MSC IN APPLIED MATHEMATICS
The two-year MSc program in Applied Mathematics ( math.bme.hu/masters/ ) provides a profound knowledge of applied mathematics, competitive both in the academic and non-academic sectors. Possible specializations are Stochastics and Financial Mathematics.
   Theoretical courses are taught mainly by internationally recognized scientists of the university, while applied courses are given mainly by experts from industry and finance who are actual appliers of mathematics. Students of our MSc program may enter leading-edge research projects of the Department of Stochastics (a cutting edge research centre in stochastics, the host of our MSc studies), or at one of the cooperating companies.
   Both specializations focus strongly on probability theory and statistics, so having an interest in these fields is essential, and a background in them is advantageous. Foundational courses are provided to those who need them.
   The students who complete our program have excellent career opportunities in the research sector (by becoming PhD students at either our university or some cutting-edge universities in the US or Europe), as well as in the commercial sector (by getting well-paid jobs at leading banks, insurance and consulting companies, or in the industry).
Important features of our MSc program: On the one hand, we put emphasis on building closer personal relationship between our students and the teaching staff of the Department via providing personal tutors to all of our MSc students. On the other hand, beside giving a very profound knowledge in mathematics, we also prepare all of our students for successful collaboration with people from industry, finance and the insurance sector who apply mathematics in their work.
   In the past years, our students have found jobs at universities e.g. in Birmingham (Alabama), Bonn, Bristol, Budapest, Eindhoven, Toronto, Zürich, Warsaw, companies like Morgan Stanley and Google, as well as the National Bank of Hungary and institutions of the European Commission.
Detailed information
CURRICULUM OF SPECIALIZATION FINANCIAL MATHEMATICS 2018
Code Title Parameters* ECTS credits
per semester
Lc Pr Lb Rq Cr I II III IV
  Theoretical foundations (20 ECTS credits). Earlier not completed, prescribed subjects from BSc in Math in order below. 10 4 2  
BMETE95AM41 Stochastic Processes 5 0 0 v 6 6      
BMETE95AM30 Probability Theory 2 3 1 0 v 4   4    
BMETE95AM33 Tools of Modern Probability Theory 4 0 0 v 4 4      
BMETE95AM42 Applied Stochastics 2 0 2 v 4 4      
BMETE92AM42 Measure Theory 4 0 0 v 4 4      
BMETE92AM40 Functional Analysis 1 4 0 0 v 4 4      
BMETE92AM45 Partial Differential Equations 2 2 0 v 4   4    
  Professional subjects (30 ECTS credits must be completed).
Courses marked by boldface letters are obligatory.
10 5 5 10
BMETE93MM00 Global Optimization 3 1 0 f 5   5    
BMETE93MM01 Linear Programming 3 1 0 v 5 5      
BMETE91MM00 Theoretical Computer Science 3 1 0 f 5       5
BMEVISZM020 General and Algebraic Combinatorics 3 1 0 f 5     5  
BMETE93MM02 Dynamical Systems 3 1 0 v 5       5
BMETE92MM00 Fourier Analysis and Function Series 3 1 0 v 5     5  
BMETE93MM03 Partial Differential Equations 2 3 1 0 f 5       5
BMETE95MM04 Stochastic Analysis and its Applications 3 1 0 v 5 5      
BMETE95MM05 Mathematical Statistics and Information Theory 3 1 0 v 5       5
BMETE91MM01 Commutative Algebra and Algebraic Geometry 3 1 0 f 5 5      
BMETE91MM02 Representation Theory 3 1 0 f 5   5    
BMETE94MM00 Differential Geometry and Topology 3 1 0 v 5     5  
  Obligatory courses of specialization (36 ECTS credits) 8 10 13 5
BMETE95MM20 Nonparametric Statistics 2 0 0 v 3     3  
BMETE95MM09 Statistical Program Packages 2 0 0 2 f 2     2  
BMETE95MM15 Multivariate Statistics 3 1 0 v 5     5  
BMETE95MM07 Markov Processes and Martingales 3 1 0 v 5 5      
BMETE95MM08 Stochastic Differential Equations 3 1 0 v 5   5    
BMETE95MM14 Financial Processes 2 0 0 f 3   3    
BMETE95MM16 Extreme Value Theory 2 0 0 v 3       3
BMETE95MM17 Insurance Mathematics 2 2 0 0 f 2   2    
BMETE95MM30 Macroeconomics and Finance for Mathmeaticians 2 0 0 v 3     3  
BMEGT30M400 Analysis of Economic Time Series 2 0 0 f 2       2
BMETE95MM26 Time Series Analysis with Applications in Finance  2 0 0 f 3 3      
  Obligatory common subjects (30 ECTS credits) 1 4 10 15
BMETE92MM01 Individual Projects 1 0 0 4 f 4   4    
BMETE92MM02 Individual Projects 2 0 0 4 f 4     4  
BMETE95MM01 Mathematical Modelling Seminar 1 2 0 0 f 1 1      
BMETE95MM02 Mathematical Modelling Seminar 2 2 0 0 f 1     1  
BMETE90MM90 Report 0 0 0 a 0   0    
BMETE90MM98 Preparatory Course for Master's Thesis 0 2 0 f 5     5  
BMETE90MM99 Master's Thesis 0 8 0 f 15       15
  Elective professional courses (8 ECTS credits must be completed)   8    
*Parameters:
 
Lc = lecture, Pr = practice, Lb = laboratory (hours per week);
 
Rq = requirement or exam type (v = examination, f = midterm exam, a = signature);
 
 Cr = ECTS credits.
CURRICULUM OF SPECIALIZATION STOCHASTICS 2018
Code Title Parameters* ECTS credits
per semester
Lc Pr Lb Rq Cr I II III IV
  Theoretical foundations (20 ECTS credits). Earlier not completed, prescribed subjects from BSc in Math in order below. 10 10    
BMETE95AM41 Stochastic Processes 5 0 0 v 6 6      
BMETE95AM30 Probability Theory 2 3 1 0 v 4   4    
BMETE95AM33 Tools of Modern Probability Theory 4 0 0 v 4 4      
BMETE95AM42 Applied Stochastics 2 0 2 v 4 4      
BMETE92AM42 Measure Theory 4 0 0 v 4 4      
BMETE92AM40 Functional Analysis 1 4 0 0 v 4 4      
BMETE92AM45 Partial Differential Equations 2 2 0 v 4   4    
  Professional subjects (30 ECTS credits must be completed).
Courses marked by boldface letters are obligatory.
10 5 5 10
BMETE93MM00 Global Optimization 3 1 0 f 5   5    
BMETE93MM01 Linear Programming 3 1 0 v 5 5      
BMETE91MM00 Theoretical Computer Science 3 1 0 f 5       5
BMEVISZM020 General and Algebraic Combinatorics 3 1 0 f 5     5  
BMETE93MM02 Dynamical Systems 3 1 0 v 5       5
BMETE92MM00 Fourier Analysis and Function Series 3 1 0 v 5     5  
BMETE93MM03 Partial Differential Equations 2 3 1 0 f 5       5
BMETE95MM04 Stochastic Analysis and its Applications 3 1 0 v 5 5      
BMETE95MM05 Mathematical Statistics and Information Theory 3 1 0 v 5       5
BMETE91MM01 Commutative Algebra and Algebraic Geometry 3 1 0 f 5 5      
BMETE91MM02 Representation Theory 3 1 0 f 5   5    
BMETE94MM00 Differential Geometry and Topology 3 1 0 v 5     5  
  Obligatory courses of specialization (30 ECTS credits)
Only one subject must be completed from the two last ones
5 10 10 5
BMETE95MM15 Multivariate Statistics 3 1 0 v 5     5  
BMETE95MM20 Nonparametric Statistics 2 0 0 v 3     3  
BMETE95MM09 Statistical Program Packages 2 0 0 2 f 2     2  
BMETE95MM07 Markov Processes and Martingales 3 1 0 v 5 5      
BMETE95MM08 Stochastic Differential Equations 3 1 0 v 5   5    
BMETE95MM14 Financial Processes 2 0 0 f 3   3    
BMETE95MM10 Limit- and Large Deviation Theorems of Probability Theory 3 1 0 v 5       5
BMETE95MM11 Stochastic Models 2 0 0 f 2   2    
BMETE95MM12 Advanced Theory of Dynamical Systems 2 0 0 f 2   2    
  Obligatory common subjects (30 ECTS credits) 1 4 10 15
BMETE92MM01 Individual Projects 1 0 0 4 f 4   4    
BMETE92MM02 Individual Projects 2 0 0 4 f 4     4  
BMETE95MM01 Mathematical Modelling Seminar 1 2 0 0 f 1 1      
BMETE95MM02 Mathematical Modelling Seminar 2 2 0 0 f 1     1  
BMETE90MM90 Report 0 0 0 a 0   0    
BMETE90MM98 Preparatory Course for Master's Thesis 0 2 0 f 5     5  
BMETE90MM99 Master's Thesis 0 8 0 f 15       15
  Elective professional courses (10 ECTS credits must be completed) 3 5 2  
*Parameters:
 
Lc = lecture, Pr = practice, Lb = laboratory (hours per week);
 
Rq = requirement or exam type (v = examination, f = midterm exam, a = signature);
 
 Cr = ECTS credits.
DESCRIPTION OF SUBJECTS
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE93MM00 Global Optimization 3 1 0 f 5   5    
Course coordinator: Dr. Boglárka Gazdag-Tóth
Different forms of global optimization problems, their transformation to each other, and their reduction to the one-dimensional problem. Comparison of the complexity of global optimization and linear programming problems. Classifications of the global optimization methods. Lagrange function, Kuhn–Tucker theorem, convex and DC programming. Basic models and methods of stochastic programming. Multi-start and stochastic methods for global optimization, their convergence properties and stopping criteria. Methods based on Lipschitz constant, and their convergence properties. Branch and Bound schema, methods based on interval analysis, automatic differentiation. Multi-objective optimization.
Literature:
– R. Horst, P. Pardalos: Handbook of Global Optimization, Kluwer, 1995.
– R. Horst, P.M. Pardalos, N.V. Thoai: Introduction to Global Optimization, Kluwer, 1995.
– A. Törn, A. Zilinskas: Global Optimization, Springer, 1989.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE93MM01 Linear Programming 3 1 0 v 5 5      
Course coordinator: Dr. Tibor Illés
System of linear equations: solution and solvability. Gauss-Jordan elimination method. System of linear inequalities. Alternative theorems, Farkas lemma and its variants. Solution of system of linear inequalities using pivot algorithms. Convex polyhedrons. Minkowski-, Farkas- and Weyl-theorems. Motzkin-theorem. Primal-dual linear programming problems. Feasible solution set of linear programming problems. Basic solution of linear programming problem. Simplex and criss-cross algorithms. Cycling, anti-cycling rules: Bland’s minimal index rule. Two phase simplex method. Revised simplex method. Sensitivity analysis. Decomposition methods: Dantzig-Wolfe. Special type of pivot algorithms: lexicographic and lexicographic dual simplex methods. Monotonic build-up simplex algorithms. Interior point methods of linear programming problems. Self-dual linear programming problem. Central path and its uniqueness. Computation of Newton-directions. Analytical centre, Sonnevend-theorem. Dikin-ellipsoid, affine scaling primal-dual interior point algorithm and its polynomial complexity. Tucker-model, Tucker theorem. Rounding procedure. Khachian’s ellipsoid algorithm. Karmarkar’s potential function method. Special interior point algorithms.
Literature:
– K.G. Murty: Linear and combinatorial programming. John Wiley & Sons Inc., New York, 1976.
– C. Roos, T. Terlaky,  J.P. Vial: Interior Point Methods for Linear Optimization. Springer US, New York, 2005.
– A. Schrijver: Theory of Linear and Integer Programming, John Wiley, New York, 1986.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE91MM00 Theoretical Computer Science 3 1 0 f 5       5
Course coordinator: Dr. Miklós Ferenczi
Foundations of logic programming and automated theorem proving. Finite models and complexity. Non classical logics in Computer Science: temporal dynamic and programming logics. Recursive functions and lambda calculus. Boole algebras, relational algebras and their applications. Some important models of computation. Basic notions of complexity theory, some important time and spaces classes. NP completeness. Randomised computation. Algorithm design techniques. Advanced data structures, amortised costs. Pattern matching in text. Data compression.
Literature:
– A. Galton: Logic for Information Technology, Wiley, 1990.
Code Title Lc Pr Lb Rq Cr I II III IV
BMEVISZM020 General and Algebraic Combinatorics 3 1 0 f 5     5  
Course coordinator: Dr. Katalin Friedl
Combinatorics of the Young tableaux, tableau rings. Pieri formulas, Schur polynomials, Kostka numbers. Robinson-Schensted-Knuth correspondence. Littlewood-Richardson numbers, Littlewood-Richardson theorem. Important symmetric polynomials, their generating functions. Cauchy-Littlewood formulas. Garsia's generalization of the fundamental theorem on symmetric polynomials. Bases of the ring of symmetric functions. Topics from combinatorial optimization: greedy algorithm, augmenting methods. Matroids, their basic properties, matroid intersection algorithm. Approximation algorithms (set cover, travelling salesman, Steiner trees). Scheduling algorithms (single machine scheduling, scheduling for parallel machines, bin packing).
Literature:
– W. Fulton, Y. Tableaux: With Applications to Representation Theory and Geometry, London Math. Soc. Student Texts, Paperback, Cambridge Univ. Press, 1996.
– R.P. Stanley: Enumerative Combinatorics I.- II., Cambridge University Press, 2001.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE93MM02 Dynamical Systems 3 1 0 v 5       5
Course coordinator: Dr. Péter Bálint
Continuous-time and discrete-time dynamical systems, continuous versus descrete: first return map, discretization. Local theory of equilibria: Grobman–Hartman lemma, stable-unstable-center manifold, Poincaré's normal form. Attractors, Liapunov functions, LaSalle principle, phase portrait. Structural stability, elementary bifurcations of equilibria, of fixed points, and of periodic orbits, bifurcation curves in biological models. Tent and logistic curves, Smale horseshoe, solenoid: properties from topological, combinatorial, and measure theoretic viewpoints. Chaos in the Lorenz model.
Literature:
– P. Glendinning: Stability, Instability and Chaos, Cambridge University Press, Cambridge, 1994.
– C. Robinson: Dynamical Systems, CRC Press, Boca Raton, 1995.
– S. Wiggins: Introduction to Applied Nonlinear Analysis and Chaos, Springer, Berlin, 1988.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE92MM00 Fourier Analysis and Function Series 3 1 0 v 5     5  
Course coordinator: Dr. Miklós Horváth
Completeness of the trigonometric system. Fourier series, Parseval identity. Systems of orthogonal functions, Legendre polynomials, Haar and Rademacher systems. Introduction to wavelets, wavelet orthonormal systems. Fourier transform, Laplace transform, applications. Convergence of Fourier series: Dirichlet kernel, Dini and Lischitz convergence tests. Fejer’s example of divergent Fourier series. Fejer and Abel-Poisson summation. Weierstrass-Stone theorem, applications. Best approximation in Hilbert spaces. Müntz theorem on the density of lacunary polynomials. Approximations by linear operators, Lagrange interpolation, Lozinski-Harshiladze theorem. Approximation by polynomials, theorems of Jackson. Positive linear operators Korovkin theorem, Bernstein polynomials, Hermite-Fejer operator. Spline approximation, convergence, B-splines.
Literature:
– G. Lorentz, M.V. Makovoz: Constructive Approximation, Springer, 1996.
– M.J.D. Powell: Approximation Theory and Methods, Cambridge University Press, 1981.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE93MM03 Partial Differential Equations 2 3 1 0 f 5       5
Course coordinator: Dr. Márton Kiss
The Laplacian in Sobolev space (revision). Weak and strong solutions to second order linear parabolic equations. Ritz-Galerkin approximation. Linear operator semigroups (According to Evans and Robinson). Weak and strong solutions to reaction-diffusion (quasilinear parabolic) equations. Ritz–Galerkin approximation. Nonlinear operator semigroups (According to Evans and Robinson). Only in examples: monotonicity, maximum principles, invariant regions, stability investigations for equilibria by linearization, travelling waves (According to Smoller). Global attractor. Inertial manifold (According to Robinson).
Literature:
– L.C. Evans: Partial Differential Equations, AMS, Providence R.I., 1998.
– J. Smoller: Shock Waves and Reaction-Diffusion Equations, Springer, Berlin, 1983.
– J.C. Robinson: Infinite-dimensional Dynamical Systems, CUP, Cambridge, 2001.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE95MM04 Stochastic Analysis and its Applications 3 1 0 v 5 5      
Course coordinator: Dr. Károly Simon
Introduction. Markov processes, stochastic semi-groups, infinitesimal generators, martingales, stopping times. Brownian motion. Brownian motion in nature. Finite dimensional distributions and continuity of Brownian motion. Constructions of the Wiener process. Strong Markov property. Self-similarity and recurrence of Brownian motion, time reversal. Reflection principle and its applications. Local properties of Brownian path: continuity, Hölder continuity, non-differenciability. Quadratic variations. Continuous martingales. Definition and basic properties. Dubbins-Schwartz theorem. Exponential martingale. Lévy processes. Processes with independent and stationary increments, Lévy-Hintchin formula. Decomposition of Lévy processes. Construction by means of Poisson processes. Subordinators, and stable processes. Examples and applications. Stochastic integration I. Discrete stochastic integrals with respect to random walks and discrete martingales. Applications, discrete Balck-Scholes formula. Stochastic integrals with respect to Poisson process. Martingales of finite state space Markov processes. Quadratic variations. Doob-Meyer decomposition. Stochastic integration II. Predictable processes. Itô integral with respect to the Wiener process, quadratic variation process. Doob-Meyer decomposition. Itô formula and its applications.
Literature:
– K.L. Chung, R. Williams: Introduction to  stochastic integration. Second edition. Birkauser, 1989.
– R. Durrett: Probability: theory and examples. Second edition. Duxbury, 1996.
– B. Oksendal: Stochastic Differential equations. Sixth edition. Springer, 2003.
– D. Revuz, M. Yor: Continuous martingales and Brownian motion. Third edition. Springer, 1999.
– G. Samorodnitsky, M.S. Taqqu: Stable Non-Gaussian Random Processes: Stochastic  Models with Infinite Variance, Chapman and Hall, New York, 1994.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE95MM05 Mathematical Statistics and Information Theory 3 1 0 v 5       5
Course coordinator: Dr. Marianna Bolla
Multivariate statistical inference in multidimensional parameter spaces: Fisher’s information matrix, likelihood ratio test. Testing hypotheses in multivariate Gauss model: Mahalanobis’ distance, Wishart’s, Hotelling’s, Wilks’ distributions. Linear statistical inference, Gauss–Markov theorem. Regression analysis, one- and two-way analysis of variance as a special case of the linear model. ANOVA tables, Fisher-Cochran theorem. Principal component and factor analysis. Estimation and rotation of factors, testing hypotheses for the effective number of factors. Hypothesis testing and  I-divergence (the discrete case). I-projections,  maximum likelihood estimate as I-projection in exponential families. The limit distribution of the I-divergence statistic. Analysis of contingency tables by information theoretical methods, loglinear models. Statistical algorithms  based on  information geometry: iterative scaling,  EM algorithm. Method of maximum entropy.
Literature:
– M. Bolla, A. Krámli: Theory of statistical inference, Typotex, Budapest, 2005.
– I. Csiszár, P.C. Shields: Information Theory and Statistics. A tutorial. In: Found. and Trends in Comm. and Info. Theory, 420-525. Now Publ. Inc., The Netherlands, 2004.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE91MM01 Commutative Algebra and Algebraic Geometry 3 1 0 f 5 5      
Course coordinator: Dr. Alex Küronya
Closed algebraic sets and their coordinate rings, morphisms, irreducibility and dimension, Hilbert Nullstellensatz, the correspondence between radical ideals and subvarieties of affine space. Monomial orders, Gröbner bases, Buchberger algorithms, computations in polynomial rings. From regular functions to rational maps, local rings, fundamentals of sheaf theory, ringed spaces. Projective space and its subvarieties, homogeneous coordinate ring, morphisms, the image of a projective variety is closed. Geometric constructions: Segre and Veronese embeddings, Grassmann varieties, projection from a point, blow-up. Dimension of affine and projective varieties, hypersurfaces. Smooth varieties, Zariski tangent space, the Jacobian condition. Hilbert function and Hilbert polynomial, examples, computer experiments. Basic notions of rings and modules, chain conditions, free modules. Finitely generated modules, Cayley-Hamilton theorem, Nakayama lemma. Localization and tensor product. Free resolutions of modules, Gröbner theory of modules, computations, Hilbert syzygy theorem.
Literature:
– A. Gathmann: Algebraic geometry, 2003, www.mathematik.uni-kl.de/~gathmann/en/pub.html
– I.R. Shafarevich: Basic Algebraic Geometry I.-II., Springer Verlag, 1995.
– M. Reid: Undergraduate Commutative Algebra, Cambridge University Press, 1996.
– R. Hartshorne: Algebraic Geometry, Springer Verlag, 1977.
– M.F. Atiyah, I.G. Macdonald: Introduction to commutative algebra, Addison Wesley Publishing, 1994.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE91MM02 Representation Theory 3 1 0 f 5   5    
Course coordinator: Dr. Alex Küronya
Differentiable manifolds, atlas, maps, immersion, submersion, submanifold, tangent space, vector field, Lie-derivative, topological background. Vector bundles, alternating forms on linear spaces, differential forms, their integration, Stokes theorem. Multilinear algebra (tensors, symmetric and alternating spaces, contraction) and applications to vector bundles. Lie groups and their basic properties; exponential map, invariant vector field, Lie algebra. Matrix Lie groups and their Lie algebras, examples. Representations of groups in general, caharcters, linear algebraic constructions. Continuous representations of Lie groups, connections among representations of Lie groups and the representations of their Lie algebras. Basics about Lie algebras,  derivations, nilpotent and solvable algebras, theorems of Engel and Lie, Jordan-Chevalley decomposition, Cartan subalgebras. Semisimple Lie algebras, Killing form, completely reducible representations. The representations of sl_2 , root systems, Cartan matrix, Dynkin diagram, classification of semisimple Lie algebras. Representations of matrix Lie groups, Weyl chambers, Borel subalgebra. The Peter-Weyl theorem.
Literature:
– G. Bredon: Topology and Geometry, Springer Verlag, 1997.
– J. Jost: Riemannian Geometry and Geometric Analysis, 4. edition, Springer Verlag, 2005.
– W. Fulton, J. Harris: Representation Theory: a First Course, Springer Verlag, 1999.
– D. Bump: Lie Groups, Springer Verlag, 2004.
– J.E. Humphreys:  Introduction to Lie Algebras and Representation Theory, Springer Verlag, 1997.
Code Title Lc Pr Lb Rq Cr I II III IV
BMETE94MM00 Differential Geometry and Topology 3 1 0 v 5     5  
Course coordinator: Dr. Szilárd Szabó
Smooth manifolds, differential forms, exterior derivation, Lie-derivation.  Stokes' theorem, de Rham cohomology, Mayer–Vietoris exact sequence, Poincaré-duality. Riemannian manifolds, Levi–Civitá connection, curvature tensor, spaces of constant curvature. Geodesics, exponential map, geodesic completeness, the Hopf–Rinow theorem, Jacobi fields, the Cartan–Hadamard theorem, Bonnet's theorem.
Literature:
– J.M. Lee: Riemannian Manifolds: an Introduction to Curvature, Graduate Texts in Mathematics 176, Springer Verlag.
– P. Petersen: Riemannian Geometry, Graduate Texts in Mathematics 171, Springer Verlag.
– J. Cheeger, D. Ebin: Comparison Theorems in Riemannian Geometry, North-Holland Publishing Company, Vol. 9, 1975.