Phd program

References:
1. K.V. Mardia, J.T. Kent, and M. Bibby, Multivariate analysis. Academic Press, New

York, 1979.

2. C.R. Rao, Linear statistical inference and its applications. Wiley, New York, 1973.

74) ERGODIC THEORY

Course coordinator: Peter Balint

No. of Credits: 3 and no. of ECTS credits: 6

Prerequisites: Topics in Analysis, Probability 1

Course Level: intermediatePhD

Brief introduction to the course:

Basic concepts of ergodic theory: measure preserving transformations, ergodic theorems, notions of ergodicity, mixing and methods for proving such properties, topological dynamics, hyperbolic phenomena, examples: eg. rotations, expanding interval maps, Bernoulli shifts, continuous automorphisms of the torus.

The goals of the course:

The main goal of the course is to give an introduction to the central ideas of ergodic theory, and to point out its relations to other fields of mathematics.

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents (week-by-week):  

Week 1: Basic definitions and examples(measure preserving transformations, examples: rotations, interval maps etc.)

Week 2: Ergodic theorems(Poincare recurrence theorem, von Neumann and Birkhoff ergodic theorems)

Week 3: Ergodicity(different characterizations, examples: rotations)

Week 4: Further examples: stationary sequences(Bernoulli shifts, doubling map, baker’s transformation)

Week 5: Mixing(different characterizations, study of examples from this point of view)

Week 6: Continuous automorphisms of the torus(definitions, proof of ergodicity via characters)

Week 7: Hopf’s method for proving ergodicity(hyperbolicity of a continuous toral automorphism, stable and unstable manifolds, Hopf chains)

Week 8: Invariant measures for continuous maps(Krylov-Bogoljubov theorem, ergodic decomposition, examples)

Week 9: Markov maps of the interval(definitions, existence and uniqueness of the absolutely continuous invariant measure)

Weeks 10-12: Further topics based on the interest of the students(eg. attractors, basic ideas of KAM theory, entropy, systems with singularities etc.)

References:

1. P. Walters:Introduction to Ergodic Theory, Springer, 2007

2. M. Brin- G.Stuck: Introduction to Dynamical Systems, Cambridge University Press 2002

75) MATHEMATICAL METHODS IN STATISTICAL PHYSICS

Course Coordinator: Balint Toth

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Topics in Analysis, Probability 1

Course Level: intermediatePhD 

Brief introduction to the course:

The main theorems of Statistical Physics are presented among others about Ising model.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of Statistical Physics.  

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents (week-by-week):  

Week 1 The object of study of statistical physics, basic notions.

Week 2-3 Curie-Weiss mean-field theory of the critical point. Anomalous fluctuations at the critical point.

Week 4-5 The Ising modell on Zd.

Week 6-7 Analiticity I: Kirkwood-Salsburg equations.

Week 8-9 Analiticity II: Lee-Yang theory.

Week 10-11 Phase transition in the Ising model: Peierls' contour method.

Week 12 Models with continuous symmetry.

76) FRACTALS AND DYNAMICAL SYSTEMS

Course Coordinator: Karoly Simon

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Topics in Analysis, Probability 1

Course Level: intermediatePhD 

Brief introduction to the course:

The main theorems about Fractals are presented among others about local dimension of invariant measures.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of the Fractals and Dynamical Systems.  

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents (week-by-week):  

Week 1-2 Fractal dimensions. Hausdorff and Packing measures.

Week 3 Basic examples of dynamically defined fractals. Horseshoe, solenoid.

Week 4-5 Young's theorem about dimension of invariant measure of a C2 hyperbolic diffeomorphism of a surface.

Week 6-7 Some applications of Leddrapier- Young theorem.

Week 8-9 Barreira, Pesin, Schmeling Theorem about the local dimension of invariant measures.

Week 10-11 Geometric measure theoretic properties of SBR measure of some uniformly hyperbolic attractors.

Week 12 Solomyak Theorem about the absolute continuous infinite Bernoulli convolutions.

References:

1. K. Falconer, Fractal geometry. Mathematical foundations and applications. John Wiley & Sons, Ltd., Chichester, 1990. 

2. K. Falconer, Techniques in fractal geometry. John Wiley & Sons, Ltd., Chichester, 1997. 

3. Y. Pesin, Dimension theory in dynamical systems. Contemporary views and applications Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997.

77) DYNAMICAL SYSTEMS

Course Coordinator: Domokos Szász

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Probability 1

Course Level: advanced PhD 

Brief introduction to the course:

The main theorems of Dynamical Systems are presented among others about the ergodic hypothesis and hard ball systems.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of the Dynamical Systems.  

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents (week-by-week):  

1. 
   Kesten-Furstenberg theorem.
   
2. 
   Kingman's subadditive ergodic theorem.
   
3. 
   Oseledec' multiplicative ergodic theorem, Lyapunov exponents.
   
4. 
   Thermodynamic formalism, Markov-partitions.
   
5. 
   Chaotic maps of the interval, expanding maps, Markov-maps.
   
6. 
   Chaotic conservatice systems.
   
7. 
   The ergodic hypothesis.
   
8. 
   Billiards, hard ball systems. The standard map.
   
9. 
   Chaotic non-conservative (dissipative) systems. Strange attractors. Fractals.
   
10. 
    Exponenets and dimensions. Map of the solenoid.
    
11. 
    Stability: invariant tori and the Kolmogorov-Arnold-Moser theorem.
    
12. 
    Anosov-maps. Invariant manifolds. SRB-measure.
    

References:

1. I.P. Cornfeld and S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, 1982

2. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, 1995

78) INVARIANCE PRINCIPLES IN PROBABILITY AND STATISTICS

Course Coordinator: Istvan Berkes

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Probability 1

Course Level: advanced PhD 

Brief introduction to the course:

The main invariance principles in Probability and Statistics are presented concentrating on strong approximation and asymptotic results.

The goals of the course:

The main goal of the course is to introduce students to some advanced methods of Probability and Statistics.  

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents (week-by-week):  

Week 1 Functional central limit theorem. Donsker's theorem via Skorokhod embedding. Weak convergence in D[0,1].

Week 2-3 Strassen's strong invariance theorem.

Week 4-5 Strong approximations of partial sums by Wiener process: Komlós-Major- Tusnády theorem and its extension (Einmahl, Sakhanenko, Zaitsev).

Week 6-7 Strong invariance principles for local time and additive functionals. Iterated processes.

Week 8-9 Strong approximation of empirical process by Brownian bridge: Komlós- Major- Tusnády theorem.

Week 10 Strong approximation of renewal process.

Week 11 Strong approximation of quantile process.

Week 12 Asymptotic results (distributions, almost sure properties) of functionals of the above processes.

References:

1. M. Csorgo-P. Revesz: Strong Approximations in Probability and Statistics. Academic Press, New York ,1981.

2. P. Revesz: Random Walk in Random and Non-Random Environments. World Scientific, Singapore , 1990.

3. M. Csorgo-L. Horvath: Weighted Approximations in Probability and Statistics. Wiley, New York , 1993.

79) STOCHASTIC ANALYSIS

Course coordinator: Miklos Rasonyi

No. of Credits: 3 and no. of ECTS credits 6

Prerequisites: Probability Theory 1

Course Level: intermediate PhD

Brief introduction to the course:

Main topics are: Brownian motion (Wiener process), martingales, stochastic (Ito) integration, stochastic differential equations, diffusion processes. These tools are heavily used in financial mathematics, biology, physics, and engineering. Thus if someone wants to enter e.g. the flourishing field of financial mathematics, it is a must to complete such a course.

The goals of the course:

Review of some calculus and probability tools. A review of the theory of stochastic processes, including continuous time Markov processes. Introducing the student to the major topics of stochastic calculus, including stochastic integration, stochastic differential equations and diffusion processes. Introducing to some applications, in particular the Black-Scholes model of financial mathematics.

The learning outcomes of the course:

A good understanding of continuous time stochastic processes, including Wiener process and other diffusion processes (Ito diffusions). Understanding and competence in stochastic integration and stochastic differential equations (SDE’s), strong and weak solutions, and conditions for existence and uniqueness. Practice in solving linear SDE’s, understanding the Ornstein-Uhlenbeck process. Understanding the relationship between weak solutions and the Stroock-Varadhan martingale problem; the notion of generator of a diffusion, and the related backward and forward partial differential equations.

More detailed display of contents:

Week 1: A review of Calculus and Probability theory topics.

Conditional expectation, main properties, continuous time stochastic processes, martingales, stopping times.

Week 2: Definition and some properties of Brownian motion.

Covariance function, quadratic variation, martingales related to Brownian motion, Markov property.

Week 3: Further properties of Brownian motion;random walks and Poisson process .

Hitting times, reflection principle, maximum and minimum, zeros: the arcsine law, Brownian motion in higher dimensions. Martingales related to random walks, discrete stochastic integrals, optional stopping in discrete setting, properties of Poisson process.

Week 4: Definition of Ito stochastic integral.

Definition and stochastic integral of simple adapted processes. Basic properties of the stochastic integral of simple processes. Stochastic integral of left-continuous, square-integrable, adapted processes. Extension to regular, adapted processes.

Week 5: Ito integrals as processes, Ito formul.,

Ito integrals as martingales, Gaussian Ito integrals, Ito formula for Brownian motion, Ito processes, their quadratic variation and the corresponding Ito formula.

Week 6: Ito processes.

Ito processes, Ito formula for Ito processes. Ito formula in higher dimensions, integration by parts formulae.

Week 7: Stochastic Differential Equations; strong solution.

The physical model and the definition of Stochastic Differential Equations (SDE), SDE of Ornstein-Uhlenbeck (OU) process, geometric Brownian motion, stochastic exponential and logarithm, explicit solution of a linear SDE, strong solution of an SDE, existence and uniqueness theorem, Markov property of solutions.

Week 8: Weak solutions of SDE’s.

Construction of weak solutions, canonical space for diffusions, the Stroock-varadhan martingale problem, generator of a diffusion, backward and forward equations, Stratonovich calculus.

Week 9: Some properties of diffusion processes.

Dynkin formula, calculation of expectation. Feynman-Kac formula.

Week 10: Further properties of diffusion processes.

Time homogeneous diffusions and their generators. Diffusions in the line: L-harmonic functions, scale function, explosion, recurrence and transience, stationary distributions.

Week 11: Multidimensional diffusions.

Existence and uniqueness. Some properties. Bessel processes.

Week 12: An application in financial mathematics.

Derivatives, arbitrage, replicating portfolio, complete market model, self-financing portfolio, the Black-Scholes model.

Optional topics: Changing of probability measure, Girsanov theorem.

Reference:

Fima C. Klebaner, Introduction to stochastic calculus with applications, Second edition, Imperial College Press, 2006.

80) PATH PROPERTIES OF STOCHASTIC PROCESSES

Course Coordinator: Peter Major

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Invariance Principles in Probability and Statistics, Stochastic Processes.

Course Level: advanced PhD 

Brief introduction to the course:

The main theorems about Path Properties of Stochastic Processes are presented among others about path properties of random walks, or branching Wiener process.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of Path Properties of Stochastic Processes.  

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents (week-by-week):  

1. 
   Constructions of Wiener process.
   
2. 
   Modulus of continuity.
   
3. 
   Laws of the iterated logarithm. Strassen's theorem.
   
4. 
   Increments of Wiener process.
   
5. 
   Local times, additive functionals and their increments.
   
6. 
   Asymptotic properties, invariance principles for local time and additive functionals.
   
7. 
   Dobrushin's theorem.
   
8. 
   Path properties of random walks, their local times and additive functionals.
   
9. 
   Random walk in random environment.
   
10. 
    Random walk in random scenery.
    
11. 
    Branching random walk and branching Wiener process.
    
12. 
    Almost sure central limit theorems.
    

References:

1. M. Csorgo-P. Revesz: Strong Approximations in Probability and Statistics. Academic Press, New York , 1981.

2. P. Revesz: Random Walk in Random and Non-Random Environments. World Scientific, Singapore , 1990.

3. P. Revesz: Random Walks of Infinitely Many Particles. World Scientific, Singapore , 1994.

4. D. Revuz-M. Yor: Continuous Martingales and Brownian Motion. Third edition. Springer, Berlin , 1999.

81) NONPARAMETRIC STATISTICS

Course Coordinator: Istvan Berkes

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Probability 1, Mathematical Statistics.

Course Level: advanced PhD 

Brief introduction to the course:

The main theorems of Nonparametric Statistics are presented like Nonparametric test and Empirical processes.

The goals of the course:

The main goal of the course is to introduce students to the main topics and methods of Nonparametric Statistics.  

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents (week-by-week):  

1. 
   Order statistics and their distribution.
   
2. 
   Empirical distribution function.
   
3. 
   Glivenko-Cantelli theorem and its extensions.
   
4. 
   Estimation of the density function. Kernel-type estimators.
   
5. 
   U-statistics.
   
6. 
   Rank correlation. Kendall-s tau.
   
7. 
   Nonparametric tests: goodness of fit, homogeneity, independence.
   
8. 
   Empirical process, approximation by Brownian bridge.
   
9. 
   Komlós-Major-Tusnády theorem.
   
10. 
    Tests based on empirical distribution: Kolmogorov-Smirnov, von Mises tests.
    
11. 
    Quantile process. Bahadur-Kiefer process.
    
12. 
    Rank tests. Wilcoxon-Mann-Whitney test.
    

References:

1. L. Takács: Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York , 1967.

2. J. Hájek: Nonparametric Statistics. Holden-Day, San Francisco , 1969.

82) TOPICS IN FINANCIAL MATHEMATICS

Course coordinator: Miklos Rasonyi

No. of Credits: 3 and no. of ECTS credits: 6

Prerequisities: Probability 1

Course Level: introductory PhD

Brief introduction to the course:

Basic concepts of stochastic calculus with respect to Brownian motion. Martingales, quadratic variation, stochastic differential equations. Fundamentals of continuous-time mathematical finance; pricing, replication, valuation using PDE methods. Exotic options, jump processes.

The goals of the course:

To obtain a solid base for applying continuous-time stochastic finance techniques; a firm knowledge of basic notions, methods. An introduction to most often used models.

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.

More detailed display of contents:

Week 1. From random walk to Brownian motion. Quadratic variation.

Week 2. Ito integral, Ito processes. Ito's formula and its applications.

Week 3. Stochastic differential equations: existence and uniqueness of solutions.

Week 4. Black-Scholes model and option pricing formula.

Week 5. Replication of contingent claims. European options.

Week 6. American options and their valuation.

Week 7. The PDE approach to hedging and pricing.

Week 8. Exotic (Asian, lookback, knock-out barrier,...) options.

Week 9. The role of the numeraire. Forward measure.

Week 10. Term-structure modelling: short rate models, affine models.

Week 11. Heath-Jarrow-Morton models. Defaultable bonds.

Week 12. Asset price models involving jumps.

Reference:

Steven E. Shreve: Stochastic calculus for finance, vols. I and II, Springer, 2004

83) NUMERICAL METHODS IN STATISTICS

Course Coordinator: Istvan Berkes

No. of Credits: 3, and no. of ECTS credits: 6

Prerequisites: Probability; Mathematical Statistics.

Course Level: advanced PhD 

Brief introduction to the course:

The main ingredients of Numerical Methods in Statistics are presented like Statistical procedures for stochastic processes, Bootstrap methods and Monte Carlos methods.

The goals of the course:

The main goal of the course is to introduce students to the Numerical Methods in Statistics.  

The learning outcomes of the course:

By the end of the course, students are enabled to do independent study and research in fields touching on the topics of the course, and how to use these methods to solve specific problems. In addition, they develop some special expertise in the topics covered, which they can use efficiently in other mathematical fields, and in applications, as well. They also learn how the topic of the course is interconnected to various other fields in mathematics, and in science, in general.