Studying at the University of Verona
Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.
Study Plan
This information is intended exclusively for students already enrolled in this course.If you are a new student interested in enrolling, you can find information about the course of study on the course page:
Laurea magistrale in Mathematics - Enrollment from 2025/2026The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University.
Please select your Study Plan based on your enrollment year.
1° Year
| Modules | Credits | TAF | SSD |
|---|
| Modules | Credits | TAF | SSD |
|---|
| Modules | Credits | TAF | SSD |
|---|
3 course to be chosen among the followingOne course to be chosen among the followingLegend | Type of training activity (TTA)
TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.
Stochastic differential equations (2016/2017)
Teaching code
4S001444
Teacher
Coordinator
Credits
6
Language
English
Scientific Disciplinary Sector (SSD)
MAT/06 - PROBABILITY AND STATISTICS
Period
II sem. dal Mar 1, 2017 al Jun 9, 2017.
Learning outcomes
This course is devoted to the presentation of basic results of theory of stochastic differential equations and their models in applied sciences.
The necessary prerequisites of Probability and Real Analysis for this course are briefly reviewed in the fi rst courses.
Program
1. Review of stochastic analysis
1.1 Stochastic processes
1.2 Brownian motion
1.3 The It^o integral
1.4 The It^o formula
2. Stochastic differential equations (SDEs)
2.1 Existence theorems for strong solutions
2.2 Martingale (weak) solutions
2.3 Linear SDEs
2.4 Backward stochasstic differential equations
2.5 Explicit solutions to SDE
3. SDEs in Physics and Economics
3.1 Langevin 's equation
3.2 The Ornstein-Uhlenbeck process
3.3 The geometric Brownian motion equations
3.4 The Black & Scholes model for the stock price model
3.5 Population growth model
4. The It^o diffusion
4.1 The Kolmogorov equation for It^o's diffusion
4.2 Examples to stock-price equation, Ornstein-Uhlenbeck equation
4.3 The Feynmann-Kac formula
4.4 The Fokker-Planck equation
5. Stochastic optimal control problems
5.1 Open loop stochastic opotimal control problems; necessary conditions
of optimality
5.2 Stochastic optimal feedback controllers
5.3 The dynamic programming equation
5.4 The linear stochastic regulator problem
6. Stochastic optimal control in Economics and Finance
6.1 The stochastic production inventory problem
6.2 Optimal portfolio selection problem
6.3 The Merton model of optimal portfolio
6.4 The stochastic advertizing problem
7. Optimal stopping
7.1 It^o's formula with stopping times
7.2 The Dirichlet problem { stochastic approach
7.3 The optimal stopping problem
7.4 Example to American option price
8. Stochastic partial differential equations
8.1 The reaction-diffusion equation
8.2 The stochastic heat equation in Rn
8.3 The stochastic wave equation
8.4 Example to population dynamics and neurophysiology
Bibliography (selective)
[1] L.C. Evans, An Introduction to Stochastic Differential Equations,
U.C. Berkeley, 2000.
[2] B. Oksendall, Stochastic Differential Equations, Springer, 2000.
[3] W. Fleming, R. Rishel, Deterministic and Stochastic Optimal Control,
Springer-Verlag, 1975.
Examination Methods
The nal result will be calculated in the following way:
1◦ 30 % will be obtained from 1 hour middle term quiz.
2◦ 70 % will be obtained from 2 hours written exam on 12th of June.
