MF002 Stochastical analysis

Faculty of Science
Spring 2014
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
Mgr. Ondřej Pokora, Ph.D. (lecturer)
Guaranteed by
prof. RNDr. Ivanka Horová, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Wed 16:00–17:50 M2,01021
  • Timetable of Seminar Groups:
MF002/01: Mon 16:00–17:50 M2,01021, Mon 16:00–17:50 MP2,01014a, O. Pokora
MF002/02: Mon 16:00–17:50 MP1,01014, O. Pokora
Prerequisites
Calculus in one and several variables, basics of probability and statistics, basics of the R language (statistical software)
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
At the end of the course students should be able to:
define the Ito and Stratonovich stochastic integrals;
solve basic types of stochastic differential equations;
prove the Ito Lemma and further properties of stochastic integral;
apply stochastic calculus to problems in financial mathematics;
Syllabus
  • Stochastic processes and their properties, L2 space, Hilbert space
  • Wiener process (Brownian motion) and its construction
  • Linear and quadratic variation
  • Ito and Stratonovich stochastic integral
  • Ito lemma, Ito process, stochastic differential equation
  • Martingales, Martingale representation theorem
  • Radon-Nikodym derivative, Cameron-Martin theorem, Girsanov theorem
  • Black-Scholes model, options, geometric Brownian motion
  • Markov processes with continuous time, diffusion, Ornstein-Uhlenbeck process
  • Stochastic interpretation of diffusion and Laplace equation, Feynman-Kac theorem
Literature
  • MELICHERČÍK, Igor, Ladislava OLŠAROVÁ and Vladimír ÚRADNÍČEK. Kapitoly z finančnej matematiky. [Bratislava: Miroslav Mračko, 2005, 242 s. ISBN 8080576513. info
  • ØKSENDAL, Bernt. Stochastic differential equations : an introduction with applications. 6th ed. Berlin: Springer, 2005, xxvii, 365. ISBN 3540047581. info
  • HULL, John. Options, futures & other derivatives. 5th ed. Upper Saddle River: Prentice Hall, 2003, xxi, 744. ISBN 0130090565. info
  • KARATZAS, Ioannis and Steven E. SHREVE. Methods of mathematical finance. New York: Springer-Verlag, 1998, xv, 415. ISBN 0387948392. info
  • KLOEDEN, Peter E., Eckhard PLATEN and Henri SCHURZ. Numerical solution of SDE through computer experiments. Berlin: Springer, 1994, xiv, 292. ISBN 3540570748. info
  • KARATZAS, Ioannis and Steven E. SHREVE. Brownian motion and stochastic calculus. New York: Springer, 1988, 23, 470. ISBN 0387976558. info
Teaching methods
Lectures (theory and applications in mathematical finance), excercises (solving problems, work with mathematical software R, homework), individual project (real financial data analysis)
Assessment methods
Conditions: active participation in seminars, individual project. Evaluation: written examination (weight 60 %) and oral examination (weight 40 %), at least 50 % of points is needed to pass.
Language of instruction
Czech
Further Comments
Study Materials
The course is taught annually.
Listed among pre-requisites of other courses
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2014, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2014/MF002