Stochastic differential equations
Stochastic differential equations
Academic year 2018/2019
- Course ID
- Prof. Enrico Priola
- 2nd year
- Teaching period
- First semester
- D.M. 270 TAF B - Distinctive
- Course disciplinary sector (SSD)
- MAT/05 - analisi matematica
- Formal authority
- Type of examination
- PROBABILITY THEORY (MAT0034) and Analysis Canale 1 (MAT0032)
Sommario del corso
The course aims to put the student in a position to understand the mathematical formulation of various models of applied sciences and financial mathematics which involve stochastic differential equations. The course uses probabilistic concepts and tools that are developed in the course ``Probability Theory'' and elements of Functional Analysis (see ``Analysis''); these concepts are briefly mentioned in the first lectures. The proofs of the main results of the course are carried out completely. They show important links between Analysis and Probability. To improve the skills of reading and study the teacher proposes the reading of some scientific articles. Together with the course ``Stochastic Processes'' it suggests an approach to the research in stochastic environments. The course also provides basic concepts on parabolic equations of Kolmogorov type.
Results of learning outcomes
At the end of the course, students will know several important methods to study stochastic models; in particular they will know the Ito stochastic integral and the related stochastic differential equations. Moreover, they will understand relations between stochastic differential equations and Kolmogorov equations. They will be able to study applications of stochastic differential equations to solve problems in applied sciences
Lectures (48 hours, 6 CFU) are given in lecture rooms.
Learning assessment methods
Oral examination. Questions on the program (theorems, remarks and examples). Concerning the proofs we require to know in details 3 important proofs. Such required proofs are given in the folder ``Teaching material'' below. This folder also contains more information on the examination.
- Reminder of basic notions on measure theory and probability theory. Multidimensional Gaussian distributions.
- Brownian motion (its construction by means of Haar functions; regularity properties of trajectories); the Wiener measure.
- The Doob L^p estimates for martingales with continuous paths.
- The Ito stochastic integral (basic properties; comparison between the stochastic integral and the Riemann-Stieltjes integral)
- The Ito formula and its applications
- Stochastic differential equations (existence and uniqueness theorems)
- Markov property of solutions of stochastic differential equations; connections between stochastic differential equations and parabolic Kolmogorov equations
- Possible applications of stochastic differential equations to Mathematical Finance and Population Dynamics
Suggested readings and bibliography
- Lectures notes
- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, Second Edition, 1991.
- Arnold, L., Stochastic Differential Equations, Theory and Applications, New York. John Wiley & Sons. 1974
- P. Baldi: Stochastic Calculus. An Introduction Through Theory and Exercises. Springer, 2017
- P. Baldi: Equazioni differenziali stocastiche e applicazioni, Pitagora Ed., Bologna, 2000.
Days Time Classroom Wednesday 11:15 - 13:15 Aula 08 - Edificio Storico (piano terra) Polo di Management ed Economia Thursday 14:00 - 16:00 Aula 08 - Edificio Storico (piano terra) Polo di Management ed Economia
Lessons: dal 26/09/2017 to 21/12/2017
This course will be delivered at the ESOMAS Department.