A generalized correlated random walk approximation to fractional brownian motion

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Tezin Türü: Yüksek Lisans

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, İstatistik Bölümü, Türkiye

Tezin Onay Tarihi: 2018

Öğrenci: BUKET COŞKUN

Danışman: CEREN VARDAR ACAR

Özet:

The application of fractional Brownian Motion (fBm) has drawn a lot of attention in a large number of areas, ranging from mathematical finance to engineering. The feature of long range dependency limited due to the value of Hurst parameter H ∈ (1/2, 1) makes fBm the desired process for stochastic modelling. The simulation of fBm is also vital for the application in such fields. Hence, the development of an algorithm to simulate an fBm is required in both theoretical and practical aspects of fBm. In this study, we mainly propose a new fBm generation method by using the Hurst parameter and the correlation structure based on this parameter and suggest an algorithm to generate correlated random walk converging to fBm, with Hurst parameter, H∈ (1/2, 1). The increments of this random walk are simulated from Bernoulli distribution with proportion p, whose density is constructed using the link between correlation of multivariate Gaussian random variables and correlation of their dichotomized binary variables. We prove that the normalized sum of trajectories of this proposed random walk yields a Gaussian process whose scaling limit is the fBm.