> 9 dbjbj"Glzvvvvz"^[[[$ [[[[[)~)~)~)~[])~[)~)~!!tzvCw!!tʄ0!Ĉ}Ĉ!)~thIndividuln studijn pln studento doktorskho studia na IES
(VyplHte elektronicky, podepsan formulY odevzdejte na sekretarit IES,
soubor ulo~te na Vaai webovou strnku)Kontaktn dajeJmno a pYjmen Pavel Dole~el`koln rok nstupu studia2007/2008Forma studiaPrezen
n`kolitelProf. RNDr. Ing. Frantiaek Turnovec, CSc.Tma doktorsk prceThree Essays on Operations Research in Economic TheoryPYedpokldan harmonogram zkouaek (kd/nzev/semestr)2007/2008
ZS: ELBF HYPERLINK "http://ies.fsv.cuni.cz/index.php?module=sylab&action=sylab&id_sylab=133&lng=cs_CZ" Economics and Law in Banking and Finance
ETPMEconomic Theory of Political Markets
LS: ELBF HYPERLINK "http://ies.fsv.cuni.cz/index.php?module=sylab&action=sylab&id_sylab=133&lng=cs_CZ" Economics and Law in Banking and Finance
ETPMEconomic Theory of Political Markets
2008/2009
ZS: ETPMEconomic Theory of Political Markets
LS: ETPMEconomic Theory of Political Markets
2009/2010
ZS: ETPMEconomic Theory of Political Markets
LS: ETPMEconomic Theory of Political Markets
Sttn doktorsk zkouka
2010/2011
ZS: ETPMEconomic Theory of Political Markets
Mal obhajoba
LS: ETPMEconomic Theory of Political Markets
Velk obhajoba
PYedpokldan harmonogram vuky (kd/nzev/semestr)2007/2008
ZS: Macroeconomics I
Microeconomics II
LS: Macroeconomics II
Microeconomics I
2008/2009
ZS: Microeconomics A I
Probability and Statistics I
LS: Probability and Statistics II
Microeconomics I
2009/2010
ZS: Microeconomics II
LS: Microeconomics I
Prce na disertaciSynopse (1 2 strany)The Ph.D. thesis will consist of three essays dealing with the multiindex transportation problem and its applications. The main topic will be the application of the general (continuous) multiindex MongeKantorovi
transportation problem (MKTP) in the area of operations research, theory of choice and theory of risk management.
The first essay Operations Research in Risk Management will focus on the theory of risk management, mainly on the statistical models predicting the probability of default, exposure at default and the loss given default, as they are defined in the Bassel II Accord. This application can be widely generalized in the area of statistical modelling, especially in general regression, validation techniques and backtesting the developed models.
The second essay Operations research in the theory of firm will consider the application of (MKTP) in operations research, more specifically in the field of stochastic schedulling and planing. The stochastic schedulling can be applied in project management, queuing theory, planning and production optimization. This essay will be mainly focused on the discrete version of (MKTP).
In the third essay Operations Research in the Choice Theory and Game Theory I will study the application of (MKTP) in the mathematical economics, utility theory and choice theory, starting with what is known, including the Richters nontransitive theory of choice and rational choice theory. I would like to show and investigate the connections of multiindex transportation probelm to the game theory. The main focus will be given to the closed preorders and continuous utility functions and special forms of choice functions.
The first essay will be based on the following concept: Suppose we have a set of random samples of the same length (so each sample is given by one probability distribution and the random variables within the sample are independent to each other). The ith variable of all the samples stands for the ith observation. Suppose, we have some output data vector of the same length as the random samples, the ith output is assigned to the ith observation. We estimate the output data using the data we have, using some kind of regression, either linear, polynomial, exponential, logit, probit, nonlinear, generalized, regression and classification trees, stochastic processes prediction and many other techniques. The question that arises as first in the area of fitting the data by a statistical model, is the question of how well it fits the data. The second arising question is, what is the models predictive power (how well it fits the newcomers and so, how it is robust). There are several methods of validating the predictive power of statistical models, such is the coefficient of determination (which is very badmainly due to its low robustness and dependancy on the number of predictors), adjusted coefficient of determination (it overcomes the problem of the determination coefficient being dependant on the number of predictors, but it is still very bad), the power curves, ROC (Operating Characteristics Curve) and CIER (Condonal Information Entropy Ratio). However, in the credit risk modeling, it can be different at the end (which is often considered to be the net profit), whether we overestimate the predicted value of the output variable, or if we underestimate it and even in what level of the output we make the error and how big it is. On the other hand, we might face another problem of having niether discrete, nor continuous probability distributions, because we must somehow solve the probelm of having many probabilites of default or loss given defaults observations equal to zero and in the case of LGD also equal to one. As is known, the proabilty that some value of absolutely continuous random variable will be reached more than once is zero whenever the support is uncountably infinite. There is also a problem of handling the estimated values as binary or categorical (banks usually use loss classes to describe the level of loss the bank should assume it will have to face with some probability). I would like to suggest another way of validating and backtesting the model using the (MKTP) which will be based on the idea, that at first we estimate (or set) the cost of underestimating or overestimating the output variable within the concept of net profit (I would like to study the situation, where the cost function which forms the (MKTP) is defined using some of the data, which we use as predictors) and then we solve the (MKTP) for the estimated cost function with given marginal distributions of the output variable and its prediction. Once we have this done, we find the absolute value of the differences between the solution of the (MKTP) and the joint probability distribution of the estimated and observed output variables. This measure can be a good way how to backtest a model with respect to the cost function. In this essay I will not study only this measure in practice and apply some sampling statistical techniques (bootstrap) to make an estimation of its distribution, but I will also suggest how to use this to enter the estimation of the output variable itself and study the effectiveness of such a suggestion and conditions for being better than the traditional way of backtesting. In this essay I would like to mention very interesting theory of Vapniks which is a rigorous way of eliminating much of what is in statistics considered normative decission. I would like to incorporate the results of this theory to be applied together with the (MKTP), so I at first choose the model using Vapniks theory for a few given families of functions and then apply all the suggested preditive power measurements and the (MKTP) for backtesting and restructuring the results of the prediction to better fit the (MKTP) resultat first randomly, then in some conducted way. I would like to study extensively the relation between the family of functions chosen and the difference
between the optimal (MKTP) given joint probability distribution and the observed distribution and so the ability to restructure the estimated values before we know the real observations.
The second essay will be focused on a different field of (MKTP) application, which is the stochastic scheduling and planning, within the thory of firm. In this essay I will study the interrelations between multiindex transportation problem and the theory of matroids and apply some results of multiindex assignment problems arising in the theory of firm.
In the third essay I will investigate the (MKTP) in the choice theory, matematical economics and game theory. In this essay I will study the closed preorders and continuous utility functions used in microeconomic theory, which eenables microeconomists to use constrained optimization techniques such as KuhnTucker conditions. I will investigate the choices and decisions made in various economic situations and economic agents behavior from the theoretical point of view. I will describe briefly the theory of rational behavior and then study some generalistions of this theory and its consequences. I will relax the transitivity assumption as is described in Richters work. Because the decission making is closely studied in game theory, I will investigate also the game theory and its concepts, such as the Nash equilibria, Pareto optimality and the stochastic games and their connection to (MKTP). It can for example solve the problem of creating the payoff function in certain games so that the game has certain properties. When we have a situation of multiple players with the same set of actions, facing the same payoff function and the game is not sequential and each player know what is the total payoff over all possible actions, but dont know what state will be exactly reached due to other players choices, we can a priori choose such payoff function, so that the expected payoff of all players is maximal possible. There are many other applications of the (MKTP) in the field of game theory and I will investigate some of them and propose some further extensions. I will apply the conclusions in the theory of political science.
In all three essays I will use theoretial concepts from different parts of mathematical theories. Due to the complexity of the (MKTP) I have to put in at least the basics of the measure theory to define the (MKTP) and describe the duality theory and explicit solutions, I also need to include basics of the functional, stochastic (to introduce quadratic variation and Brownian motion and its applications in financial modelling) and real calculus, simple algebra (at least to handle matrices and describe some interesting concepts, such as matroids and their connection to (MKTP)), theory of neural networks to be able to build some of the algorithms, theory of genetic and evolutionary algorithms, theory of probability and stochastic processes (which will be the most important one), statistics, Vapniks theory of statistical learning which can be placed somewhere between the classical statistics and neural networks theory, theory of deterministic and stochastic schedulling, combinatorial optimization, linear and nonlinear programming and optimization method (SIMPLEX algorithm at least) and the wide range of theory of network flows and connected graph theory. I will never use more theory than I need and that will be indispensable to understand what is going on in the different applications. Therefore I omit many definitions and theorems, that I will use, but I always try to put reference of where these results can be found.
Within the work I would like to at least in one of the essays to create some algorithms, that solves the real problems and implment it in VBA (Visual Basics for Applications) and then compare them from the point of view of the time effectivity and solution effectivity.
The conclusions of the three essays should contain theoretical extensions of the theory, new applications and algorithms to solve the problems more effectively and/or faster and suggestions for modelling and backtesting the scoring models in risk management, suggestions for creating systems within the framework of game theory to attain the most effective payoffs and avoid adverse behavior of individuals and finaly suggestions for the firms to better conduct the production and schedulling problems.Zkladn literaturaS. T. Rachev and L. Ruschendorf. HYPERLINK "http://www.amazon.com/MassTransportationProblemsProbabilityApplications/dp/0387983503/ref=sr_1_6/10235065836144921?ie=UTF8&s=books&qid=1189524578&sr=86" Mass Transportation Problems: Volume I: Theory (Probability and its Applications). Springer; 1 edition, 1998.
S. T. Rachev and L. Ruschendorf. HYPERLINK "http://www.amazon.com/MassTransportationProblemsProbabilityApplications/dp/0387983503/ref=sr_1_6/10235065836144921?ie=UTF8&s=books&qid=1189524578&sr=86" Mass Transportation Problems: Volume II: Applications (Probability and its Applications). Springer; 1 edition, 1998.
E. Aarts and J. Lenstra. Local Search in Combinatorial Optimization.
Princeton University Press, 2003.
T. S. Arthanari and Y. Dodge. Mathematical Programming in Statistics.
Willey, 1981.
F. Bock. An algorithm for solving travelingsalesman and related network
optimization problems. Operations Research. Society of America,
October 1958. The fourteen National Meeting.
B. Korda. U
ebnice linern1ho programovn. SNTL, 1962.
HYPERLINK "http://www.amazon.com/exec/obidos/searchhandleurl/10235065836144921?%5Fencoding=UTF8&searchtype=ss&index=books&fieldauthor=Cedric%20Villani" C. Villani. Topics in Optimal Transportation (Graduate Studies in Mathematics, Vol. 58). American Mathematical Society, 2003.
R. L. Rardin. Optimization in Operations Research. Prentice Hall; 1st edition, 1997.
The list will be updated.Harmonogram prac 2008/2009
ZS: Working on disertation thesis, finishing the first part of the first essay named Operations Research in Risk Management, which will deal with the application and theory of the general transportation problem in credit risk modelling, validating and becktesting the models.
LS: Working on the disertation thesis, finishing the second part of the first essay Operations Research in the Risk Management, which will deal with the application of the general transportation problem in the validation of statistical scoring models. Working on the GAUK 66409 grant Ekonomick teorie voleb a nov volebn historie R recieved for 20092010 and on the GAR 402/09/1066 grant Political economy of voting: Rational voter theory and models of strategic voting and manipulation . Finishing the paper dealing with the efficiency of voting systems intended for submittion in AUCO.
2009/2010
ZS: Working on the disertation thesis, finishing the first part of the second essay named Operations Research in the Theory of Firm dealing with the theory of stochastic schedulling. Finishing the work on GAUK 66409 grant for the year 2009 and on the GAR 402/09/1066 grant with outputs prepared for publication.
LS: Working on the disertation thesis, finishing the second part of the second essay named Operations Research in the Theory of Firm dealing with the applications of MongeKantorovi
problem in the stochastic schedulling and its applications. Working on GAUK 66409 grant conclusions and preparing the final version of grant paper. Preparing for the state doctoral exams.
2010/2011
ZS: Working on disertation thesis, finishing the first part of the third essay named Operation Research in the Choice Theory and Game Theory delaing with the applications of MongeKantorovi
Problem in the theory of choice and its applications. Finishing the GAUK 66409 grant. Preparing for the small defense of disertation thesis.
LS: Working on disertation thesis, finishing the second part of the third essay named Operation Research in the Choice Theory and Game Theory delaing with the applications of MongeKantorovi
Problem in the game theory. Preparing for the final defense of the disertation thesis.
PYedpokldan publikace vsledko2008/2009
ZS: Sending a contribution to AUCO with a review of recently published book Voting, Power and Freedom .
LS: Sending a contribution to AUCO with assumed title On the Efficiency of Voting in the IC, IAC and GIAC models .
2009/2010
ZS: Sending a contribution to IES WP with assumed title MongeKantorovi
Problem in the Theory of FirmApplications .
Sending a contribution to AUCO with assumed title Gerrymandering in the Czech Parliamentary Elections .
Sending a contribution to AUCO with assumed title New voting history of Czech Republic .
LS: Sending a contribution to EconLit Journal with assumed title MongeKantorovi
Problem in the Theory of FirmApplications .
Sending a contribution to AUCO with assumed title Rational, Semirational and Irrational Voter in the Czech Elections .
2010/2011
ZS: Sending a contribution to EconLit Journal with assumed title MongeKantorovi
Problem in the Choice TheoryApplications .
LS: Sending a contribution to IES WP with assumed title MongeKantorovi
Problem in Game TheoryApplications .
Sending a contribution to EconLit Journal with assumed title MongeKantorovi
Problem in Game TheoryApplications .
Sending a contribution to AUCO with assumed title Geometry of Voting .
Konkretizace studijnho plnu pro 1. rok studia
Vuka
ast na doktorskch seminYch
Za
tek prce na disertaci
PYedpokldan zkouaky
Dala aktivityStarting the work on disertation thesis
Grant request: Grant request to the Grant Agency of UK with the topic MultiIndex MongeKantorovi
Transportation Problem in Credit Risk Modelling .
Publications:
ZS: Sending a contribution to IES WP with assumed title MongeKantorovi
Problem in Credit Risk Management .
LS: Sending a contribution to EconLit Journal withassumed title MongeKantorovi
Problem in Credit Risk Management .
LS: Sending a contribution to IES WP with assumed title Application of MongeKantorovi
Problem in the Choice Theory .
Participation on doctoral seminars: ELBF (MongeKantorovi
Transportation Problem in Credit Risk Management), ETPM (Application of MongeKantorovi
Transportation Problem to Choice and Game Theory)
Teaching:
ZS: Microeconomics II, Macroeconomics I
LS: Microeconomics I, Macroeconomics II
Other activities:
ZS, LS: Attending at least 50% of the doctoral students defenses.
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