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master's thesis
Markovljevi lanci u neprekidnom vremenu
2020. urn:nbn:hr:126:920394
Popović, Ana
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Pure Mathematics and Mathematics Teaching
Probability, Mathematical Statistics and Data Science Research Group

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Popović, A. (2020). Markovljevi lanci u neprekidnom vremenu (Master's thesis). Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics. Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:920394

Popović, Ana. "Markovljevi lanci u neprekidnom vremenu." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:920394

Popović, Ana. "Markovljevi lanci u neprekidnom vremenu." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:920394

Popović, A. (2020). 'Markovljevi lanci u neprekidnom vremenu', Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 30 September 2024, https://urn.nsk.hr/urn:nbn:hr:126:920394

Popović A. Markovljevi lanci u neprekidnom vremenu [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2020 [cited 2024 September 30] Available at: https://urn.nsk.hr/urn:nbn:hr:126:920394

A. Popović, "Markovljevi lanci u neprekidnom vremenu", Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2020. Available at: https://urn.nsk.hr/urn:nbn:hr:126:920394

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Metadata
TitleMarkovljevi lanci u neprekidnom vremenu
Title (english)Continuous-Time Markov Chains
AuthorAna Popović
MentorNenad Šuvak (mentor)
Committee memberMirta Benšić (predsjednik povjerenstva)
Committee memberNenad Šuvak (član povjerenstva)
Committee memberDanijel Grahovac (član povjerenstva)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
(Chair of Pure Mathematics and Mathematics Teaching)
(Probability, Mathematical Statistics and Data Science Research Group)
Osijek
Defense date and country2020-06-09, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Probability Theory and Statistics
Abstract
U ovom radu proučavamo Markovljeve lance u neprekidnom vremenu. Kako bi shvatili Markovljeve lance u neprekidnom vremenu najprije smo se upoznali s definicijama slučajnog procesa u neprekidnom vremenu, vremena skokova, vremena čekanja, procesa skokova te naveli neka svojstva eksponencijalne distribucije. Zatim definicija Markovljevog lanca u neprekidnom vremenu \(\left ( X_{t}, t\geqslant 0\right)\) koja nam kaže da je to slučajan proces u neprekidnom vremenu s prebrojivim skupom stanja za koje vrijedi da buduće stanje ne ovisi o prošlom stanju nego samo o sadašnjem. Nadalje, konstruiramo Markovljev lanac u neprekidnom vremenu pomoću Markovljevog lanca u diskretnom vremenu \(\left ( \mathbb{Z}_{n}, n\in \mathbb{N}\right )\) gdje nakon konstrukcije zaključujemo da je \(\left ( \mathbb{Z}_{n}, n\in \mathbb{N}\right )\) proces skokova procesa \(\left ( X_{t}, t\geqslant 0\right )\). Pokazali smo da ukoliko je proces regularan tada je ujedno i Markovljev lanac u neprekidnom vremenu. Primjeri takvih lanaca kojima se bavimo u radu su Poissonov proces i proces radanja. Na kraju rada se bavimo svojstvima i strukturom Markovljevih lanaca u neprekidnom vremenu, objašnjavamo apsorpcijske vjerojatnosti, povratnost i prolaznost, stacionarne distribucije, granične distribucije te navodimo i dokazujemo Ergodski teorem.
Abstract (english)
In this thesis we study continuous-time Markov chains. To understand continuous-time Markov chains, firstly we need to define terms continuous-time random process, jump times, holding times, jump processes, and introduce some properties of exponential distribution Then, there is the definition of the continuous-time Markov chain \(\left ( X_{t}, t\geqslant 0\right)\) which describes them as a continuous-time random process with a finite state-space for which the future state does not depend on the past state but only on the present one. Furthermore, we construct a continuous-time Markov chain using the discrete-time Markov chain \(\left ( \mathbb{Z}_{n}, n\in \mathbb{N}\right )\) so that \(\left ( \mathbb{Z}_{n}, n\in \mathbb{N}\right )\) is jump process of \(\left ( X_{t}, t\geqslant 0\right)\) . It has been shown that if a process is regular, then it is a continuous-time Markov chain. Examples of such chains that we deal with in this thesis are the Poisson process and the birth process. At the end of the thesis, we deal with the properties and structure of continuous-time Markov chains, we explain absorption probabilities, recurrence and transience, invariant distributions and limiting distributions, and state and prove the Ergodic theorem.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:920394
Study programmeTitle: Mathematics; specializations in: Financial and Statistical Mathematics, Mathematics and Computer Science, Industrial and Applied Mathematics
Course: Financial and Statistical Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2020-06-18 09:17:11