Unit STOCHASTIC DIFFERENTIAL EQUATIONS
- Course
- Mathematics
- Study-unit Code
- A005439
- Curriculum
- Matematica per analisi di sistemi complessi e dati
- Teacher
- Irene Benedetti
- Teachers
-
- Irene Benedetti
- Hours
- 42 ore - Irene Benedetti
- CFU
- 6
- Course Regulation
- Coorte 2025
- Offered
- 2025/26
- Learning activities
- Caratterizzante
- Area
- Formazione matematica teorica avanzata
- Academic discipline
- MAT/05
- Type of study-unit
- Opzionale (Optional)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Stationary Processes, Martingales, Gaussian Processes. Brownian Motion and elements of Stochastic Calculus: Riemann-Stiljes integral, Ito integral, Ito process, stochastic differential equations.
- Reference texts
- Grimmett-Stirzaker: Probability and Random Processes; Clarendon Press, Oxford (1982). A. Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi University Press, Springer 2011
- Educational objectives
- Generally, after passing the exam, the student has a deep knowledge of the general properties of the main stochastic processes, and skillness in the methods of studying and connecting them, together with some ability in stochastic calculus. The students particularly motivated could be invited to face also some first-level research problems.
- Prerequisites
- Some basic notions of Elementary Probability and Measure Theory should be already known to the students.
- Teaching methods
- Lectures in classroom
- Other information
- Student office: https://www.unipg.it/personale/irene.benedetti/didattica visit the webpage: www.unistudium.unipg.it For info on support services for students with Specific Learning Disorders and/or Disabilities please refer to the web page: http://www.unipg.it/disabilita-e-dsa
- Learning verification modality
- Oral exam: the test usually lasts about 40 minutes. The student should give definitions, theorems and proofs contained in the program as well as solve some very simple exercises. The aim of the colloquium is to evaluate if and to what extent the student is acquainted with the main topics studied, and check his/her capability in handling them, establishing connections and consequences. For info on support services for students with Specific Learning Disorders and/or Disabilities please refer to the web page: http://www.unipg.it/disabilita-e-dsa
- Extended program
- Martingales: general properties, convergence theorems, characterization in L_2. Optional theorem and Wald Formula. Gaussian processes: general theory, examples, Wiener process and its properties. Brownian Motion: existence and approximation, properties if its trajectories, scale invariance, Iterated Logarithm Theorem and the Arcsin Law. Stochastic Integration: Stieltjes and Ito integrals. Ito formulas and stochastic differentials. Stochastic differential equations: existence and uniqueness theorem, methods of solution in the linear case.