Uvod u stohastičke parcijalne diferencijalne jednadžbe

master's thesis

• Title (english): Introduction to stochastic partial differential equations
• Author: Mate Poljak
• Mentor: Marko Erceg (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Marko Horvat (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-12-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu upoznali smo se s pojmom stohastičkih diferencijalnih jednadžbi i njenim primjenama. Podsjetili smo se gradiva teorije vjerojatnosti potrebnog za razumijevanje ovog rada. Upoznali smo se s Brownovim gibanjem, stohastičkim procesom čija je ”derivacija“ upravo ”bijeli šum “ podataka koji se često pojavljuje prilikom modeliranja parcijalnim diferencijalnim jednadžbama nekog problema iz stvarnoga svijeta. Kao osnovne pojmove za ovu tematiku uveli su se Itov integral i Itova formula. Opisano je kako izgledaju stohastičke diferencijalne jednadžbe, te je stavljen naglasak na egzistenciji i jedinstvenosti njihovog rješenja, te dokaz navedene tvrdnje. Na samome kraju rada obrađeno je nekoliko primjena teorije gdje se prikazao jedan dio upotrebe stohastičkih diferencijalnih jednadžbi u rješavanju problema koji se pojavljuju u praksi.

Abstract (english)

In this paper, we were introduced to the term stochastic differential equation and its applications. We have recalled the foundations of probability theory necessary for understanding this thesis. We were introduced to Brownian motion, a stochastic process whose ”derivative” is the ”white noise” of data that often appears in modeling real-world problems with partial differential equations. The Ito integral and the Ito formula were introduced as basic concepts for this topic. It is described what stochastic differential equations look like and emphasis is placed on the existence and uniqueness of their solution, as well as the proof of the stated statement. At the very end of the paper, several applications of the theory were discussed, where one part of the use of stochastic differential equations in solving problems arising from the real world was presented.

• Keywords: teorija vjerojatnosti; Brownovo gibanje; ”bijeli šum “ podataka; Itov integral; Itova formula
• Keywords (english): probability theory; Brownian motion; "white noise" of data; Ito integral; Ito formula
• Language: croatian
• URN:NBN: urn:nbn:hr:217:174396
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-12-19 11:51:01