MF002 Stochastical analysis

Faculty of Science
Spring 2015
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
Mon 16:00–17:50 M2,01021
  • Timetable of Seminar Groups:
MF002/01: Mon 10:00–11:50 M6,01011, Mon 10:00–11:50 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.
Abstract
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;
use the Ito Lemma and further properties of stochastic integral for calculations with Ito processes;
use the change of probability measure to transform the stochastic processů
apply stochastic calculus to practical problems (mainly in financial mathematics);
Key topics
  • 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
Study resources and 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
Approaches, practices, and methods used in teaching
Lectures (theory and applications in mathematical finance), excercises (solving problems, work with mathematical software R, homework), individual project (real financial data analysis)
Method of verifying learning outcomes and course completion requirements
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
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 2014, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025, Spring 2026, Spring 2027.
  • Enrolment Statistics (Spring 2015, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2015/MF002