MSc in Applied Mathematics | Budapest University of Technology and Economics

MSC IN APPLIED MATHEMATICS
The two-year MSc program in Applied Mathematics (math.bme.hu/masters/) provides a profound knowledge of applied mathematics which is competitive both in the academic and non-academic sectors. Possible specializations are Stochastics and Financial Mathematics. Students of our MSc program may enter leading-edge research projects of the Department of Stochastics (a cutting edge research center in stochastics, the host of our MSc studies). The most important features of our MSc programs are as follows: 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, we do not only give a very profound knowledge in Stochastics but we also prepare all of the students for successful collaboration with those people from industry, finance or insurance sectors who apply mathematics in their work. Some of the best students of our MSc program continue their studies and become Ph.D. students at either our university or some cutting-edge universities in the US or Europe. Others get well-paid jobs at leading banks, insurance, and consulting companies, like Morgan Stanley or in the industry.
Detailed information
CURRICULUM OF SPECIALIZATION FINANCIAL MATHEMATICS
Code	Title	Parameters*					ECTS credits per semester
Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
	Theoretical foundations (14 ECTS credits). Prescribed subjects from BSc in Math, or elective professional courses.						6	8
	Professional subjects (30 ECTS credits must be completed). Courses marked by boldface letters are obligatory.						10	10	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)						13	5	8	10
BMETE95MM25	Biostatistics	0	2	0	f	3	3
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
BMETE93MM14	Dynamic Programming in Financial Mathematics	2	0	0	v	3	3
BMETE95MM16	Extreme Value Theory	2	0	0	v	3		3
BMETE95MM17	Insurance Mathematics 2	2	0	0	f	2				2
BMETE95MM18	Multivariate Statistics with Applications in Economy	2	0	0	f	2	2
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 (10 ECTS credits must be completed)							3	2	5
*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
Code	Title	Parameters*					ECTS credits per semester
Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
	Theoretical foundations (20 ECTS credits). Prescribed subjects from BSc in Math, or elective professional courses.						10	10
	Professional subjects (30 ECTS credits must be completed). Courses marked by boldface letters are obligatory.						10	10	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						10	5	5	10
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)								5	5
*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.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM25	Biostatistics	0	2	0	f	3	3
Course coordinator: Dr. Marianna Bolla
Introduction to epidemiology. Classical epidemiological study designs. Predictive models. Multivariate logistic regression. Survival analysis. Biases in epidemiological studies. Examples, case studies, usage of SAS software.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM15	Multivariate Statistics	3	1	0	v	5	5
Course coordinator: Dr. Marianna Bolla
Multivariate central limit theorem and its applications. Density, spectra and asymptotic distribution of random matrices in multivariate statistics (Wishart-, Wigner-matrices). How to use separation theorems for eigenvalues and singular values in the principal component, factor, and correspondence analysis. Factor analysis as low rank representation, relatios between representations and metric clustering algorithms. Methods of classification: discriminatory analysis, hierarchical, k-means, and graph theoretical methods of cluster analysis. Spectra and testable parameters of graphs. Algorithmic models, statistical learning. EM algorithm, ACE algorithm, Kaplan–Meier estimates. Resampling methods: bootstrap and jackknife. Applications in data mining, randomized methods for large matrices. Mastering the multivariate statistical methods and their nomenclature by means of a program package (SPSS or S+), application oriented interpretation of the output data.
Literature: – K.V. Mardia, J.T. Kent, J.M. Bibby: Multivariate Analysis, Academic Press, Elsevier Science, 1979.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM20	Nonparametric Statistics	2	0	0	v	3	3
Course coordinator: Dr. László Györfi
Density function estimation. Distribution estimation, L1 error. Histogram. Estimates by kernel function. Regression function estimation. Least square error. Regression function. Partition, kernel function, nearest neighbour estimates. Empirical error minimization. Pattern recognition. Error probability. Bayes decision rule. Partition, kernel function, nearest neighbour methods. Empirical error minimization. Portfolio strategies. Log-optimal, empirical portfolio strategies. Transaction cost.
Literature: – L. Devroye, L. Györfi: Nonparametric Density Estimation, Wiley, 1985. – L. Devroye, L. Györfi, G. Lugosi: Probability Theory of Pattern Recognition, Springer, New York, 1996. – L. Györfi, M. Kohler, A. Krzyzak, H. Walk: A Distribution-Free Theory of Nonparametric Regression, Springer, New York, 2002.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM09	Statistical Program Packages 2	0	0	2	f	2	2
Course coordinator: Dr. Csaba Sándor
The goal of the course is to provide an overview of contemporary computer-based methods of statistics with a review of the necessary theoretical background. How to use the SPSS (Statistical Package for Social Sciences) in program mode. Writing user’s macros. Interpretation of the output data and setting the parameter values accordingly. Definition and English nomenclature of the dispalyed statistics. Introduction to the S+ and R Program Packages and surveying the novel algorithmic models not available in the SPSS (bootstrap, jackknife, ACE). Practical application. Detailed analysis of a concrete data set in S+.
Literature: – K.V. Mardia, J.T. Kent, M. Bibby: Multivariate analysis, Academic Press, New York, 1979. – L. Ketskeméty, L. Izsó: Introduction to the SPSS Program Package, in Hungarian, ELTE Publishers, Budapest, 2005. – S+ or R User's Guide (together with the program package).

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM07	Markov Processes and Martingales	3	1	0	v	5			5
Course coordinator: Dr. Károly Simon
Martingales: Review (conditional expectations and tower rule, types of probabilistic convergences and their connections, martingales, stopped martingales, Doob decomposition, quadratic variation, maximal inequalities, martingale convergence theorems, optional stopping theorem, local martingales). Sets of convergence of martingales, the quadratic integrable case. Applications (e.g. Gambler's ruin, urn models, gambling, Wald identities, exponential martingales). Martingale CLT. Azuma-Höffding inequality and applications (e.g. travelling salesman problem). Markov chains: Review (definitions, characterization of states, stationary distribution, reversibility, transience-(null-)recurrence). Absorbtion probabilites. Applications of martingales, Markov chain CLT. Markov chains and dynamical systems; ergodic theorems for Markov chains. Random walks and electric networks. Renewal processes: Laplace transform, convolution. Renewal processes, renewal equation. Renewal theorems, regenerative processes. Stationary renewal processes, renewal paradox. Examples: Poisson process, applications in queueing. Point processes: Definition of point processes. The Poisson point process in one and more dimensions. Transformations of the Poisson point process (marking and thinning, transforming by a function, applications). Point processes derived from the Poisson point process. Discrete state Markov processes: Review (infinitesimal generator, connection to Markov chains, Kolmogorov forward and backward equations, characterization of states, transience-(null-)recurrence, stationary distribution). Reversibility, MCMC. Absorption probabilities and hitting times. Applications of martingales (e.g. compensators of jump processes). Markov processes and dynamical systems; ergodic theorems for Markov processes. Markov chains with locally discrete state space: infinitesimal generator on test functions.
Literature: – T. Lindvall: Lectures on the Coupling Method, Dover Publications, Inc., Mineola, NY, 2002. – J.R. Norris: Markov chains. Cambridge University Press, Cambridge, 1998. – S. Resnick: Adventures in Stochastic Processes, Birkhäuser Boston, 1992. – M. Rosenblatt: Markov processes: Structure and Asymptotic Behavior. Springer-Verlag, New York-Heidelberg, 1971. – D. Williams: Probability with Martingales. Cambridge University Press, 1991.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM08	Stochastic Differential Equations	3	1	0	v	5				5
Course coordinator: Dr. Bálint Tóth
Introduction. Itô integral with respect to the Wiener process and continuous martingale, multi-dimensional stochastic integral. Local time. Local time of random walks on the line. Inverse local time, discrete Ray-Knight theorem. Local time of Brownian motion and Ray-Knight theorem. Tanaka formula and its applications. Skorohod reflection, reflected Brownian motion, a theorem by P. Lévy. Stochastic differential equations. SDEs of diffusions: Ornstein-Uhlenbeck, Bessel, Bessel-squared, exponential Brownian motion. SDE of transformed diffusions. Weak and strong solutions, existence and uniqueness. SDE with boundary conditions. Interpretation of the infinitesimal generator. Applications to physics, population dynamics, and finance. Duffusions. Basic examples: Ornstein-Uhlenbeck, Bessel, Bessel-squared, geometrical Brownian motion. Interpretation as stochastic integrals, and Markov processes. Infinitesimal generator, stochastic semi-groups. Martingale problem. Connection with parabolic and elliptic partial differential equations. Feyman-Kac formula. Time-change. Cameron-Martin-Girsanov formula. One-dimensional diffusions. Scale function and speed measure. Boundary conditions. Time-inversion. Application to special processes. Special selected topics. Brownian excursion. Two-dimensional Brownian motion, Brownian sheet. SLE. Additive functionals of Markov processes.
Literature: – K.L. Chung, R. Williams: Introduction to stochastic integration, 2nd edition, Birkauser, 1989. – N. Ikeda, S. Watanabe: Stochastic differential equations and diffusion processes, 2nd edition, North Holland, 1989. – K. Ito, H.P. McKean: Diffusion processes and their sample paths, Springer, 1965. – J. Jacod, S.N. Shiryaev: Limit theorems for stochastic processes, Springer, 1987. – S. Karlin, H.M. Taylor: A second course in stochastic processes, Academic, 1981. – D. Revuz, M. Yor: Continuous martingales and Brownian motion, 3rd edition, Springer, 1999.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM14	Financial Processes	2	0	0	f	3				3
Course coordinator: Dr. Péter Móra
Discrete models. Optimal parking, strategy in advantageous and disadvantageous situations. Self-financing portfolio, arbitrage, completeness of a market model. American, European, Asian option. Binary model. Pricing non-complete market in discrete model. Balck–Scholes' theory: B-S formula via martingales. Itô representation theorem. Applications, admissible strategies. Capital Asset Pricing Model (CAPM). Portfolios. The beta coefficient, security market line, market and capital-market equilibrium. Option pricing by using GARCH models. Problems of optimal investments. Extreme value theory, maxima, records.
Literature: – J.M. Steele: Stochastic Calculus and Fiancial Applications, Springer, New York, 2001. – B.C. Arnold, N. Balakrishnan, H.N. Nagaraja: Records, John Wiley and Sons, 1998.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE93MM14	Dynamic Programming in Financial Mathematics	2	0	0	v	3	3
Course coordinator: Dr. Katalin Nagy
Optimal strategies, discrete models. Fundamental principle of dynamic programming. Favourable and unfavourable games, brave and cautious strategies. Optimal parking, planning of large purchase. Lagrangean mechanics, Hamilton-Jacobi equation. Viscous approximation, Hopf-Cole transformation, Hopf-Lax infimum-convolution formula. Deterministic optimal control, startegy of optimal investment, viscous solutions of generalized Hamilton-Jacobi equations. Pontryagin’s maximum principle, searching conditional extreme values in function spaces. Optimal control of stochastic systems, Hamilton-Jacobi-Bellman equation.
Literature: – L.C. Evans: Partial Differential Equations, AMS, Providence, R.I., 1998.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM16	Extreme Value Theory	2	0	0	v	3		3
Course coordinator: Dr. Bálint Vető
Review of the limit theorems, normal domain of attraction, stable low of distributions, alpha-stable domain of attractions. Max-stable distributions, Fisher-Tippet theorem, standard extreme value distributions, regularly varying functions and their properties, Frechet and Weibull distributions and characterization of their domain of attraction. Gumbel distribution. Generalized Pareto distribution. Peak over threshold. Methods of parameter estimations. Applications in economy and finance.
Literature: – A.J. McNeil, R. Frey, P. Embrechts: Quantitative Risk Management Priceton University Press, 2005. – B.C. Arnold, N. Balakrishnan, H.N. Nagaraja: Records, John Wiley and Sons.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM17	Insurance Mathematics 2	2	0	0	f	2				2
Course coordinator: Dr. Attila Gerényi
Fundamental types of insurance: life and non-life. Standard types of non-life insurance, models. Individual risk model. Claim calculation and approximations. Most important distributions of the number of claim. Most important distributions of the claims payments. Complex risk model, recursive method of Panjer, compound Poisson distributions. Classical principles: expected value, maximum loss, quantile, standard deviation, variance; theoretical premium principles: zero utilizes, Swiss, loss-function. Mathematical properties of premium principles. Credibility theory, Bühlmann model. Bonus, premium return. Reserves, IBNR models.
Literature: – G.E. Rejda: Principles of Risk Management and Insurance.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM18	Multivariate Statistics with Applications in Economy	2	0	0	f	2	2
Course coordinator: Dr. Marianna Bolla
Multivariate central limit theorem and its applications. Density, spectra and asymptotic distribution of random matrices in multivariate statistics (Wishart-, Wigner-matrices). How to use separation theorems for eigenvalues and singular values in the principal component, factor, and correspondence analysis. Factor analysis as low rank representation, relatios between representations and metric clustering algorithms. Methods of classification: discriminatory analysis, hierarchical, k-means, and graph theoretical methods of cluster analysis. Spectra and testable parameters of graphs. Algorithmic models, statistical learning. EM algorithm, ACE algorithm, Kaplan–Meier estimates. Resampling methods: bootstrap and jackknife. Applications in data mining, randomized methods for large matrices. Mastering the multivariate statistical methods and their nomenclature by means of a program package (SPSS or S+), application oriented interpretation of the output data.
Literature: – K.V. Mardia, J.T. Kent, J.M. Bibby: Multivariate Analysis, Academic Press, Elsevier Science, 1979.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMEGT30M400	Analysis of Economic Time Series	2	0	0	f	2		2
Course coordinator: Dr. Dietmar Meyer
The course starts with a short introduction, which is followed by the generalization of the already known growth and conjuncture models. We discuss the issues of financing growth, the role of human capital, the dynamics of the budget deficit, endogenous population growth, healthcare economics and renewable resources. It is followed by the problem of the time consistency (both in finance and in budget policy), which – through different expectations – lead to the dynamic game theoretical approaches. This allows us to give the microeconomic background of the discussed macroeceonomic events. The course concludes with the discussion of the models of economic evolution.
Literature: – S. Dowrick, R. Pitchford, S. Turnovsky (ed.): Economic Growth and Macroeconomic Dynamics, Cambridge University Press, Cambridge, 2004. – B. Huber: Optimale Finanzpolitik und zeitliche Inkonsistenz. Physica-Verlag, Heidelberg, 1996. – S. Turnovsky: Methods of Macroeconomic Dynamics, MIT Press, Cambridge (Mass.), 2000. – F. Vega-Redondo: Evolution, Games, and Economic Behaviour, Oxford University Press, Oxford, 1996.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM26	Time Series Analysis with Applications in Finance	2	0	0	f	3			3
Course coordinator: Dr. Károly Simon
White noise and basic ARMA models, lag operators and polynomials, auto- and crosscorrelation, autocovariance, fundamental representation, state space representation, predicting ARMA models, impulse-response function, stationary ARMA models, Wold Decomposition, vector autoregression (VAR): Sims and Blanchard-Quah orthogonalization, variance decomposition, VARs in state space notation, Granger causality, spectral representation, spectral density, filtering, spectrum of the filtered series, constructing filters, Hodrick-Prescott filter, random walks and unit root time series, cointegration, Beveridge-Nelson decomposition, Bayesian Vector Autoregression (BVAR) models, Gibbs Sampling, coding practice and application to financial and macroeconomic data.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM10	Limit- and Large Deviation Theorems of Probability Theory	3	1	0	v	5		5
Course coordinator: Dr. Bálint Tóth
Part I.: Limit theorems: Weak convergence of probability measures and distributions. Tightness: Helly-Ptohorov theorem. Limit theorems proved with bare hands: Applications of the reflection principle to random walks: Paul Lévy’s arcsine laws, limit theorems for the maximum, local time and hitting times of random walks. Limit theorems for maxima of i.i.d. random variables, extremal distributions. Limit theorems for the coupon collector problem. Proof of limit theorem with method of momenta. Limit theorem proved by the method of characteristic function. Lindeberg’s theorem and its applications: Erdős-Kac theorem: CLT for the number of prime factors. Stable distributions. Stable limit law of normed sums of i.i.d. random variables. Characterization of the characteristic function of symmetric stable laws. Weak convergence to symmetric stable laws. Applications. Characterization of characteristic function of general (non-symmetric) stable distributions, skewness. Weak convergence in non-symmetric case. Infinitely divisible distributions:. Lévy-Hinchin formula and Lévy measure. Lévy measure of stable distributions, self-similarity. Poisson point processes and infinitely divisible laws. Infinitely divisible distributions as weak limits for triangular arrays. Applications. Introduction to Lévy processes: Lévy-Hinchin formula and decomposition of Lévy processes. Construction with Poisson point processes (a la Ito). Subordinators and Lévy processes with finite total variation, examples. Stable processes. Examples and applications. Part II.: Large deviation theorems: Introduction: Rare events and large deviations. Large deviation principle. Computation of large deviation probabilities with bare hands: application of Stirling’s formula. Combinatorial methods: The method of types. Sanov’s theorem for finite alphabet. Large deviations in finite dimension: Bernstein’s inequality, Chernoff’s bound, Cramer’s theorem. Elements of convex analysis, convex conjugation in finite dimension, Cramer’s theorem in R^d. Gartner-Ellis theorem. Applications: large deviation theorems for random walks, empirical distribution of the trajectories of finite state Markov chains, statistical applications. The general theory: general large deviation principles. The contraction principle and Varadhan’s lemma. large deviations in topological vector spaces and function spaces. Elements of abstract convex analysis. Applications: Schilder’s theorem, Gibbs conditional measures, elements of statistical physics.
Literature: – A. Dembo, O. Zeitouni: Large deviation techniques and application. Springer, 1998. – R. Durrett: Probability: theory and examples. Second edition. Duxbury, 1996. – B.V. Gnedenko, A.N. Kolmogorov: Limit theorems for sums of independent random variables, 1951. – W. Feller: An introduction to probability theory and its applications. Vol.2. Wiley, 1970. – D.W. Stroock: An introduction to the theory of large deviations. Springer, 1984. – S.R.S. Varadhan: Large deviations and application . SIAM Publications, 1984. – D. Williams: Probability with martingales. Cambridge UP, 1990.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM11	Stochastic Models	2	0	0	f	2				2
Course coordinator: Dr. Gábor Pete
Coupling methods (stochastic dominance, coupling random variables and stochastic processes, examples: connectivity using dual graphs, optimization problems, combinatorial probability problems) Percolation (definitions, correlation inequalities, duality, contour methods) Strongly dependent percolation: Winkler percolation, compatible 0-1 sequences Basics of statistical physics (Gibbs measure, a few basic models). Card shuffling (completely shuffled deck, how many times should one shuffle?) Random graph models (Erdős–Rényi, Barabási–Albert; basic phenomena). Variants of random walks: scenery reconstruction, self-avoiding és self-repelling walks, loop-erased walks, random walk in random environment). Queueing models and basic behavior; stationary distribution and reversibility, Burke Theorem; systems of queues. Interacting particle systems (simple exclusion on the torus and on the infinite lattice, stationary distribution, Palm distributions, couplings, other models). Graphical construction of continuous time Markov processes (Yule model, Hammersley's process, particle systems). Self organized criticality: sandpile models (questions of construction, commutative dynamics, stationary distribution in finite volume, power law decay of correlations). Linear theory of stationary processes: strongly and weakly stationary processes, spectral properties, autoregressive and moving average processes. Analysis of time series, long memory processes. Models of risk processes.
Literature: – G. Grimmett: Percolation, Springer-Verlag, Berlin, 1999. – T. Liggett: Interacting Particle Systems, Springer-Verlag, Berlin, 2005. – T. Lindvall: Lectures on the Coupling Method, Dover Publications, Inc., Mineola, NY, 2002. – H. Thorisson: Coupling, Stationarity, and Regeneration, Springer-Verlag, New York, 2000. – J. Walrand: An Introduction to Queueing Networks, Prentice Hall, 1988. – W. Werner: Lectures on Two-dimensional Critical Percolation, http://arxiv.org/abs/0710.0856 – W. Werner: Random Planar Curves and Schramm–Loewner Evolutions, http://arxiv.org/abs/math/0303354 – O. Zeitouni: Lecture Notes on Random Walks in Random Environment, XXXI summer school in probability, St Flour, France, Volume 1837 of Springer's Lecture notes in Mathematics

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM12	Advanced Theory of Dynamical Systems	2	0	0	f	2				2
Course coordinator: Dr. Domokos Szász
Subadditive and multiplicative ergodic theorems. Lyapunov exponents. Spectral properties of measure preserving transformations. Shadowing lemma. Markov partitions and their construction for uniformly hyperbolic systems. Perron-Frobenius operator and its spectrum. Doeblin-Fortet inequality.Stochastic properties of hyperbolic dynamical systems. Kolmogorov-Sinai entropy. Ornstein’s isomorphy theorem (without proof).
Literature: – M. Pollicott: Lectures on Ergodic theory and Pesin Theory on compact manifolds, CUP, 1993. – R. Bowen: Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer LNM 470, 1975. – M. Brin, G. Stuck: Introduction to Dynamical Systems, CUP, 2002.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE92MM01	Individual Projects 1	0	0	4	f	4		4
Course coordinator: Dr. Márta Lángné Lázi
Within the framework of the subject the student is working on an application oriented research subject based on stochastic mathematics lead by an external supervisor. At the end of each semester the student writes a report about his results which will be also presented by him to the other students in a lecture. The activities to be exercised: literature research, modelling, computer aided problem solving, mathematical problem solving.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE92MM02	Individual Projects 2	0	0	4	f	4			4
Course coordinator: Dr. Márta Lángné Lázi
Within the framework of the subject the student is working on an application oriented research subject based on stochastic mathematics lead by an external supervisor. At the end of each semester the student writes a report about his results which will be also presented by him to the other students in a lecture. The activities to be exercised: literature research, modelling, computer aided problem solving, mathematical problem solving.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM01	Mathematical Modelling Seminar 1	2	0	0	f	1	1
Course coordinator: Dr. Domokos Szász
The aim of the seminar to present case studies on results, methods and problems from applied mathematics for promoting: the spreading of knowledge and culture of applied mathematics; the development of the connections and cooperation of students and professors of the Mathematical Institute, on the one hand, and of personal, researchers of other departments of the university or of other firms, interested in the applications of mathematics. The speakers talk about problems arising in their work. They are either applied mathematicians or non-mathematicians, during whose work the mathematical problems arise. An additional aim of this course to make it possible for interested students to get involved in the works presented for also promoting their long-range carrier by building contacts that can lead for finding appropriate jobs after finishing the university.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE95MM02	Mathematical Modelling Seminar 2	2	0	0	f	1			1
Course coordinator: Dr. Domokos Szász
The aim of the seminar to present case studies on results, methods and problems from applied mathematics for promoting: the spreading of knowledge and culture of applied mathematics; the development of the connections and cooperation of students and professors of the Mathematical Institute, on the one hand, and of personal, researchers of other departments of the university or of other firms, interested in the applications of mathematics. The speakers talk about problems arising in their work. They are either applied mathematicians or non-mathematicians, during whose work the mathematical problems arise. An additional aim of this course to make it possible for interested students to get involved in the works presented for also promoting their long-range carrier by building contacts that can lead for finding appropriate jobs after finishing the university.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE90MM90	Report	0	0	0	a	0		0
Course coordinator: Dr. Attila Andai
A tárgyat akkor tekintjük teljesítettnek, ha a hallgató a felvételi során megkövetelt alapképzésbeli tárgyak elvégzésével az előírt legalább 65 kreditet teljesítette, továbbá a hallgatónak van elfogadott diplomatémája és témavezetője.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE90MM98	Preparatory Course for Master's Thesis	0	2	0	f	5			5
Course coordinator: Dr. Attila Andai
A diplomamunka a matematikushallgatóknak a témavezető irányításával elért önálló kutatási, kutatás-fejlesztési eredményeit tartalmazó írásbeli beszámoló (dolgozat). A hallgató a dolgozatban mutassa be a vizsgált témát, fejtse ki a problémákat, és részletesen ismertesse eredményeit. A munkának a matematikus tanulmányok ismeretanyagára kell épülnie és a szerző önálló, saját munkája legyen. A diplomamunkának arról kell tanúskodnia, hogy a hallgató az egyetemi tanulmányai során szerzett matematikai ismereteit, képességeit a gyakorlati életben vagy az elméleti kutatásokban egy több hónapra kiterjedő munka folyamán önállóan tudja alkalmazni oly módon, hogy a megoldandó problémát felismeri, a megoldáshoz vezető út nehézségeivel megbirkózik, a megfelelő színvonalú megoldást megtalálja, és azt mások számára érthetően leírja. A dolgozat legyen tömör, de a témában nem járatos matematikus olvasó számára is érthető. A Diplomamunka előkészítés c. tárgy keretében a hallgató összegyüjti mindazokat az információkat és matematikai eredményeket, amelyek a diplomamunka megírásához szükségesek.

Code	Title	Lc	Pr	Lb	Rq	Cr	I	II	III	IV
BMETE90MM99	Master's Thesis	0	8	0	f	15				15
Course coordinator: Dr. Attila Andai
A diplomamunka a matematikushallgatóknak a témavezető irányításával elért önálló kutatási, kutatás-fejlesztési eredményeit tartalmazó írásbeli beszámoló (dolgozat). A hallgató a dolgozatban mutassa be a vizsgált témát, fejtse ki a problémákat, és részletesen ismertesse eredményeit. A munkának a matematikus tanulmányok ismeretanyagára kell épülnie és a szerző önálló, saját munkája legyen. A diplomamunkának arról kell tanúskodnia, hogy a hallgató az egyetemi tanulmányai során szerzett matematikai ismereteit, képességeit a gyakorlati életben vagy az elméleti kutatásokban egy több hónapra kiterjedő munka folyamán önállóan tudja alkalmazni oly módon, hogy a megoldandó problémát felismeri, a megoldáshoz vezető út nehézségeivel megbirkózik, a megfelelő színvonalú megoldást megtalálja, és azt mások számára érthetően leírja. A dolgozat legyen tömör, de a témában nem járatos matematikus olvasó számára is érthető. A Diplomamunka-készítés c. tárgy keretében a hallgató megírja a diplomamunkáját.