Browsing Mathematical Sciences by Title
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Valjarević, Dragana (Kragujevac , 2013)[more][less]
Abstract: Jedan od važnih i osnovnih ciljeva nauke je da se među događajima i pojavama utvrde uzročno-posledične veze. Šta se podrazumeva pod pojmom uzročnosti i kako se ona može meriti bila je tema mnogih rasprava. U ekonomiji, Grendžerova uzročnost (C. nj. Granger, 1969) je veoma dobro poznat koncept i jedna od najprimenjivanijih metoda u istraživanjima. Definicija uzročnosti u smislu Grendžera zasniva se na ideji da sadašnjost ili budućnost ne mogu uzrokovati prošlost. Udrugim naučnim oblastima se, takođe, dugo raspravljalo o pojmu uzročnosti. Međutim, do bitnijeg napretka dolazi tek poslednjih dekada. Danas, pojam uzročnosti ima široku primenu u fizici, biološkim i sociološkim naukama, istoriji, medicini posebno u epidemiologiji, ekonomiji i dr. Predmetistraživanjaove doktorske disertacije je statistička teorija uzročnostii njena primena na slaba rešenja stohastičkih diferencijalnih jednačina i martingalnu reprezentaciju. Pokazuje se da je ovaj koncept ekvivalentan sa slabom jedinstvenošću slabih rešenja stohastičkih diferencijalnih jednačina i ekstremnim rešenjima martingalnog problema. Na ovaj koncept se mogu primeniti i vremena zaustavljanja, pa se u skladu s tim dokazuje i ekvivalencija sa ekstremnim rešenjem martingalnog problema za zaustavljene procese, kao i sa lokalno jedinstvenim slabim lokalnim rešenjima. Takođe, koncept uzročnosti se može primeniti i u Teoriji martingala. Naime, ovaj koncept se može dovesti u vezu sa očuvanjem svojstva martingalnosti, ortogonalnim martingalima, stabilnim potprostorima, kao i martingalnim reprezentacijama, koje imaju primenu, naročito u finansijskoj matematici. U glavi 1 dati su osnovni pojmovi iz teorije verovatnoća, definicija slučajnog procesa i pregled njihovih osnovnih osobina. Takođe, ovde su date definicije martingala i semimartingala kao i stohastička integracija u odnosu na semimartingale. U ovoj glavi su date i stohastičke diferencijalne jednačine sa semimartingalima kao i njihova stroga rešenja, dok će o slabim rešenjima više biti reči u glavi 3. Daje se i definicija uzročnosti, koja se zasniva na Grendžerovoj uzročnosti, koju uvodi Mikland (P. A. Mykland š32]), a kasnije njeno uopštenje daje lj. Petrović (š34, 36, 37, 38, 39, 40, 42]). Takođe, data definicija uzročnosti je sa fiksnog vremena proširena na vremena zaustavljanja. Na kraju su dati neki osnovni rezultati koji se odnose na navedene pojmove uzročnosti. U glavi 2 je prikazano kako se koncept statističke uzročnosti može primeniti u Teoriji martingala. Za svojstvo martingalnosti, koje je vezano za filtracije, dokazano je da je očuvanje tog svojstva, kada σ- algebra informacija raste, direktno vezano za koncept uzročnosti. Takođe, može se uspostaviti ekvivalencija između ortogonalnosti martingala i koncepta uzročnosti (64). Isto se može dokazati i za lokalne martingale i za zaustavljene lokalne martingale. Koncept uzročnosti se može povezati i sa stabilnim potprostorima koji sadrže zdesna neprekidne modifikacije martingala oblikaMt = P(A j Ft) (videti 44). U glavi 3 data su definisana slaba rešenja različitih tipova diferencijalnih jednačina koje su generisane semimartingalima. Naime, dokazano je da je koncept uzročnosti ekvivalentan sa slabom jedinstvenošću slabih rešenja (39). Takođe, na slaba rešenja su primenjena i vremena zaustavljanja, pa je proučavana i veza između lokalne jedinstvenosti slabih lokalnih rešenja i koncepta uzročnosti sa vremenima zaustavljanja. Drugačiji pristup rešavanju stohastičkih diferencijalnih jednačina, tj. martingalniproblemjetakođerazmatranidokazanajeekvivalencijaizmeđu ekstremnog rešenja martingalnog problema i uzročnosti (41). Uglavi 4 je razmatrana veza između koncepta uzročnosti i martingalne reprezentacije u odnosu na različite filtracije sa posebnim osvrtom na već poznate rezultate. Pokazano je da se dati rezultati mogu primeniti na slaba rešenja stohastičkih diferencijalnih jednačina generisanih semimartingalima, kao i na martingalni problem koji je pridružen datoj jednačini. Delovi disertacije 2.2, 2.3, 2.4, 3.1, 3.2, 3.4, 3.6 sadrže nove rezultate koji su objavljeni u radovima 39, 41, 44, 64. URI: http://hdl.handle.net/123456789/2788 Files in this item: 1
Dragana_Valjarevic_PhD.pdf ( 553.7Kb ) -
Jeremić, Katarina (Beograd , 2017)[more][less]
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Parezanović, Nedeljko (Belgrade)[more][less]
URI: http://hdl.handle.net/123456789/161 Files in this item: 1
phdNedeljkoSParezanovic.pdf ( 3.500Mb ) -
Stanojević, Jelena (Beograd , 2015)[more][less]
Abstract: The main goal of the thesis is the development of a new suggested trans- form con¯dence intervals for the ratio of the variances of the two samples. Since now, the methods based on the F statistic have been suggested in the literature. However, the defect of that intervals is the huge sensitivity in re- lation with assumption of parameters distribution. Suggested statistic could be modi¯ed. Edgeworth expansion of the t-statistic has found the place in the thesis and based on that intervals have been compared. Also, on the base of the simulation it was point out that Johnsons transformation give better result in the sense of probability covering in regard to F interval and interval based on Halls transformation. Moreover, the con¯dence intervals for the mean and variances for the one and two sample problems have been considered in the dissertation. Especially, the problem of the di®erence of the proportions for the two samples, with the numerical results and data from the insurance. In addition, the existing methods for the estimation of the extreme value index and the high quantiles have been reviewed. Particularly, the direct simulation estimation of the quantile and probability covering of its deviation from the rights value, for Pareto and Gamma distributions, and also for general Pareto distribution have been discussed. The results were obtained by large deviation theory and their generalization on the topological spaces is stated. In this research, beside the probability theory and elemen- tary principles of the classical analysis, methods of the statistical theory and statistical conclusions have been applied. URI: http://hdl.handle.net/123456789/4261 Files in this item: 1
phdJelenaStanojevic.pdf ( 1.063Mb ) -
Manojlović, Mladen (Beograd , 2013)[more][less]
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Popović, Aleksandra (Beograd , 2023)[more][less]
URI: http://hdl.handle.net/123456789/5670 Files in this item: 1
v1_masterAleksandraPopovic.pdf ( 2.125Mb ) -
Pilipović, Predrag (Beograd , 2021)[more][less]
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Vrećica, Ilija (Beograd , 2022)[more][less]
Abstract: First part of dissertation examines sumsets hA = {a1 + · · · + ah ∈ Z d : a1, . . . , ah ∈ A}, where A is a finite set in Z d . It is known that there exists a constant h0 ∈ N and a polynomial pA(X) such that pA(h) = |hA| for h ⩾ h0. However, little is known of polynomial pA and constant h0. Cone CA over the set A contains information about hA, for all h ∈ N. When A has d + 2 elements, polynomial pA and constant h0 can be explicitly described. When A has d + 3 elements, an upper bound is found for the number of elements of hA. Second part of dissertation examines Selmer groups of elliptic curves in the con gruent number family. A squarefree natural number is congruent if and only if there exists a right triangle with area n whose sides all have integer lengths. It is known that n is a congruent number if and only if elliptic curve En : y 2 = x 3 − n 2x has nonzero rank as an algebraic group. Selmer groups of isogenies on En are interesting, because their rank is not smaller than the rank of En, so when the Selmer groups have rank zero, then the elliptic curve En also has rank zero. Elements of these Selmer groups can be represented as partitions of a particular graph, from which one may find the distribution of ranks of Selmer groups. URI: http://hdl.handle.net/123456789/5533 Files in this item: 1
Teza_Ilija_Vrecica.pdf ( 1.817Mb ) -
Aranđelović, Ivan (Beograd , 1999)[more][less]
URI: http://hdl.handle.net/123456789/4137 Files in this item: 1
Stavovi_o_presecanju.PDF ( 2.592Mb ) -
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 ) -
Milošević, Petar (Beograd , 2015)[more][less]
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Konaković, Aleksandra (Beograd , 2014)[more][less]
URI: http://hdl.handle.net/123456789/3857 Files in this item: 1
MasterRadMinaAleksandraKonakovic.pdf ( 913.2Kb ) -
Jocković, Jelena (Beograd , 2012)[more][less]
Abstract: Statistical methodology for dealing with extremes depend on how extreme values are defined. One way to extract extremes from a given sample x1, x2, ..., xN is to consider maxima (minima). The other way is to consider values y1 = x1 − u, y2 = x2 − u, . . . , yn = xn − u, where y1, y2, . . . , yn are sample members above (below) a given predetermined threshold u. These two methods lead to two different approaches in extreme value theory. This doctoral dissertation has two main goals. One of them is to apply the techniques from extreme value framework to certain type of combinatorial problems. The other goal is to contribute to the field of statistical modeling of extremes. The dissertation consists of three chapters. In the first chapter, we introduce generalized extreme value distributions and generalized Pareto distributions (GPD). These two families play key roles in the two approaches to modeling extremes. We set out the theoretical background for both approaches. In the second chapter, we apply the extremal techniques to combinatorial waiting time problems. Precisely, we consider Coupon collector’s problem, defined as follows: elements are sampled with replacement from the set Nn = {1, 2, . . . , n} under assumption that each element has probability 1/n of being drawn. The subject of interest is the waiting time Mn until all elements of Nn or some other pattern are sampled. We focus our attention to the following two cases: 1. Mn is the waiting time until all elements of Nn are sampled at least r times, where r is a positive integer; 2. Mn is the waiting time until all pairs of elements jj, j ∈ Nn are sampled. We present new results related to the asymptotic behavior of the waiting time Mn, if it is known that a large number of trials was performed and the experiment is not over. For both cases, we determine the limiting distribution of exceedances of Mn over high thresholds, and answer some related questions: how to choose a suitable high threshold (depending on n) in order to obtain a limiting distribution; under what conditions the limit does not depend on the threshold; are the generalized Pareto distributions the only possible limits. We also estimate the speed of convergence in both cases. The third chapter of the dissertation is devoted to estimation of parameters and quantiles of the generalized Pareto distributions. We restrict the attention to the two-parameter version of GPD, defined as: Wγ,σ(x) = 1 − e−x , x ≥ 0, γ = 0 1 − 1 + γ σx −1 , x ≥ 0, γ > 0 1 − 1 + γ σx −1 , x ∈ h 0,−σ γ i , γ < 0. Well known problem with this model is inconsistency with the sample data, which is that one or more sample observations exceed the estimated upper bound in case when γ < 0. We propose a new, general technique to overcome the inconsistency problem and improve performance of the existing GPD estimation methods. We apply the proposed technique to methodof- moments and method-of-probability-weighted-moments estimates, investigate its performance through computer simulation and provide some real data examples. Finally, we address the problem of estimating high GPD quantiles. We evaluate the robustness of some estimation methods through simulation study and present a case study from finance (value-at-risk estimation), with special emphasis to certain difficulties related to this field of application. URI: http://hdl.handle.net/123456789/4271 Files in this item: 1
phdJockovic_Jelena.pdf ( 1.687Mb ) -
Jocković, M. Jelena (Belgrade , 2012)[more][less]
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Petruševski, Ljiljana (Belgrade , 1986)[more][less]
URI: http://hdl.handle.net/123456789/51 Files in this item: 1
phdLjiljanaPetrusevski.pdf ( 1.651Mb ) -
Kurepa, Đuro (Zagreb , 1960)[more][less]
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Kulezić, Dragan (Beograd , 1996)[more][less]
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Todorović, Nataša (Beograd , 2012)[more][less]
Abstract: Nehoroševljeva teorema (1977) igra veoma značajnu ulogu u razumevanju prirode hamiltonijanskih sistema. Pored činjenice da daje prikaz celokupne dinamike faznog prostora, ova teorema obezbeđuje i stabilnost kvazi-integrabilnih sistema u smislu da se vremena stabilnosti dejstava eksponencijalno produžavaju smanjenjem parametra poremećaja. Jedan od uslova teoreme je da integrabilna aproksimacija hamiltonijanske funkcije zadovoljava takozvani uslov strmosti. Pretpostavka izneta u dokazu teoreme je da pored parametra poremećaja, i strmost utiče na stabilizaciju sistema. Cilj doktorske teze bio je da se ova pretpostavka proveri numeričkim putem, kao i da se ilustruje uticaj strmosti na dinamiku sistema. Korišćen je model četvorodimenzione kvazi-integrabilne strme simplektičke mape, na kojoj se intenzitet strmosti lako podešava pomoću odgovarajućeg parametra. Oslanjajući se na tzv. Brzi indikator Ljapunova (Fast Lyapunov Indicator- FLI), numeričku metodu za detekciju haosa, prikazana je rezonantna struktura modela i merena je difuzija Arnoldovog tipa na odabranoj rezonanci. U prvom delu eksperimenta su za fiksirane vrednosti parametra strmosti merene promene difuzije u odnosu na smanjenje poremećaja, i potvrđeno je da se vrednost eksponenata fitovane funkcije povećava (sistem se stabilizuje) što je sistem strmiji. Otkriveno je i da eksponencijalna funkcija u strmoj nekonveksnoj oblasti u izvesnom smislu osciluje, što je rezultat koji još uvek nema svoju teorijsku interpretaciju. Takođe, primećeno je da se sa povećanjem strmosti, hiperboličke tačke zajedno sa svojom okolinom izmeštaju sa rezonantne krive. I konačno, u eksperimentu u kojem je meren direktan uticaj strmosti na brzinu difuzije, potvrđena je pretpostavka iz dokaza Nehoroševljeve teoreme, o postojanju kritične vrednosti parametra strmosti. Za vrednosti koje su manje od kritične, strmost nema uticaj na brzinu difuzije, dok je za vrednosti koje su veće od kritične, ovaj uticaj eksponencijalan. Štaviše, vrednost eksponenta ove funkcije u izvesnom smislu ima opšti karakter to jest ne zavisi od ostalih parametara sistema. Detaljnije analitičko objašnjenje ovog rezulata, svakako je jedan od budućih zadataka u interpretaciji Nehoroševljeve teoreme. URI: http://hdl.handle.net/123456789/3569 Files in this item: 1
Ntodorovic_Doktorat_1R.pdf ( 2.955Mb ) -
Opačić, Jasna (Beograd , 2012)[more][less]
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Srećković, Dijana (Beograd , 2012)[more][less]