several variables, with particular emphasis on matrix calculus and differential and integral calculation in multiple variables. In addition, the course introduces the basic elements of stochastic processes with particular focus on the Poisson process and Markov chains and financial mathematical applications.
The main arguments are:
1) Vector spaces. Operations between vectors. Matrix operations. Determinant. Linear transformations and matrices. Eigenvalues and eigenvectors. Quadratic Forms.
2) N-dimensional euclidean vector space.Taylor Series. Functions in multiple variables. Riemann Integral. Double Integral. Total Derivatives and Partial Derivatives. Maximum and minimum for multiple variable functions. Newton-Raphson method.
Linear differential equations of first and second order.
3) Random variables, distributions and laws, computation of probabilistic laws. Convergence and approximation, limit theorems.
4) Stochastic processes: Poisson processes, Markov chains. Random walks.
5) Financial transactions and markets. Interest rates. Non-arbitrage principle. Definition of economic risk. Decisions under uncertainty and risk aversion. Risk Measures. First notions about portfolio selection problem.