MF002 Stochastical analysis

Faculty of Science
Spring 2017
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
doc. PaedDr. RNDr. Stanislav Katina, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 20. 2. to Mon 22. 5. Wed 16:00–17:50 M6,01011
  • Timetable of Seminar Groups:
MF002/01: Mon 20. 2. to Mon 22. 5. Wed 18:00–19:50 MP1,01014, Wed 18:00–19:50 M6,01011, O. Pokora
Prerequisites
Calculus: derivative, limit, Riemann integral, Taylor expansion, calculus for functions.
Basics of linear algebra: vector space, basis, norm, inner product.
Probability and statistics, basics of random processes: axiomatic theory of probability, probability space, random variable, normal probability distribution, expected value, variance, correlation, estimations of parameter, confidence intervals, random process, examples of random processes.
Basics of statistical software R: syntax of basic commands, matrices, vectors, basics of 2D graphics.
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: (1) define the Ito and Stratonovich stochastic integrals, (2) solve basic types of stochastic differential equations, (3) use the Ito Lemma and further properties of stochastic integral for calculations with Ito processes, (4) use the change of probability measure to transform the stochastic process, (5) 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
  • KARATZAS, Ioannis and Steven E. SHREVE. Brownian motion and stochastic calculus. New York: Springer, 1988, 23, 470. ISBN 0387976558. info
  • ØKSENDAL, Bernt. Stochastic differential equations : an introduction with applications. 6th ed. Berlin: Springer, 2005, xxvii, 365. ISBN 3540047581. 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. Methods of mathematical finance. New York: Springer-Verlag, 1998, xv, 415. ISBN 0387948392. info
  • HULL, John. Options, futures & other derivatives. 5th ed. Upper Saddle River: Prentice Hall, 2003, xxi, 744. ISBN 0130090565. info
  • MELICHERČÍK, Igor; Ladislava OLŠAROVÁ and Vladimír ÚRADNÍČEK. Kapitoly z finančnej matematiky. [Bratislava: Miroslav Mračko, 2005, 242 s. ISBN 8080576513. info
Approaches, practices, and methods used in teaching
Lectures: 2 hours a week. Exercises: 2 hours a week, partial work with mathematical software R, homeworks and project.
Method of verifying learning outcomes and course completion requirements
Exercises: active participation, homeworks and individual project. Final exam: written and oral part, at least 25 % of points in each part and at least 50 % of points in total (written:oral part weights ca. 2:1) 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 2014, Spring 2015, Spring 2016, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025, Spring 2026, Spring 2027.
  • Enrolment Statistics (Spring 2017, recent)
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