MSC IN APPLIED MATHEMATICS 

The twoyear MSc program in Applied Mathematics (math.bme.hu/masters/) provides a profound knowledge of applied mathematics which is competitive both in the academic and nonacademic sectors. Possible specializations are Stochastics and Financial Mathematics. Students of our MSc program may enter leadingedge research projects of the Department of Stochastics (a cutting edge research center in stochastics, the host of our MSc studies). Some of the best students of our MSc program continue their studies and become Ph.D. students at either our university or some cuttingedge universities in the US or Europe. Others get wellpaid jobs at leading banks, insurance, and consulting companies, like Morgan Stanley or in the industry. 

CURRICULUM OF SPECIALIZATION FINANCIAL MATHEMATICS 

Code 
Title 
Parameters* 
ECTS credits 

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). 
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: 

CURRICULUM OF SPECIALIZATION STOCHASTICS 

Code 
Title 
Parameters* 
ECTS credits 

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). 
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) 
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: 

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 GazdagTóth 

Different forms of global optimization problems, their transformation to each other, and their reduction to the onedimensional 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. Multistart 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. Multiobjective optimization. 

Literature: 

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. GaussJordan 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 Weyltheorems. Motzkintheorem. Primaldual linear programming problems. Feasible solution set of linear programming problems. Basic solution of linear programming problem. Simplex and crisscross algorithms. Cycling, anticycling rules: Bland’s minimal index rule. Two phase simplex method. Revised simplex method. Sensitivity analysis. Decomposition methods: DantzigWolfe. Special type of pivot algorithms: lexicographic and lexicographic dual simplex methods. Monotonic buildup simplex algorithms. Interior point methods of linear programming problems. Selfdual linear programming problem. Central path and its uniqueness. Computation of Newtondirections. Analytical centre, Sonnevendtheorem. Dikinellipsoid, affine scaling primaldual interior point algorithm and its polynomial complexity. Tuckermodel, Tucker theorem. Rounding procedure. Khachian’s ellipsoid algorithm. Karmarkar’s potential function method. Special interior point algorithms. 

Literature: 

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: 

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. RobinsonSchenstedKnuth correspondence. LittlewoodRichardson numbers, LittlewoodRichardson theorem. Important symmetric polynomials, their generating functions. CauchyLittlewood 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: 

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 

Continuoustime and discretetime dynamical systems, continuous versus descrete: first return map, discretization. Local theory of equilibria: Grobman–Hartman lemma, stableunstablecenter 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: 

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 AbelPoisson summation. WeierstrassStone theorem, applications. Best approximation in Hilbert spaces. Müntz theorem on the density of lacunary polynomials. Approximations by linear operators, Lagrange interpolation, LozinskiHarshiladze theorem. Approximation by polynomials, theorems of Jackson. Positive linear operators Korovkin theorem, Bernstein polynomials, HermiteFejer operator. Spline approximation, convergence, Bsplines. 

Literature: 

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. RitzGalerkin approximation. Linear operator semigroups (According to Evans and Robinson). Weak and strong solutions to reactiondiffusion (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: 

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 semigroups, 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. Selfsimilarity and recurrence of Brownian motion, time reversal. Reflection principle and its applications. Local properties of Brownian path: continuity, Hölder continuity, nondifferenciability. Quadratic variations. Continuous martingales. Definition and basic properties. DubbinsSchwartz theorem. Exponential martingale. Lévy processes. Processes with independent and stationary increments, LévyHintchin 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 BalckScholes formula. Stochastic integrals with respect to Poisson process. Martingales of finite state space Markov processes. Quadratic variations. DoobMeyer decomposition. Stochastic integration II. Predictable processes. Itô integral with respect to the Wiener process, quadratic variation process. DoobMeyer decomposition. Itô formula and its applications. 

Literature: 

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 twoway analysis of variance as a special case of the linear model. ANOVA tables, FisherCochran theorem. Principal component and factor analysis. Estimation and rotation of factors, testing hypotheses for the effective number of factors. Hypothesis testing and Idivergence (the discrete case). Iprojections, maximum likelihood estimate as Iprojection in exponential families. The limit distribution of the Idivergence 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: 

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, blowup. 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, CayleyHamilton theorem, Nakayama lemma. Localization and tensor product. Free resolutions of modules, Gröbner theory of modules, computations, Hilbert syzygy theorem. 

Literature: 

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, Liederivative, 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, JordanChevalley 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 PeterWeyl theorem. 

Literature: 

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, Liederivation. Stokes' theorem, de Rham cohomology, Mayer–Vietoris exact sequence, Poincaréduality. 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: 

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, Wignermatrices). 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, kmeans, 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: 

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. Logoptimal, empirical portfolio strategies. Transaction cost. 

Literature: 

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 computerbased 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: 

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. AzumaHöffding 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: 

Code 
Title 
Lc 
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Lb 
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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, multidimensional stochastic integral. Local time. Local time of random walks on the line. Inverse local time, discrete RayKnight theorem. Local time of Brownian motion and RayKnight theorem. Tanaka formula and its applications. Skorohod reflection, reflected Brownian motion, a theorem by P. Lévy. Stochastic differential equations. SDEs of diffusions: OrnsteinUhlenbeck, Bessel, Besselsquared, 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: OrnsteinUhlenbeck, Bessel, Besselsquared, geometrical Brownian motion. Interpretation as stochastic integrals, and Markov processes. Infinitesimal generator, stochastic semigroups. Martingale problem. Connection with parabolic and elliptic partial differential equations. FeymanKac formula. Timechange. CameronMartinGirsanov formula. Onedimensional diffusions. Scale function and speed measure. Boundary conditions. Timeinversion. Application to special processes. Special selected topics. Brownian excursion. Twodimensional Brownian motion, Brownian sheet. SLE. Additive functionals of Markov processes. 

Literature: 

Code 
Title 
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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. Selffinancing portfolio, arbitrage, completeness of a market model. American, European, Asian option. Binary model. Pricing noncomplete market in discrete model. Balck–Scholes' theory: BS formula via martingales. Itô representation theorem. Applications, admissible strategies. Capital Asset Pricing Model (CAPM). Portfolios. The beta coefficient, security market line, market and capitalmarket equilibrium. Option pricing by using GARCH models. Problems of optimal investments. Extreme value theory, maxima, records. 

Literature: 

Code 
Title 
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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, HamiltonJacobi equation. Viscous approximation, HopfCole transformation, HopfLax infimumconvolution formula. Deterministic optimal control, startegy of optimal investment, viscous solutions of generalized HamiltonJacobi equations. Pontryagin’s maximum principle, searching conditional extreme values in function spaces. Optimal control of stochastic systems, HamiltonJacobiBellman equation. 

Literature: 

Code 
Title 
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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, alphastable domain of attractions. Maxstable distributions, FisherTippet 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: 

Code 
Title 
Lc 
Pr 
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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 nonlife. Standard types of nonlife 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, lossfunction. Mathematical properties of premium principles. Credibility theory, Bühlmann model. Bonus, premium return. Reserves, IBNR models. 

Literature: 

Code 
Title 
Lc 
Pr 
Lb 
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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, Wignermatrices). 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, kmeans, 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: 

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BMEGT30M400 
Analysis of Economic Time Series 
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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: 

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BMETE95MM26 
Time Series Analysis with Applications in Finance 
2 
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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, impulseresponse function, stationary ARMA models, Wold Decomposition, vector autoregression (VAR): Sims and BlanchardQuah orthogonalization, variance decomposition, VARs in state space notation, Granger causality, spectral representation, spectral density, filtering, spectrum of the filtered series, constructing filters, HodrickPrescott filter, random walks and unit root time series, cointegration, BeveridgeNelson decomposition, Bayesian Vector Autoregression (BVAR) models, Gibbs Sampling, coding practice and application to financial and macroeconomic data. 

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BMETE95MM10 
Limit and Large Deviation Theorems of Probability Theory 
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5 


Course coordinator: Dr. Bálint Tóth 

Part I.: Limit theorems: Weak convergence of probability measures and distributions. Tightness: HellyPtohorov 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: ErdősKac 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 (nonsymmetric) stable distributions, skewness. Weak convergence in nonsymmetric case. Infinitely divisible distributions:. LévyHinchin formula and Lévy measure. Lévy measure of stable distributions, selfsimilarity. Poisson point processes and infinitely divisible laws. Infinitely divisible distributions as weak limits for triangular arrays. Applications. Introduction to Lévy processes: LévyHinchin 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. 

Literature: 

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BMETE95MM11 
Stochastic Models 
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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) 

Literature: 

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BMETE95MM12 
Advanced Theory of Dynamical Systems 
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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. PerronFrobenius operator and its spectrum. DoeblinFortet inequality.Stochastic properties of hyperbolic dynamical systems. KolmogorovSinai entropy. Ornstein’s isomorphy theorem (without proof). 

Literature: 

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BMETE92MM01 
Individual Projects 1 
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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. 

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BMETE92MM02 
Individual Projects 2 
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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. 

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BMETE95MM01 
Mathematical Modelling Seminar 1 
2 
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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 nonmathematicians, 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 longrange carrier by building contacts that can lead for finding appropriate jobs after finishing the university. 

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BMETE95MM02 
Mathematical Modelling Seminar 2 
2 
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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 nonmathematicians, 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 longrange carrier by building contacts that can lead for finding appropriate jobs after finishing the university. 

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BMETE90MM90 
Report 
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a 
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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. 

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BMETE90MM98 
Preparatory Course for Master's Thesis 
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2 
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5 

Course coordinator: Dr. Attila Andai 

A diplomamunka a matematikushallgatóknak a témavezető irányításával elért önálló kutatási, kutatásfejlesztési 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. 

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BMETE90MM99 
Master's Thesis 
0 
8 
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15 



15 
Course coordinator: Dr. Attila Andai 

A diplomamunka a matematikushallgatóknak a témavezető irányításával elért önálló kutatási, kutatásfejlesztési 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 Diplomamunkakészítés c. tárgy keretében a hallgató megírja a diplomamunkáját. 
