## MF002 Stochastical analysis

Faculty of Science
Spring 2022
Extent and Intensity
2/2/0. 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
Prerequisites
Calculus: derivative, limit, Riemann integral, Taylor expansion.
Basics of linear algebra: vector space, norm, inner product.
Probability and statistics: probability space, random variable, normal probability distribution, expected value, variance, correlation, point and interval estimators of parameters, definitons and basic properties of random processes.
Software: at least basic experience with R, statistical analysis of a dataset.
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 and diffusion processes. In the theoretical part, the student learns how to understand the Wiener process, 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, valuation of some financial derivatives).
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.
• 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. Practical classes: 2 hour a week, partial work with mathematical software R. Distance form: online lectures, practical classes and discussions.
Assessment methods
Exercises: active participation in online course and discussions, solving homeworks and project. Depending on the current epidemiological situation, the form of the final will be decided. Full-time form of the final exam: (1) written and (2) oral part. Distance form of the final exam: (1) written online electronic part and (2) oral part (one-to-one online talk). For successful completion, it is necessary to achieve at least 50 % of the maximum total points.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Listed among pre-requisites of other courses
Teacher's information
https://is.muni.cz/auth/el/sci/jaro2021/MF002/index.qwarp
The lessons are usually in Czech or in English as needed, and the relevant terminology is always given with English equivalents.

The target skills of the study include the ability to use the English language passively and actively in their own expertise and also in potential areas of application of mathematics.

Assessment in all cases may be in Czech and English, at the student's choice.

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 2019, Spring 2020, Spring 2021.
• Enrolment Statistics (Spring 2022, recent)