Egzaktna simulacija trajektorija difuzija

master's thesis

• Title (english): Exact simulation of diffusion trajectories
• Author: Marin Knežević
• Mentor: Vanja Wagner (mentor)
• Committee member: Vanja Wagner (predsjednik povjerenstva)
• Committee member: Snježana Lubura Strunjak (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-03-03, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu proučavamo metode simulacija trajektorija difuzija određenih stohastičkim diferencijalnim jednadžbama s posebnim naglaskom na egzaktnu metodu koja se temelji na uzorkovanju s odbacivanjem. U prvom poglavlju dajemo pregled teorije slučajnih procesa i Itovog diferencijalnog računa potrebnog za sva daljnja razmatranja. U drugom poglavlju predstavljamo klasičnu metodu prihvaćanja i odbacivanja odnosno “rejection sampling” uvodeći pritom i alternativan pristup korišten u egzaktnoj metodi. Nadalje, u trećem poglavlju iz teoretske perspektive prezentiramo metode za simulaciju trajektorija difuzija, prvo Eulerovu metodu i zatim egzaktnu metodu. Također, dajemo ocjenu efikasnosti egzaktnog algoritma. Konačno, u četvrtom poglavlju obje metode implementiramo u matematičkom programskom okruženju R te vršimo usporedbe na konkretnoj stohastičkoj diferencijalnoj jednadžbi.

Abstract (english)

In this thesis, we study different methods of simulating diffusion trajectories determined by stochastic differential equations with special emphasis on the exact method based on rejection sampling. In the first chapter, we give an overview of the theory behind stochastic processes and Ito's differential calculus necessary for all further considerations. In the second chapter, we present the standard method of rejection sampling, while also introducing an alternative approach used in the exact method. Furthermore, in the third chapter, from a theoretical perspective, we present methods for simulating diffusion trajectories, first the Euler method and then the exact method. We also evaluate the efficiency of the exact algorithm. Finally, in the fourth chapter, we implement both methods using the mathematical programming environment R and perform comparisons given a fixed stochastic differential equation.

• Keywords: metode simulacija trajektorija difuzija; stohastičke diferencijale jednadžbe; uzorkovanje s odbacivanjem; Itov diferencijalni račun; matematičko programsko okruženje R
• Keywords (english): methods of simulating diffusion trajectories; stochastic differential equations; rejection sampling; Ito's differential calculus; mathematical programming environment R
• Language: croatian
• URN:NBN: urn:nbn:hr:217:856220
• Study programme: Title: Finance and Business 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: 2023-07-26 08:47:05