Studies on Fractional Brownian Motion: Its Representations and Properties

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姓名

吳恕銘(Shu-Ming Wu) 查詢紙本館藏

畢業系所

數學系

論文名稱

(Studies on Fractional Brownian Motion: Its Representations and Properties)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 分數布朗運動
★ 分數微積分

關鍵字(英)

★ fractional calculus
★ fBm

論文目次

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

參考文獻

[1] Larry C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York, 1985.
[2] Francesca Biagini, et al., Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London, 2008.
[3] John B. Conway, Functions of One Complex Variables I , Springer-Verlag, New York, 1978.
[4] Harry Hochstadt, The Function of Mathematical Physics, Pure and Applied Mathematics, Vol. 23, John Wiley & Sons, Canada, 1971.
[5] Ioannis Karatzas, Steven E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Springer-Verlag, New York, 1991.
[6] Hui-Hsiung Kuo, Introduction to Stochastic Integration, Springer, 2006.
[7] Tom Lindstrom, “Fractional Brownian fields as integrals of white noise”, Bull. London Math. Soc. Vol.25, No.1, pp.83-88, 1992.
[8] Bernt Oksendal , Stochastic Differential Equations-An Introduction with Application, 6th edition, Springer, Berlin Heidelberg, 2003.
[9] Vladas Pipiras and Murad S. Taqqu, “Deconvolution of fractional Brownian motion”, Journal of Time Series analysis, Vol. 23, No. 4, pp. 487-501, 1999.
[10]Vladas Pipiras and Murad S. Taqqu, “Are classes of diereministic integrands for fractional Brownian motion on an interval complete? ”, Bernoulli, Vol. 7, No. 6, pp. 873-897, Dec. 2001.
[11]Vladas Pipiras and Murad S. Taqqu, “Fractional calculus and its connection to fractional Brownian motion”, Theory and Applications of Long-Range Dependence, Birkhäuser Boston, Boston, MA, pp. 165-201, 2003.
[12]Igor Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation”, Fractional Calculus & Applied Analysis, Vol. 5, No. 4, pp. 367-386, 2002.
[13]L. C. G. Rogers, “Arbitrage with fractional Brownian motion”, Mathematical Finance, Vol. 7, No. 1, pp. 95-105, Jan. 1997.
[14]Stefan G. Samko, et al., Fractional Integrals and Derivatives-Theory and Application, OPA(Amsterdam) B.V., 1993.
[15]Gennady Samorodnitsky, Murad S. Taqqu, Stable Non-Gaussian Random Processes-Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994
[16]Richard L. Wheeden, Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis, Pure and Applied Mathematics, Vol. 43, Marcel Dekker, 1977.

指導教授

趙一峰(I-F. Chao)

審核日期

2009-7-2

推文