- Docente: Daniele Ritelli
- Credits: 8
- SSD: MAT/05
- Language: Italian
- 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
Prelininary notions
Differential and integral one dimensionale calculus, infinite series linear algebra (rank, solution of liner systems, determinant). The student come consult chaptersi 4, 5, 6, 7 and 8i of "Lezioni di Analisi Matematica" publisher Esculapio, author Daniele Ritelli and chapter 5, 7, 11 of "Sistemi lineari e matrici" publisher Pitagora, authors Barnabei e Bonetti
Contents
Elements of calculus in several variables. Lebesgue measure and Lebesgue integral. Abstract measure. Probability measure. Beppo Levi's Theorem (monotone convergence), Fatou's Lemma, 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. Probability integral. Ordinary differential equations of first order: linear and separable. Second order linear differential equations. Partial differential equations: the heat equation integrated by means of Fourier transform. Constant coefficients Parabolic equations. Black Schoies equation and its integration
Readings/Bibliography
Lecture notes prepared by the instructor are availalble. The main sources are:
W. Rudin: Principles of Mathematical Analysis. Chap. 10. Mac Graw Hill 1986
B. Osgood: The Fourier Transform and its Applications.
http://arni.epfl.ch/_media/courses/circuitsandsystems2011/book-2009.pdf
H. Hsu: Probability, Random Variables and Random Processes,
MacGraw Hill
F. Coppex: Solving the Black-Scholes equation: a
demystification. www.francoiscoppex.com/blackscholes.pdf
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.
Teaching tools
Video beamer. Blackboard.
Links to further information
http://www.ams.org/mathscinet/MRAuthorID/618511
Office hours
See the website of Daniele Ritelli