Problem optimalnega ustavljanja Brownovega mostu : magistrsko delo

Avsenik, Ana Marija (author), Bernik, Janez (mentor)

V delu obravnavamo optimalen čas ustavljanja Brownovega mostu in funkcionala Brownovega mostu. Spomnimo se nekaterih osnovnih pojmov, kot so filtracije, martingali in Brownovo gibanje. S pomočjo slučajnih procesov poskušamo poiskati lastnosti Brownovega mostu ter si s tem pomagati pri izračunu optimalnega časa ustavljanja, pri čemer uporabimo tudi znanje reševanja diferencialnih enačb. Pogledamo si še optimalen čas ustavljanja lihe potence, preslikanega in integrala Brownovega mostu. Omenjene krivulje nato simuliramo v programu R.

• Language: Slovenian
• Keywords: Brownov most, Brownovo gibanje, Itova formula, martingal, lokalni martingal, navadna diferencialna enačba, optimalen čas ustavljanja
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-159095
• UDC: 519.2
• COBISS.SI-ID: 200147971
• Publication date in RUL: 29.06.2024

Secondary language

• Language: English
• Title: The problem of optimal stopping of a Brownian bridge

Abstract

This work studies optimal stopping time of a Brownian bridge and functional of a Brownian bridge. We recall some basic terms as filtrations, martingales and Brownian motion. We study their properties using stochastic processes. We solve optimal stopping problem, and several related problems, using knowledge of solving differential equations. We also study some related problems where the gain functions are certain functionals of a Brownian bridge. Odd powers, reflected and integral of a Brownian bridge is considered. For simulation we use program R.

• Keywords: Brownian bridge, Brownian motion, Ito's lemma, martingal, local martingal, ordinary differential equation, optimal stopping time