- Docente: Daniele Ritelli
- Credits: 6
- SSD: MAT/05
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Financial Markets and Institutions (cod. 0901)
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from Sep 09, 2024 to Oct 24, 2024
Learning outcomes
Mathematical tools in measure theory and differential equations in view of applications in Mathematical finance
Course contents
Preliminary notions
It is crucial that the student has a firm grasp of the basic notions of one-variable Calculus, Linear Algebra and Multivariable Calculus. This includes differential and integral one-dimensional calculus, infinite series linear algebra (rank, solution of liner systems, determinant, eigenvalues and eigenvectors), and elements of calculus in several variables. Sequences of functions and complex numbers are also essential. Mastering these prerequisites will ensure that you are well-prepared and confident in your understanding of the material.
Here, it is possible to download an entry test.
Contents
Ordinary differential equations
Ordinary differential equations of first order: linear and separable. Second-order linear differential equations.
Lebesgue measure and Lebesgue integral.
Abstract measure. Probability measure. Beppo Levi's Theorem (monotone convergence) dominated the convergence Theorem. Derivation of integrals dependent on a parameter. Integration in product spaces: Fubini's Theorem. Decomposition of integrals on R^2. Change of variable. Applications to the two-dimensional random variables. Absolute continuity: Radon-Nikodym Theorem. Review some useful special functions: Euler Gamma and Beta, Error function—probability integral.
Partial differential equations: the heat equation integrated using Fourier transform. Constant coefficients Parabolic equations. Black Scholes equation and its integration
Readings/Bibliography
D. Ritelli, G. Spaletta: Introductory Mathematical Analysis for Quantitative Finance CRC press 2020 https://www.taylorfrancis.com/books/9781351245111
Teaching methods
Lessons ex Cathedra aided by tablet and Comèuter Algebra. Homework
Assessment methods
The written examination is 2 hours when it is possible to use calculators and self-made formularies. The exam aims to detect the student's capability to face theoretical and practical problems in the branch of Mathematical Analysis, which is useful in Finance.
During the course, homework will be assigned (optional), even in groups of up to 3 students, to be returned to mandatory deadlines that, if carried out correctly, will affect the assessment of the written test.
Evaluation Criteria
18-23: minimal preparation and ability to analyse, related to only instrumental mastery of the problems presented in the course, correct use of the methods at the elementary level;
24-27: technically adequate preparation but with some limits concerning the topics covered; good analytical skills, although not particularly articulate, correct use of methods at an intermediate level;
28-30: technically adequate preparation concerning the topics dealt with, good analytical skills in complex problems, correct use of methods at standard level
30L: Excellent, very thorough, and exhaustive knowledge of the topics covered in the course, as well as a full mastery of the methods from a theoretical point of view.
In the (unfortunate) case of an online exam, the teacher will provide a Dropbox link for downloading the text and one for uploading, where the student will have to upload a PDF file with the grid of the answers given and the solution of the proposed problems.
Teaching tools
Video beamer and blackboard. Computer algebra to illustrate important topics.
The teaching material presented in class will be available to the student in electronic format through the university's institutional portal. Username and password are reserved for students enrolled at the University of Bologna.
Office hours can be delivered using Teams.
The teacher responds to e-mail messages duly signed by the student with Name, Surname and enrollment number, which concern appointment requests or topics not covered by the course information presented here.
Links to further information
Office hours
See the website of Daniele Ritelli
SDGs
This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.