Educate students about recent mathematically advanced fields of financial mathematics.

Wilmott, P., Dewynne, J, Howison, S.: Option Pricing, Mathematical Models and Computations, UK: Oxford Financial Press, 1995

Kwok, Y.K.: Mathematical models of financial derivatives, Berlin, Springer Verlag, 1998.

Hull, J.: Options, futures and other derivative securities, New York, Prentice Hall, 1989.

individual task

- Selected interest rate models

- Multi-period models of portfolio theory

- Derivative pricing with transaction costs

- Portfolio risk management

examination

Provide a comprehensive overview of recent trends in biomathematics

A primer on mathematical models in biology / Lee A. Segel, Leah Edelstein-Keshet. Philadelphia, Pa. : Society for Industrial and Applied Mathematics, 2013

Mathematical biology : 1. : An introduction / J. D. Murray. New York : Springer, 2002

Mathematical biology : 2. : Spatial models and biomedical applications / J. D. Murray. New York : Springer, 2003

Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering / Steven H. Strogatz. Cambridge : Perseus Books, 1994

2-MAT-111 Dynamical Systems OR 2-MAT-112 Partial Differential Equations (1) OR 2-MAT-121 Partial Differential Equations (2)

individual work

Continuous assessment: individual work

Exam: final exam and project

Assessment grade scale: A 90%, B 80%, C 70%, D 60%, E 50%

Principles of mathematical modeling, modeling goals, model building, model simulations, parameter selection, non-dimensionalization, model robustness, results analysis. Biochemical kinetics, enzymatic reactions, cooperativity, quasi-stationary approximation. Epidemiological models. Dynamics on neural and other cell membranes, Hodgkin-Huxley model, Fitzhugh-Nagumo model.

50/50

Students will become familiar with the theory and techniques used in current research in mathematical biology and in mathematical models in the natural and social sciences in general. At the same time, students try to work on a separate project in this area. They will also gain new knowledge from population models, chemical kinetics and cell biology.

examination

The student actively participates in the course and gives a talk on an agreed topic.

Conic convex programming, duality theory on conic convex programming, applications of conic programming in various areas, Theory of interior point methods, interior-point algorithms vor large scale conic problems, conic relaxations.

100% /0

Students become familiar with new fields of convex optimization and with modern algorithmic approaches for solving these problems. They will become familiar with the newest results, trends and open problems in convex optimization.

D. J. Acheson: Elementary Fluid Dynamics, Oxford, Clarendon Press, 1990

G. K. Batchelor: An Introduction to Fluid Dynamics, Cambridge University Press, 2000

P. Guba: Dynamika tekutín, skriptá, FMFI UK, 2021

Interim assessment during the semester has a weight of 30% (homeworks 20%, bonus exercises 10%). The two semester exam papers have a total weight of 70% (the first paper taken in the middle of the semester, the second paper taken at the end of the semester). The student must obtain at least half of the points from each semester exam paper. The final evaluation can be adjusted by an oral exam (theoretical questions, written preparation).

Grading: A (100-91), B (90-81), C (80-71), D (70-61), E (60-51), FX (50-0)

English

Equations of motion for ideal fluids. Vorticity. Irrotational flow. Vorticity equation.

Equations of motion for viscous fluids. Examples of simple viscous flows. Flows with circular streamlines.

Convection and diffusion of vorticity. Gravity waves. Dispersion and group velocity. Surface tension effects and capillary waves.

Internal gravity waves. Waves with finite amplitude. Hydraulic shocks and solitary waves. Kelvin--Helmholtz instability. Thermal convection. Centrifugal instability. Theorem on the stability of shear flow. General theorem on the stability of viscous flow. Uniqueness of steady viscous flow. Transition to turbulence.

30/70

Teach the students to derive and analyse basic equations describing fluid flow.

E. J. Hinch: Perturbation Methods, Cambridge University Press, 1991

J. Kevorkian, J. D. Cole: Multiple Scale and Singular Perturbation Methods, Springer, 1996

Interim assessment during the semester has a weight of 30% (homeworks 20%, bonus exercises 10%). The two semester exam papers have a total weight of 70% (the first paper taken in the middle of the semester, the second paper taken at the end of the semester). The student must obtain at least half of the points from each semester exam paper. The final evaluation can be adjusted by an oral exam (theoretical questions, written preparation).

Grading: A (100-91), B (90-81), C (80-71), D (70-61), E (60-51), FX (50-0)

English

Algebraic equations: Iterative method. Algebraic equations: Expansion method. Singular perturbations and rescaling. Logarithmic Poincare's expansions. Convergence and asymptoticity. Asymptotic approximation of integrals. Watson's lemma. The steepest descent method. Regular perturbation problems in differential equations. Singular perturbation problems in differential equations. Method of matched asymptotic expansions. Multiple scale method. WKBJ method. Poincare-Lindstedt method. Radius of convergence and Domb-Sykes plots.

30/70

To give an overview of basic asymptotic methods for solving algebraic and differential problems in applied mathematics.

Mathematical Methods in Risk Theory / Hans Bühlmann. New York : Springer, 1996;

Modern Actuarial Risk Theory Using R / Rob Kaas, Marc Goovaerts, Jan Dhaene, Michel Denuit. Second Edition, Heidelberg : Springer-Verlag, 2008;

Insurance Risk and Ruin / David C. M. Dickson. First Edition, New York : Cambridge University Press, 2005;

Non-Life Insurance Mathematics / Thomas Mikosch. Second Edition, Heidelberg : Springer-Verlag, 2009.

During the semester the student can get 100% points for one written exam, which is usually written at the end of the semester.

Grade thresholds: A: 100.00% – 90.00%; B: 89.99% – 80.00%; C: 79.99% – 70.00%; D: 69.99% – 60.00%; E: 59.99% – 50.00%; Fx: 49.99% – 0.00%.

Slovak, English

Probability distributions of small and large claims. Heavy tailed probability distributions: subexponential distributions, long tailed distributions, regularly varying distributions, distributions with fat tails. Risk process in non-life insurance. Cramér-Lundberg model, Cramér-Lundberg theorem. Probability of ruin of an insurance company in finite and infinite time. Generalized versions of the strong law of large numbers. Law of the iterated logarithm.

Practical work during semester 100% / final exam 0%.

After completing the course, the student will gain a detailed overview of ruin theory and the Cramér-Lundberg model. He or she will be able to approximate the ruin probability of the insurance company even if the claim size distribution has a heavy right tail. In addition, the student will know the various generalizations of the strong law of large numbers that are used in the field of non-life insurance.

Ross S: Simulation, Elsevier Academic Press 2006

Fishman GS: Monte Carlo: Concepts, Algorithms and Applications, Springer 1996

Selected scientific papers

Preliminary semester evaluation: projects

Final examination: oral examination

Approximate grade thresholds: A 90%, B 80%, C 70%, D 60%, E 50%

Slovak, English

Selected methods of simulation sampling of random vectors and random processes (uniform and non-uniform generation on the surface and in the interior of multi-dimensional spheres and polyhedrons, sampling of processes used in financial mathematics). Advanced variance reduction techniques and simulation of rare events.

http://www.iam.fmph.uniba.sk/ospm/Harman/teaching.htm

50/50

After completing the course, students will be able to use advanced techniques of variance reduction and simulation generation of random vectors and processes.

lectures

Collett, David. Modelling survival data in medical research. CRC press, 2015.

Klein, John P., et al., eds. Handbook of survival analysis. Boca Raton, FL:: CRC Press, 2014.

Broström, G. Event history analysis with R. Chapman and Hall/CRC, 2021

project

Slovak, English

Censoring, survival function and hazard function. Discrete time models. Nonparametric estimates. Kaplan-Meier estimate. Comparing two survival datasets. Log-rank test. Wilcoxon test. Proportional hazards model. Likelihood function for proportional hazards model parameters. Weibull model as a proportional hazards model. Log-linear form of the Weibull distribution. More distributions used in survival analysis. Diagnostics in proportional hazards model. Cox-Snell residuals. Martingale residuals. Aft (accelerated failure time) models. Cox regression model. Models with time-dependent explanatory variables.

100/0

Students will gain knowledge on selected statistical methods in survival analysis and how to use them when analyzing censored data

2 (lectures)

1. Bohner, M. and Allan Peterson Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser, 2001.

2. Bohner, Martin, and Allan Peterson. "Advances in Dynamic Equations on Time Scales, Birkhauser, 2003

3. Agarwal, Ravi, Donal O'Regan, and Samir Saker. Dynamic inequalities on time scales. Vol. 2014. Cham: Springer, 2014.

Exam: oral

1. Fundamentals of time scale calculus.

2. Differential calculus for single-variable functions on time scales.

3. Integral calculus for single-variable functions on time scales.

4. Introduction to dynamic equations: basic concepts, existence and uniqueness theorems.

5. Introduction to dynamic inequalities: overview of fundamental inequalities on time scales and their significance.

0/100

Students will learn the basics of the dynamic calculus on time scales with a focus on the principles underlying dynamic equations and inequalities on time scales.

2 (seminar)

James G., Witten D., Hastie T., Tibshirani R. An introduction to statistical learning: with applications in R, Springer Nature, 2021, 622 strán, ISBN 978-10716114181

Final exam will be oral.

The questions on the final exam will be from the topics presented during the course.

Grade scale: A 90%, B 80%, C 70%, D 60%, E 50%

English

This course is possible to complete remotely.

Students will present shorter parts of a monograph, and thus for continuity they will need to understand the preceding parts as well. Individual parts will have scope of not exceeding one meeting. In case of smaller number of enrolled students, the teacher will present material from the monograph as well.

Covered topics include: concept of a classification model, evaluation of models, linear and tree models, linear discriminant analysis (LDA), logistic regression, inference about statistical models, crossentropy, support vector machines.

Weight of work during semester / final exam: Practical work during semester 80% / final exam 20%.

After completion of this class the students will be acquainted with classification methods from statistics and machine learning, as well as with mathematical principles on which they are based.

2 (seminar)

Recommended literature will change on yearly basis.

For year 2025/2026 the following monograph will be used:

Krebs M., Shaheen A., Expander families and Cayley Graphs: A beginner’s guide, Oxford University Press, 288 pages, ISBN 978-0199767113

Final exam will be oral.

The questions on the final exam will be from the topics presented during the course.

Grade scale: A 90%, B 80%, C 70%, D 60%, E 50%

English

This course is possible to complete remotely.

Students will present shorter parts of a monograph, and thus for continuity they will need to understand the preceding parts as well. Individual parts will have scope of not exceeding one meeting. In case of smaller number of enrolled students, the teacher will present material from the monograph as well.

Weight of work during semester / final exam: Practical work during semester 80% / final exam 20%.

Continual evaluation is based on quality of presentations and participation in discussions.

After completion of this course the students will be acquainted with advanced topics from graph theory with interdisciplinary emphasis.

Type of activities: course

Number of hours:

per week: 2 per level/semester: 26

S. V. Patankar: Numerical Heat Transfer and Fluid Flow, CRC Press, 2018

F. Moukalled, L. Mangani and M. Darwish: The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM® and Matlab®, Springer,2016

H. K. Versteeg and W. Malalasekera: An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Pearson, 2007

2-EFM-107/15 Partial differential equations

2-EFM-101/15 Numerical modelling

3-MAM-009/15 Fluid flow models

Exam: oral

Grading: A (100-91), B (90-81), C (80-71), D (70-61), E (60-51), FX (50-0)

Slovak, English

Introduction to the finite volume method

1. Discretization schemes for advection-diffusion equations (central, upwind and hybrid differencing schemes) 2. Properties of the discretization schemes (conservativeness, boundedness and transportiveness) 3. Solution algorithms for pressure-velocity coupling (SIMPLE and PISO algorithms) 4. Direct and iterative methods for solving discretised equations (TDMA, Jacobi and Gauss–Seidel methods) Application of OpenFOAM® software in solving fluid dynamics problems 5. The heat transfer equation 6. The Navier–Stokes equations for single-phase incompressible flows 7. Lid-driven cavity flow 8. Poiseuille flow Meshing for complex geometries with OpenFOAM® 9. Flow around a cylinder. Von Kármán vortex street 10. Single-phase flow in the pore space. Scalar and particulate transport in the pore space Application of OpenFOAM® software in solving problems in financial mathematics and biomathematics 11. The Black–Scholes equation for option pricing 12. The Fisher–KPP equation for population dynamics

30/70

Students are introduced to the finite volume method for the numerical solution of partial

differential equations describing diffusion and advection problems in applied mathematics.