幾何布朗運動之推廣與應用

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吳政訓(Cheng-Hsun Wu) 查詢紙本館藏

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論文名稱

幾何布朗運動之推廣與應用
(A generalization of geometric Brownian motion with applications)

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摘要(中)

布朗運動（Brownian motion）是一個實用的數學模型（Wiener (1923), Levy
(1948), Ciesielski (1961)），在生物（Brown(1827)）、物理（Eistein (1905),
Mazo (2002)）、經濟與財務工程（Bachelier (1900), Black and Scholes (1973)）
隨機微積分（Ito (1944)）及許多領域上廣為研究及應用，成果豐碩，影響深
遠。
雖然幾何布朗運動有著多元化的應用，但是無法涵蓋所有的隨機現象。因此推廣幾何布朗運動, 可以擴展適用範圍，此為本文之主要的目的。本文研究下列幾何布朗運動所推廣的隨機過程及其變化型式。
我們將研究此隨機過程之數學性質，討論其在財務工程的應用，並提出參數之統計推論。

摘要(英)

Brownian motion is a rigorous mathematical model (Wiener (1923), Levy (1948),
Ciesielski (1961)) with fruitful applications ranging from biology (Brown (1827)),
physics (Einstein (1905), Mazo (2002)), economy and financial engineering
(Bachelier (1900), Black and Scholes (1973)), to stochastic calculus (Ito (1944)),
among others.
Functional of Brownian motion is also useful in stochastic modeling. This is
particularly true for geometric Brownian motion. For instance, it has been applied to
model prices of stock (page 365 in Karlin and Taylor (1975), Black and
Scholes(1973)), rice (Yoshimoto el al. (1996)), labor (page 363 in Karlin and Taylor
(1975)) and others (Shoji (1995)). See Yor (2001) for more details.
Although geometric Brownian motion has a great variety of applications, it can not
cover all the random phenomena. It is then desirable to find a general model with
geometric Brownian motion as a special model. The purpose of this paper is to
investigate the generalizations of geometric Brownian motion
and its variants.
For the processes mentioned above, we will first study their mathematical properties. Next, we will discuss their applications in financial engineering. In practice, the parameters are unknown and have to be inferred from
realizations of processes. We will present estimation and test procedures.

關鍵字(中)

★ 布朗運動
★ 幾何布朗運動
★ 永續憑證問題
★ 選擇權定價
★ 隨機過程之統計推論
★ 財務工程

關鍵字(英)

★ Brownian motion
★ geometric Brownian motion
★ perpetual warrants
★ option pricing
★ statistical inference for stochastic processes
★ financial engineering

論文目次

Contents
1 Introduction 1
2 Mathematical Properties of U(t) and X(t) 5
2.1 Distributional Properties . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Martingale Properties . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 First Passage Times . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Distributions of ¯ Tu and Tx . . . . . . . . . . . . . . . . . 16
2.3.2 Laplace Transforms of ¯ Tu and Tx . . . . . . . . . . . . . 21
3 Applications of X(t) in Financial Engineering 25
3.1 PerpetualWarrant Problem . . . . . . . . . . . . . . . . . . . . 27
3.2 Option Pricing Problem . . . . . . . . . . . . . . . . . . . . . . 30
3.3 The Optimal Portfolio SelectionProblem . . . . . . . . . . . . . 33
4 Statistical Inferences Based on U(t) and X(t) 39
4.1 Normal and Likelihood Based Inferences . . . . . . . . . . . . . 40
4.1.1 Statistical Inferences for a and b Based on Sampling
Scheme A . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.2 Statistical Inferences for c and d Based on Sampling
Scheme A . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.3 Statistical Inferences for c and d Based on Sampling
Scheme B . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.4 Statistical Inferences for a and b Based on Sampling
Scheme C . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Quadratic Variation Based Inferences . . . . . . . . . . . . . . . 54
4.2.1 Statistical Inferences Based on Sampling Scheme C . . . 54
4.2.2 Statistical Inferences Based on Sampling Scheme D . . . 62
5 Theory and Applications of Y (t) 71
5.1 Mathematical Properties of Y (t) . . . . . . . . . . . . . . . . . . 72
5.1.1 Distributions and Moments of Y (t) . . . . . . . . . . . . 72
5.1.2 Martingale Properties of Y (t) . . . . . . . . . . . . . . . 78
5.1.3 First Passage Time of Y (t) . . . . . . . . . . . . . . . . . 79
5.2 Applications of Y (t) in Financial Engineering . . . . . . . . . . 81
5.2.1 The Properties of the Bankrupt Time . . . . . . . . . . . 82
5.2.2 The PerpetualWarrant Problem. . . . . . . . . . . . . . 84
5.2.3 The Option Pricing Problem . . . . . . . . . . . . . . . . 87
5.3 Statistical Inferences Based on Y (t) . . . . . . . . . . . . . . . . 94
5.3.1 Method ofMoment Based Inferences . . . . . . . . . . . 94
5.3.2 Quadratic Variation Based Inferences . . . . . . . . . . . 95
6 Theory and Applications of Zx(t) and Zy(t) 97
6.1 Mathematical Properties of Zx(t) and Zy(t) . . . . . . . . . . . 98
6.1.1 Distributional Properties . . . . . . . . . . . . . . . . . . 98
6.1.2 Martingale Properties . . . . . . . . . . . . . . . . . . . 102
6.1.3 First Passage Time . . . . . . . . . . . . . . . . . . . . . 103
6.2 Applications of Zx(t) and Zy(t) in Financial Engineering . . . . 106
6.2.1 PerpetualWarrants . . . . . . . . . . . . . . . . . . . . . 106
6.2.2 Option Pricing Problem . . . . . . . . . . . . . . . . . . 106
6.3 Statistical Inferences Based on Zx(t) and Zy(t) . . . . . . . . . . 107
6.3.1 Statistical Inferences for a and b . . . . . . . . . . . . . . 107
6.3.2 Statistical Inferences for c and d . . . . . . . . . . . . . . 109
6.3.3 Quadratic Variation Based Inferences . . . . . . . . . . . 111
7 Conclusions 113
A 119
B 121
C 125
D 129
E 131
F 133
G 137
H 141
I 143

參考文獻

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指導教授

許玉生(Yu-Sheng Hsu)

審核日期

2009-7-4

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