# 87575 - METODI MATEMATICI PER LA FINANZA

## 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.

These notions will be verified by the instructor and in the absence of them it will be difficult to take the exam. Here it possibile to download an entry test https://www.dropbox.com/s/zjyzh5pc0vnqib4/acess.pdf?dl=0

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 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

Lecture notes prepared by the instructor, which cover all the teaching are available. The main sources are:

D. Ritelli, G. Spaletta: Introductory Mathematical Analysis for Quantitative Finance CRC press 2020 https://www.taylorfrancis.com/books/9781351245111

W. Rudin: Principles of Mathematical Analysis. Chap. 10. Mac Graw Hill 1986

B. Osgood: The Fourier Transform and its Applications.

F. Coppex: Solving the Black-Scholes equation: a demystification.

## Teaching methods

Lessons ex Cathedra. 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 5 students, to be returned to mandatory deadlines that, if carried out correctly, will affect the assessment of the written test.

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## Teaching tools

Video beamer. Blackboard.