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master's thesis
Model difuzije s primjerima
Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2019. urn:nbn:hr:126:695053
Ivanković, Marija Magdalena
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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Ivanković, M. M. (2019). Model difuzije s primjerima (Master's thesis). Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics. Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:695053

Ivanković, Marija Magdalena. "Model difuzije s primjerima." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2019. https://urn.nsk.hr/urn:nbn:hr:126:695053

Ivanković, Marija Magdalena. "Model difuzije s primjerima." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2019. https://urn.nsk.hr/urn:nbn:hr:126:695053

Ivanković, M. M. (2019). 'Model difuzije s primjerima', Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 11 April 2025, https://urn.nsk.hr/urn:nbn:hr:126:695053

Ivanković MM. Model difuzije s primjerima [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2019 [cited 2025 April 11] Available at: https://urn.nsk.hr/urn:nbn:hr:126:695053

M. M. Ivanković, "Model difuzije s primjerima", Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2019. Available at: https://urn.nsk.hr/urn:nbn:hr:126:695053

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Metadata
TitleModel difuzije s primjerima
Title (english)Diffusion Model with Examples
AuthorMarija Magdalena Ivanković
MentorMirta Benšić (mentor)
Committee memberDragana Jankov Maširević (predsjednik povjerenstva)
Committee memberMirta Benšić (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 country2019-12-20, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Other Mathematical Disciplines
Abstract
U ovom diplomskom radu upoznali smo se s procesom difuzija. U prvom poglavlju smo definirali stohastičke diferencijalne jednadžbe. Krenuli smo od Riemannovog integrala, definirali Riemann-Stieltjesov integral nakon čega smo došli do Itôvog integrala, Itôve formule i važnijih svojstava za razumijevanje koncepta stohastičkih diferencijalnih jednadžbi. U drugom poglavlju, koji je i glavni dio rada detaljnije smo razradili difuzije. Definirali smo difuzije preko stohastičkih diferencijalnih... More jednadžbi te preko Markovljevih procesa. Opisali smo infinitezimalne parametre, definirali vrijeme prvog zaustavljanja i difuzije sa zaustavljanjem koje smo objasnili pomoću primjera. Zatim smo naveli uvjete pomoću kojih se može provjeriti je li stohastički proces difuzija. Slijedi definiranje gustoće skaliranja i gustoće mjere brzine pomoću kojih smo definiral funkcije S<l, x] , M<l, x], Σ(l), N(l). Pomoću tih funkcija smo odredili vrste granica intervala na kojem je difuzija definirana. Granica može biti privlačeća, dostižna, nedostižna te ih po Felleru dijelimo na regularnu, izlaznu, ulaznu i prirodnu. Iduće smo definirali infinitezimalni generator, uz koji se vežu i Kolmogorovljeve jednadžbe. Također smo uočili i da je rješenje Kolmogorovljeve jednadžbe unazad prijelazna funkcija gustoće procesa, dok je rješenje vremenski nezavisne Kolmogorovljeve jednadžbe unaprijed funkcija gustoće stacionarne distribucije procesa. Na Ornstein-Uhlenbeckovom procesu smo pokazali kako izgledaju Kolmogorovljeve jednadžbe, uočili smo da je Vasičekov model specijalan slučaj Ornstein-Uhlenbeckova procesa koji se često koristi za modeliranje kamatnih stopa. Treće poglavlje temelji se na Pearsonovim difuzijama. Može se reći da Pearsonove difuzije možemo definirati na dva načina, pomoću Pearsonove diferencijalne jednadžbe do koje je Kolmogorovljev došao proučavanjem Kolmogorovljeve jednadžbe unaprijed te kao rješenje stohastičke diferencijalne jednadžbe određenog oblika. Na kraju smo naveli tri Pearsonove difuzije, njihove stohastičke diferencijalne jednadžbe, parametre, funkciju prijelazne gustoće te grafički prikazali simulacije trajektorija. Less
Abstract (english)
In this thesis, we introduced the diffusion process. In the first chapter, we defined stochastic differential equations. We started from the Riemann integral, defined the Riemann-Stieltjes integral, after which we came to the Itô integral, Itô formula and more important properties for understanding the concept of stochastic differential equations. In the second chapter, which is also the main part of the paper, we have elaborated diffusions. We defined diffusions via stochastic... More differential equations and Markov processes. We have described the infinitesimal parameters, defined the hitting time and diffusion with killing, which we explained with the help of example. We have then outlined the conditions by which the stochastic process can be verified as diffusion. The following is to define the scale function and the speed density by which we have defined the functions S<l, x], M<l, x], Σ(l), N(l). Using these functions, we have determined the types of boundaries. The boundary can be attracting, attainable, unattainable and, according to Feller, we divide them into regular, exit, entrance and natural. Next, we have defined the infinitesimal generator, which is linked to the Kolmogorov equations. We have also noticed that the solution of the Kolmogorov backward equation is a transition density, while the solution of the time-independent Kolmogorov forward equation is a stationary distribution of the corresponding diffusion. In the Ornstein-Uhlenbeck process, we have showed what the Kolmogorov equations look like, we have noticed that the Vasicek model is a special case of the Ornstein-Uhlenbeck process, which is often used for interest rate modeling. The third chapter is based on Pearson diffusions. It can be said that we can define Pearson diffusions in two ways, using the Pearson differential equation that Kolmogorov obtained by studying the Kolmogorov forward equation and as a solution of a stochastic differential equation of a certain form. Finally, we have listed the three Pearson diffusions, their stochastic differential equations, the parameters, the transition density and graphically depicted the simulations of the sample paths. Less
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:695053
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
RepositoryRepository of School of Applied Mathematics and Informatics
Created on2020-01-10 08:34:35
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