MF002 Stochastic analysis

Faculty of Science
Spring 2019
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 18. 2. to Fri 17. 5. Mon 12:00–13:50 M4,01024
  • Timetable of Seminar Groups:
MF002/01: Mon 18. 2. to Fri 17. 5. Wed 8:00–9:50 MP1,01014, Wed 8:00–9:50 M4,01024, 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.
Course objectives
This course introduces the basic principles and methods of the stochastic analysis and modeling of real phenomenons (in economy, financial mathematics, biology, engineering) using the Wiener process. Well-known methods of the mathematical analysis are extended to random processes, to the Wiener process in particular and to diffusion processes in general. In the theoretical part, the student learns how to understand and apply the L2-space, L2-convergence of random variables, to calculate stochastic integrals, to solve stochastic differential equations, to use martingals and to realize the connection between the diffusion processes and the partial differential equations. In the practical classes, the student learns how to simulate the Wiener process and diffusion process using computers, how to estimate parameters using the simulation studies and how to model real phenomenons (option price, neuronal membrane potential, quality measure) or how to calculate the price of some financial derivatives (European and binary barrier options).
Learning outcomes
After completing this course, the student will be able to:
- describe the Wiener process and its properties and apply it in mathematical modeling;
- solve basic stochastic differential equations;
- describe the principle of the equivalent (e. g., risk-neutral) probability;
- model the option price and of the neuron membrane potential in time using simulations of the trajectories of the Wiener process;
- apply the fundamental principle of the pricing of financial derivatives and calculate the price of the European and binary barrier option.
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
  • 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
Teaching methods
Lectures: 2 hours a week. Exercises: 2 hours a week, partial work with mathematical software R, homeworks and project.
Assessment methods
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 2017, spring 2018, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2019, recent)
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