Mathematics MS Thesis Defense by Abdullah Harun Karakuş

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KOÇ UNIVERSITY

GRADUATE SCHOOL OF SCIENCES & ENGINEERING

MATHEMATICS

MS THESIS DEFENSE BY ABDULLAH HARUN KARAKUŞ

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Title: Two Coalescing Brownian Flows: Correlated and Rank-Based

Speaker: Abdullah Harun Karakuş

Time: August 4, 2017, 11:00

Place: Room ENG 208

Koç University

Rumeli Feneri Yolu

Sariyer, Istanbul

Thesis Committee Members:

Assoc. Prof. Mine Çağlar (Advisor, Koc University)

Assoc. Prof. Atilla Yılmaz (Koc University)

Asst. Prof. Mehmet Öz (Özyeğin University)

Abstract:

We consider a stochastic differential equation on the real line which is driven by two correlated Brownian motions. We prove it has a unique flow solution. Then, we generalize this flow to a flow on the circle, which represents an oriented graph with two edges and two vertices. We prove that both flows are coalescing. Coalescence leads to the study of a correlated reflected Brownian motion on the quadrant. Moreover, we find the distribution of the hitting time to the origin of a reflected Brownian motion. This has implications for the effect of the correlation coefficient on the coalescence time of our flows. Then, we study a stochastic differential equation with rank-based coefficients on the plane. The flow solution of this SDE follows from the first SDE. We also study coalescence of this SDE and its kernel solutions.