Academic Year 2023/2024

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Financial Markets and Institutions (cod. 0901)

Learning outcomes

Mathematical tools in measure theory and differential equations in view of applications in Mathematical finance

Course contents

Preliminary notions
It is assumed that the basic notions of one variable Calculus, Linear Algebra and Multivariable Calculus are mastered by the student, that is differential and integral one dimensional calculus, infinite series linear algebra (rank, solution of liner systems, determinant, eigenvalues and eigenvectors), elements of calculus in several variables. Suequences of functions. Complex numbers.

Here it possibile to download an entry test https://www.dropbox.com/s/zjyzh5pc0vnqib4/acess.pdf?dl=0


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 convergence Theorem. Derivation of integrals dependent by 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 of some useful special functions: Euler Gamma and Beta, Error function. Probability integral. 

Partial differential equations: the heat equation integrated by means of Fourier transform. Constant coefficients Parabolic equations. Black Scholes equation and its integration


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

Written examination of 2 hours, where is possible to use calculators and self made formularies. The aim of the exam is to detect the capability of the student to face both theoretical and practical problems in the branch Mathematical Analysis which is useful in Finance.


During the course will be assigned (optional) homework, 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 analyze, related to an only instrumental mastery of the problems presented in the course, correct use of the methods at elementary level;
24-27: technically adequate preparation but with some limits with respect to the topics covered, good analytical skills, although not particularly articulate, correct use of methods at an intermediate level;
28-30: technically adequate preparation with respect to the topics dealt with, good analytical skills in complex problems, correct use of methods at standard level
30L: excellent and very thorough and exhaustive knowledge of the topics covered in the course, full mastery of the methods also from a theoretical point of view.


In the 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.

During the course will be assigned tasks (optional) to be carried out at home, even in groups of up to 5 items, to be returned to mandatory deadlines that if carried out correctly will affect the assessment of the written test
During the course will be assigned tasks (optional) to be carried out at home, even in groups of up to 5 items, to be returned to mandatory deadlines that if carried out correctly will affect the assessment of the written test

Teaching tools

Teaching resources

Video beamer and blackboard. Computer algebra to illustrate important topics.

The teaching material presented in class will be made 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 matriculation number, and which concern appointment requests or topics that are not covered by the course information presented here.

Links to further information


Office hours

See the website of Daniele Ritelli


Quality education Gender equality

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.