CODE 109055 ACADEMIC YEAR 2024/2025 CREDITS 6 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA SCIENTIFIC DISCIPLINARY SECTOR MAT/06 LANGUAGE Italian (English on demand) TEACHING LOCATION GENOVA SEMESTER 2° Semester TEACHING MATERIALS AULAWEB OVERVIEW The course contains an introduction to stochastic calculus and martingale theory that naturally attends in applications in economics. The teaching contributes to the achievement of Sustainable Development Goals 4 and 5 of the UN 2030 Agenda. AIMS AND CONTENT LEARNING OUTCOMES Introduction to Stochastic Calculus and Martingale theory. Applications in mathematical finance. AIMS AND LEARNING OUTCOMES Upon completion of the course, the student will know the fundamentals of the theory of stochastic processes in discrete and continuous time, and will learn the mathematical foundations of Itô stochastic calculus by seeing some of its applications to the financial world. He/she will also be able to handle advanced tools of probability and the theory of stochastic processes. PREREQUISITES Program of the course "Probability. TEACHING METHODS Lectures + laboratory SYLLABUS/CONTENT Conditional hope/filtrations and stopping times/discrete-time martingale (Doob's inequalities, convergence results; Doob-Meyer decomposition)/continuous-time martingale (Continuous or continuous right-hand 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 VERONICA UMANITA' Ricevimento: By appointment by email. EMANUELA SASSO Ricevimento: By appointment by email. DAMIANO POLETTI Exam Board VERONICA UMANITA' (President) DAMIANO POLETTI EMANUELA SASSO (President Substitute) LESSONS LESSONS START According to the academic calendar approved by the Course Council. Class schedule The timetable for this course is available here: Portale EasyAcademy EXAMS EXAM DESCRIPTION Oral examination. Students with certified DSA, disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination arrangements that, while respecting the teaching objectives, take into account individual learning patterns and provide suitable compensatory tools. In order to request exam accommodations, the instructions detailed on Aulaweb https://2023.aulaweb.unige.it/course/view.php?id=12490#section-3 should be followed. In particular, accommodations should be requested significantly in advance (at least 10 days) of the exam date by writing to the teacher with a copy of the School Referring Teacher and the appropriate Office (see instructions). ASSESSMENT METHODS Verification of learning is through oral examination only and will focus on topics covered in class. The student will be expected to show correctness of language and mathematical formalism, to have a deep knowledge of mathematical objects and results of the course, and be able to use them naturally. Exam schedule Data appello Orario Luogo Degree type Note 19/09/2025 09:00 GENOVA Esame su appuntamento