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研究生： 盧亞鴻
Lu, Ya-Hung
論文名稱： 二維度幾何布朗運動轉折點之估計
Estimation of a Bivariate Geometric Brownian Motion with Change-points
指導教授： 任眉眉
Zen, Mei-Mei
學位類別： 碩士
Master
系所名稱： 管理學院 - 統計學系
Department of Statistics
論文出版年： 2010
畢業學年度： 98
語文別： 英文
論文頁數： 39
中文關鍵詞： 二維度幾何布朗運動 、轉折點 、黃金價格 、石油價格
外文關鍵詞： Bivariate geometric Brownian motion, Change-point, Gold prices, Crude oil prices
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    • 探討模型中轉折點的工作被應用在許多領域，本文主要針對二維度幾何布朗運動模型探討轉折點及相關參數之估計問題。二維度幾何布朗運動廣泛被應用在經濟價格資料。我們考量在此模型之下，利用最大概似進行轉折點之估計以及提出一個檢定流程來尋找轉折點發生後改變的參數估計。此外，我們也考量轉折點個數未知的情況。文中針對金價以及原油（布蘭特原油）價格資料進行實證分析，探進而研究本估計法之表現。本轉折點及相關參數之估計問題可推廣到任意d-維度幾何布朗運動模型上。

    Bivariate geometric Brownian motion is widely used in modeling bavariate correlated economical data. Based on this model, the task of detecting change-points is considered. A likelihood-based approach for estimating the positions of change-points and a testing procedure for testing which parameter in the process is changed after the estimated change-point are proposed. In addition, both known and unknown the number of change-points are considered. Finally, gold and crede-oil (Brent crude) data are analyzed to illustrate the proposed method. Numerical results show that the proposed procedure is
    useful.

    1. Introduction .........................................10 1.1 Motivation and Background ...........................10 1.2 Objective ...........................................10 1.3 Organization ........................................11 2. Literature Review ....................................12 2.1 Geometric Brownian Motions ..........................12 2.2 Estimation of a GBM with a Change-point .............13 2.3 Bivariate Geometric Brownian Motions ................14 2.4 Distribution of Return Rates with Change-points .....16 3. Methodology ..........................................17 3.1 Estimation Procedure ................................17 3.2 Determination of k ..................................18 3.3 Testing Procedure ...................................19 3.3.1 Equality of Covariances ...........................19 3.3.2 Equality of Means .................................20 4. Empirical Application ................................22 4.1 Data Description ....................................22 4.2 Data Analysis .......................................22 4.2.1 Estimation Result .................................26 4.2.2 Testing Procedure Result ..........................26 4.2.3 Model Diagonostic .................................32 4.2.4 Determination of k ................................32 4.3 Summary .............................................36 5. Concluding Remarks ...................................38

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    [2] Hsu, Y. J. (2001), “Change-Point Estimation for the Stock Market Prices”, Master Thesis, Institute of Statistics, NCKU.
    [3] Johnson, R. A. and Wichern, D. W. (2007), “Appied Multivariate Statistical Analysis.”, 6th Edition, Pearson.
    [4] Kendall, M. G. (1953), “ The analysis of economic time-series. Part 1. Prices. ,” Journal of the Royal Statistical Society, 96, 11-25.
    [5] Kutner, M. H., Nachtsheim, C. J., Neter, J. and Li, W. (2005), “ Applied Linear Statistical Models.”, 5th Edition, McGrawHill.
    [6] Ross, S. M. (2003), “ Introdution to Probability Models.”, 8th Edition, ACADEMIC PRESS.
    [7] Zhou, C. H. (1997), “ Default Correlation: An Analytical Result.”, Technical report, Federal Reserve Board.

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