Introduction to Stochastic Analysis (MAE818)
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General
School 
School of Science 

Academic Unit 
Department of Mathematics 
Level of Studies 
Undergraduate 
Course Code 
ΜΑΕ818 
Semester 
5 
Course Title 
Introduction to Stochastic Analysis 
Independent Teaching Activities 
Lectures (Weekly Teaching Hours: 3, Credits: 6) 
Course Type 
Special background 
Prerequisite Courses   
Language of Instruction and Examinations 
Greek 
Is the Course Offered to Erasmus Students 
Yes (in English) 
Course Website (URL)  See eCourse, the Learning Management System maintained by the University of Ioannina. 
Learning Outcomes
Learning outcomes 
Stochastic Analysis is the branch of mathematics whose objective is the study of Stochastic processes with nondifferentiable paths. One of the basic tools is the stochastic integral of Ito via which stochastic differential equations are defined. Stochastic differential equations are used to model and study random phenomena evolving in continuous time. Stochastic analysis has applications in areas as physics and finance. The aim of the course is to introduce the students to the basic notions, tools and applications of Stochastic Analysis. After the course the students will know:


General Competences 

Syllabus
Basic notions of stochastic processes in discrete and continuous time: Definition, notions of equality, distributions of stochastic processes, processes with continuous paths. Filtrations, stopping times, conditional expectation. Fundamental classes of stochastic processes: Martingales, Levy processes, Markov processes, transition probabilities and generators. Brownian motion: Definition, existence and uniqueness, basic properties (e.g.~analytic properties of paths, reflection principle, strong Markov property, relation with the heat equation), martingales related to Brown motion and stopping times. Stochastic calculus: Construction of the Ito integral with respect to Brownian motion, the integral as a stochastic process, Ito's formula. Stochastic differential equations (SDE), existence and uniqueness, solutions of some special SDE. Applications to PDE: Harmonic functions and the exit problem for Brownian motion, probabilistic interpretation of solutions, FeynmanKac formula. The Laplace operator as the generator of Brownian motion. Ito processes and their generator. Applications in financial mathematics: Portfolios and arbitrage, European options, BlackScholes formula.
Teaching and Learning Methods  Evaluation
Delivery 
Facetoface  

Use of Information and Communications Technology  
Teaching Methods 
 
Student Performance Evaluation 
Greek or English

Attached Bibliography
See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:
 Lawrence C. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, 2013.
 Bernt Oksendal, Stochastic Differential Equations: An Introduction with Applications of Univesitext, SpringerVerlag, Berlin, 6th edition, 2003.
 S.N. Cohen and R.J. Elliott, Stochastic Calculus and Applications, Second Edition of Probability and Its Applications, Birkhauser, 2015.
 I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition volume 113 of Graduate Texts in Mathematics, Springer, 1991.
 D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd Edition volume 293 of Grundlehren der mathematischen Wissenschaften [A Series of Comprehensive Studies in Mathematics], Springer, 2005.