PřF:MF002 Stochastical analysis - Course Information

MF002 Stochastical analysis

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 16:00–17:50 M1,01017

• Timetable of Seminar Groups:

MF002/01: Mon 12:00–13:50 MP1,01014, Mon 12:00–13:50 M3,01023, O. Pokora
MF002/02: Mon 18:00–19:50 MP1,01014, Mon 18:00–19:50 M1,01017, 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

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).

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

• MF003 Derivatives pricing
  MF001 && MF002  

• Enrolment Statistics (Spring 2016, recent)
  
• Permalink: https://is.muni.cz/course/sci/spring2016/MF002