Mathematics
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Todić, Bojana (Beograd , 2024)[more][less]
Abstract: This dissertation deals with the coupon collector problem, which in its simplest (classical) form can be formulated as follows: A collector wants to collect a set of n distinct coupons, by buying a single coupon each day. The random variable of interest is the waiting time until the collection is completed. The goal of the dissertation is to propose and analyze three new generalizations of the classical coupon collector problem obtained by introducing additional coupons with special purposes into the set of n standard coupons. The first two chapters are devoted to the results on the classical coupon collector problem and the known generalizations obtained by introducing additional coupons into the coupon set ([1], [2], [39], [54]). New results are presented in chapters 3,4, and 5. The third chapter of the dissertation is dedicated to the case where, in addition to the standard coupons, the coupon set consists of a null coupon (which can be drawn, but does not belong to any collection), and an additional universal coupon, that can replace any standard coupon. For the case of equal probabilities of standard coupons, the asymptotic behavior (as n → ∞) of the expectated value and variance of the waiting time for a fixed size subcollection of a collection of coupons is obtained when one or both probabilities of additional coupons are fixed, and the remaining coupons have equal small probabilities. These results, published in [27], generalize part of the results in [2]. The same problem is analyzed using a Markov chain approach, which led to the determination of the fundamental matrix and some related features of the collection process (probability that the coupon collection process ends in a certin way). These results are contained in the paper [24]. For the case of unequal probabilities of standard coupons, a class of bounds is derived for the first and second moments of the waiting time until the end of the experiment by using majorization techniques and refining the bounds proposed in [51]. The quality of the proposed bounds is tested in numerical experiments, and the specific bounds from the class with the most desirable properties are given. These results are published in [26]. The fourth chapter of the dissertation deals with the generalization in which the additional coupon (so called, penalty coupon) interferes with the collection of standard coupons in the sense that the collection process ends when the absolute difference between the number of collected standard coupons and the number of collected penalty coupons is equal to n. This generalization can be seen as a special case of the random walk with two absorbing barriers. The distribution and a simple upper bound on the first moment of the corresponding waiting time are determined by combinatorial considerations. The application of the Markov chain approach led to obtaining the fundamental matrix. These results are published in [53]. In the fifth chapter of the dissertation another additional coupon (so called reset coupon) is introduced, which acts as a reset button, in the sense that the set of coupons drawn up to time (day) t becomes empty if the reset coupon is drawn on day t+1. In the case of unequal probabilities of standard coupons, the distribution of the corresponding waiting time is obtained by combinatorial considerations. For the case of equal probabilities of obtaining standard coupons, For the case of equal probabilities, applying the first step analysis for the correspondingly constructed Markov chains led to the expressions for the expected waiting time and its simple form in terms of the beta function. These results are used for analysing the asymptotic behavior (when the size of the collection tends to infinity) of the expected waiting time, taking into account possible values of the probability of obtaining a reset coupon. These results are published in [25]. Setting the probabilities of the additional coupons to zero, all three generalizations of the coupon collector problem defined and analyzed in this dissertation as well as the obtained results reduce to the corresponding results for the classical coupon collector problem. URI: http://hdl.handle.net/123456789/5677 Files in this item: 1
Todic_Bojana_disertacija.pdf ( 1.004Mb ) 
Vicanović, Jelena (Beograd , 2024)[more][less]
Abstract: A convex continuoustime maximization problem is formulated and the nec essary optimality conditions in the infinitedimensional case are obtained. As a main tool for obtaining optimal conditions in this dissertation we use the new theorem of the alternative. Since there’s no a differentiability assumption, we perform a linearization of the problem using subdifferentials. It is proved that the multiplier with the objective function won’t be equal to zero. It was also shown that if the linear and nonlinear constraints are separated, with additional assumptions it can be guaranteed that the multiplier with nonlinear constraints will also be nonzero. In the following, an integral constraint is added to the original convex problem, so that a Lyapunovtype problem, i.e. an isoperimetric problem, is considered. Lin earization of the problem using subdifferentials proved to be a practical way to ignore the lack of differentiability, so the optimality conditions were derived in a similar way. It is shown that the obtained results will also be valid for the vector case of the isoperimetric problem. Additionally, the optimality conditions for the smooth problem were considered. On the minimization problem, it was shown that the necessary conditions of KarushKuhnTucker type will be valid with the additional regularity constraint condition. Also, any point that satisfies the mentioned optimality conditions will be a global minimum. URI: http://hdl.handle.net/123456789/5676 Files in this item: 1
J.Vicanovic_doktorska_disertacija.pdf ( 2.198Mb ) 
Babanić, Mirko (Beograd , 2022)[more][less]
Abstract: The preparatory part of the dissertation, which leads to the basic one, is based on returnvariance parameters that represent two key random variables of the model devised by Markowitz. The research used historical data that in themselves reflect all available information absorbed by the financial market, and therefore, we can consider them not only homogeneous but also absolute (for reasons of realization). Therefore, an analytical procedure of approximation by a sixthdegree polynomial was performed on such data, which represent combinations of values of average returns and variances of portfolio returns, thus establishing a relation that is explicitly expressed by an algebraic sixthdegree polynomial equation. After that, further analytical procedure determined the conditions for the existence of both the minimum and the tangent portfolio and redefined the terms: efficient portfolio set, preference toward risk, risk aversion, and indifference line. The central topic of the dissertation, the revision of Tobin 's separation theorem, is formulated and proved through three theorems, one basic and two auxiliary. URI: http://hdl.handle.net/123456789/5593 Files in this item: 1
Mirko Babanic  Disertacija.pdf ( 1.504Mb ) 
Mutavdžić, Nikola (Beograd , 2023)[more][less]
Abstract: In this PhD thesis we investigate bounds of the gradient of harmonic and harmonic quasiconformal mappings. We also discuss such bounds for functions that are harmonic with respect to the hyperbolic metric or certain other metrics. This research has been motivated by some recent results about Lipschitzcontinuity of quasiconformal mappings that satisfy the Laplace gradient inequality. More precisely, the mappings we consider are solutions of the Dirichlet problem for the Poisson equation and can be considered as a generalization of harmonic mappings. Besides the ball, we also work with general domains on which solutions of the Dirichlet problem are defined, as well as general codomains. Finally, we announce new results that have been formulated for regions of C1,αsmoothness, both as the domain and the codomain. Besides presenting the main results, we give an overview of general notions from differential geometry and recall some of the properties of hyperbolic metric in an ndimensional ball. We also state properties of harmonic and subharmonic functions with respect to the hyperbolic metric, which are analogous to some classical results from the theory if harmonic functions and Hardy’s theory. It turns out that the gradients of hyperbolic harmonic functions behave differently from those of euclidean harmonic functions. A similar conclusion is obtained for the family of Tαharmonic functions. Namely, unlike the space of harmonic functions, the solution of the Dirichlet problem in the space of Tαharmonic functions is shown to be Lipschitzcontinuous when so is the boundary function. In addition, we investigate Höldercontinuity of the solution of the Dirichlet problem for the Poisson equation in the euclidean and hyperbolic metric. We will present versions of the Schwarz lemma on the boundary for pluriharmonic mappings in Hilbert and Banach spaces. These results will follow from the version of the Schwarz lemma for harmonic mappings from the unit disc to the interval (1, 1) without the assumption that the point z = 0 maps to itself. Furthermore, we show a version of the boundary Schwarz lemma for harmonic mappings from a ball to a ball, not necessarily of the same dimension. The proof uses a version of the Schwarz lemma for multivariable functions, first considered by Burget. This result is obtained by integrating the Poisson kernel over socalled polar caps. The assumption that point z = 0 maps to itself is again not needed, thus yielding a generalization of a recent result by D. Kalaj. At the end of this section, it is demonstrated that the analogous result is false in the case of hyperbolic harmonic functions. In a certain sense, this means that the Hopf lemma is not valid for hyperbolic harmonic functions. Amongst various versions of the Schwarz lemma, we have been investigating bounds of the modulus for classes of holomorphic functions f on the unit disc whose index If fulfils certain geometric conditions. These classes are a generalization of the star and αstar functions, previously investigated by B. N. Örnek. Our method is based on using Jack’s lemma and can be applied in certain more general cases. As an illustration, we derive the sharp bounds for the modulus of a holomorphic function f with index If whose codomain is a vertical strip, as well as bounds for the modulus of the derivative of f at point z = 0. Moreover, we give a bound for the rate of growth of the modulus of holomorphic functions on disk U that map point z = 0 to itself and whose codomain is a vertical strip. URI: http://hdl.handle.net/123456789/5582 Files in this item: 1
Doktorska_Disertacija_Nikola_Mutavdzic.pdf ( 939.2Kb ) 
Merkle, Ana (Beograd , 2023)[more][less]
Abstract: Many new developments in the filed of probability and statistics focus on finding causal connections between observed processes. This leads to considering dependence relations and investigating how the past influence the present and the future. The well known concept of Granger (1969) causality is closely related to the idea of local dependence introduced by Schweder (1970). Granger studied time series, while Schweder considered Markov chains. The concept was later extended to more general stochastic processes by Mykland (1986). All this concepts incorporate the time into consideration dependence. The dissertation consist of four chapters. New results are presented in the fourth chap ter. The main aim of this doctoral dissertation is to determine di↵erent concepts of stochastic predictability using the well known tool of conditional independence. Follow Granger’s idea, relationships between family of sigma  algebras (filtrations) and between processes in continuous ti me were considered since continuous time models dependence represent the first step in various applications, such as in finance, econometric practice, neuroscience, epidemiology, climatology, demographic, etc. In this dissertation the concept of dependence between stochastic processes and filtration is introduced. This concept is named causal predictability since it is focused on prediction. Some major characteristics of the given concept are shown and connections with known concept of dependence are explained. Finally, the concept of causal predictability is applied to the processes of di↵usion type, more precisely, to the uniqueness of weak solutions of Ito stochastic di↵erential equations and stochastic di↵erential equations with driving semi martingales. Also, the representation theorem in terms of causal predictability is established and numerous examples of applications of the given concept are presented such as application in financial mathematics in the view of modeling default risk, in Bayesian statistics. The idea for the future might be to deal with the case of progressive stochastic predictability, i.e. the generalization of stochastic predictability from fixed time to stopping time. URI: http://hdl.handle.net/123456789/5572 Files in this item: 1
DOKTORAT_finalnaVerzija.pdf ( 1.785Mb )