CODE 109055 ACADEMIC YEAR 2022/2023 CREDITS 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA SCIENTIFIC DISCIPLINARY SECTOR MAT/06 TEACHING LOCATION GENOVA SEMESTER 2° Semester TEACHING MATERIALS AULAWEB OVERVIEW The course introducts stochastic calculus and martingale theory, that naturally intervene in applications in economics. AIMS AND CONTENT LEARNING OUTCOMES Introduction to Stochastic Calculus and Martingale Theory. Applications in finance. AIMS AND LEARNING OUTCOMES Upon completion of the course, the student knows the fundamentals of the theory of stochastic processes in discrete and continuous time; he also learns the mathematical basis of stochastic calculus according to Itô. He/she is able to handle advanced tools of probability and stochastic process theory and sees some applications to the financial world. PREREQUISITES Probability Theory TEACHING METHODS Letures. SYLLABUS/CONTENT Conditional Expectation/filtrations and stopping times/discrete-time martingale (Doob's inequalities, convergence results; Doob-Meyer decomposition)/continuous-time martingale (Continuous or continuous right-handed trajectory versions with limits from the left. Kolmogorov's theorem. Continuous-time martingales and their properties. Martingales closed by an integrable or integrable square random variable) Brownian motion (scale change invariance property, strong Markov property, reflection principle, law of maximum, level crossing times, geometric Brownian motion, multidimensional Brownian motion, recurrence and transience. Brownian motion with drift. Ornstein-Uhlenbeck process, Bessel process, Brownian bridge.)/ Stochastic integration/Simulation MB/ Applications to finance. RECOMMENDED READING/BIBLIOGRAPHY A. Pascucci, PDE and Martingale methods in Option Pricing, Bocconi & Springer Series (2010) P. Baldi, S. Shreve, Stochastic Calculus and Finance P. Baldi, Equazioni differenziali stocastiche e applicazioni, Bologna, Pitagora (2000) Mörters, Peres, Brownian Motion F. Caravenna, Moto browniano e analisi stocastica TEACHERS AND EXAM BOARD EMANUELA SASSO Ricevimento: By appointment by email. VERONICA UMANITA' Ricevimento: By appointment by email. Exam Board VERONICA UMANITA' (President) EMANUELA SASSO LESSONS LESSONS START The class will start according to the academic calendar. Class schedule The timetable for this course is available here: Portale EasyAcademy EXAMS EXAM DESCRIPTION Oral test. Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools. ASSESSMENT METHODS Verification of learning is by oral examination only and will focus on topics covered in class. The student will be expected to show correctness in mathematical language and formalism, be well acquainted with the mathematical objects and results of the course, and be able to use them naturally.
