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研究生： 吳恕銘
Shu-Ming Wu
論文名稱：
Studies on Fractional Brownian Motion: Its Representations and Properties
指導教授： 趙一峰
I-F. Chao
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 97
語文別： 中文
論文頁數： 51
中文關鍵詞： 分數布朗運動 、分數微積分
外文關鍵詞： fractional calculus, fBm
相關次數： 點閱：12  下載：0 
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    • 分數布朗運動的表現是以分數微積分作基礎。在本篇文章裡我們
    會先討論到分數微積分的定義以及性質，包括存在性、函數的微分與
    積分的不同表示法、互為反算子、以及半群性質。再來，利用維納積
    分定義分數布朗運動，並且利用分數微積分展示出分數布朗運動的核
    的不同的表示法。文章的末了我們也會討論一些有關分數布朗運動的
    一些性質。

    In this thesis, we discuss the representations of fractional Brownian
    motion (fBm) and properties of fBm. We name it “fractional” because the
    kernels can be represented by fractional calculus. My thesis has three
    contributions. First, we show some properties about fractional calculus,
    including existence, various expressions of derivative and integral of
    function, inverse operators, and semigroup property. Second, we propose
    to use fractional calculus to indicate fBm and different representations of
    kernels. Finally, we show some properties of fBm in the end of this thesis.

    中文摘要------------------------------------------------------------------------------------------------- i 英文摘要------------------------------------------------------------------------------------------------ ii 致謝------------------------------------------------------------------------------------------------------ iii 目錄------------------------------------------------------------------------------------------------------ iv 圖目錄---------------------------------------------------------------------------------------------------- v 第零章 介紹-------------------------------------------------------------------------------------------1 第一章 分數微積分---------------------------------------------------------------------------------2 一、 介紹---------------------------------------------------------------------------------------2 二、 基本符號定義及一些基本定理 -------------------------------------------------2 三、 有限區間內的分數積分及分數微分--------------------------------------------4 四、 實數線上的分數積分及分數微分----------------------------------------------10 五、 分數微積分的幾何表現-----------------------------------------------------------17 第二章 維納積分以及碎形布朗運動介紹--------------------------------------------------20 一、 維納積分-------------------------------------------------------------------------------20 二、 實數線上的碎形布朗運動--------------------------------------------------------24 三、 有限時間區間上的碎形布朗運動----------------------------------------------28 四、 結論-------------------------------------------------------------------------------------31 第三章 碎形布朗運動的一些特性------------------------------------------------------------33 一、 時間域上不同H的轉換------------------------------------------------------------33 二、 長範圍相依性------------------------------------------------------------------------34 三、 Holder 連續性------------------------------------------------------------------------35 四、 半鞅過程-------------------------------------------------------------------------------36 參考文獻-------------------------------------------------------------------------38 附錄 一些特殊函數介紹---------------------------------------------40 圖 目 錄 圖一、分數積分0.75 0 (I f )(10) + ，f(t) = t + 0.5sin(t)的圖形--------------------------------------18

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