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Stochastic differential equations
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Stochastic differential equations
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Academic year 2016/2017
- Course ID
- MAT0044
- Teacher
- Prof. Enrico Priola
- Year
- 2nd year
- Teaching period
- First semester
- Type
- D.M. 270 TAF B - Distinctive
- Credits/Recognition
- 6
- Course disciplinary sector (SSD)
- MAT/05 - analisi matematica
- Delivery
- Class Lecture
- Language
- English
- Attendance
- Mandatory
- Type of examination
- Oral
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Sommario del corso
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Course objectives
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.
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Results of learning outcomes
Knowledge of the Ito stochastic integral and the related stochastic differential equations. Knowledge of the relations between stochastic differential equations and Kolmogorov equations. Ability to apply stochastic differential equations to solve problems in applied sciences.
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Course delivery
Lessons in the classroom.
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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.
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Program
- 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
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- 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: Equazioni differenziali stocastiche e applicazioni, Pitagora Ed., Bologna, 2000.
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Class schedule
Days Time Classroom Monday 14:00 - 16:00 Aula 8 - Edificio Storico Polo di Management ed Economia Tuesday 16:00 - 18:00 Aula 9 - Edificio Storico Polo di Management ed Economia Lessons: dal 26/09/2016 to 09/12/2016
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Note
This course will be delivered at the ESOMAS Department.
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