Itov račun za jednostavne procese sa skokovima

master's thesis

• Title (english): It's account for simple processes with jumps
• Author: Dino Milat
• Mentor: Vanja Wagner (mentor)
• Committee member: Vanja Wagner (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Goranka Nogo (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu proučavamo Itov račun, proširujući teoriju s klasičnog integratora, Brownovog gibanja, na jednostavne procese sa skokovima. U uvodnim poglavljima bavimo se jednom od osnovnih klasa slučajnih procesa sa skokovima - Poissonovim procesom i složenim Poissonovim procesom, a uz to navodimo i dokazujemo neka od njihovih svojstava koja će nam biti korisna u daljnjim razmatranjima. Nakon toga uvodimo tipične pojmove teorije Itovog računa, ali obzirom na složeni Poissonov proces. Bavimo se Itovim integralom i njegovim svojstvima te Itovom formulom, a na kraju tog poglavlja dajemo kratak osvrt na procese beskonačne aktivnosti, koji predstavljaju generalizaciju dosad uvedenih procesa sa skokovima. U četvrtom poglavlju izdvajamo neke od stohastičkih diferencijalnih jednadžbi sa skokovima te njihova rješenja koja mogu poslužiti u modeliranju tržišnih povrata. Konačno, u zadnjem poglavlju bavimo se za procese Girsanovljevim teorem za procese sa skokovima, jednim od najvažnijih teorema stohastičkog računa, koji opisuje ponašanje slučajnih procesa prilikom zamjene mjere. Podsjećamo se tog teorema za Brownovo gibanje, a zatim ga navodimo za standardni i složeni Poissonov proces, uz neke korisne primjere vezane uz modeliranje diskontiranog procesa cijena u financijama.

Abstract (english)

In this thesis, we take a look at Ito’s calculus, expanding the theory from the classic integrator, the Brownian motion, to jump processes. The introductory chapters are dedicated to the fundamental class of stochastic processes with jumps - the Poisson process and the compound Poisson process. We present and prove some of their basic properties, which are useful in our further analysis. Furthermore, we introduce some basic concepts of Ito’s calculus, with respect to the compound Poisson process. Ito’s integral and its properties are examined, along with Ito’s formula and, in the end, we give a short overview of infinity activity processes. These processes are a generalization of the processes we have studied up to that point. In the fourth section, we take a look at stochastic differential equations with jumps and their solutions, which serve as a great tool to model market returns. We conclude the thesis by introducing the Girsanov theorem for jump processes, which is one of the most important results of stochastic calculus. It describes the behaviour of stochastic processes under changes of measure. We remind ourselves of the Girsanov theorem for Brownian motion, and then we state the theorem for standard and compound Poisson processes. Finally, we give some useful examples regarding the modeling of discount prices of financial instruments.

• Keywords: Brownovo gibanje; Poissonov proces; Itov integral; Girsanov teorem za procese sa skokovima
• Keywords (english): Brownian motion; Poisson process; Girsanov theorem for jump processes
• Language: croatian
• URN:NBN: urn:nbn:hr:217:731432
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access; Embargo expiration date: 2024-04-01
• Created on: 2024-02-02 12:10:03