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姓名 吳淑萍(Shu-Ping Wu)  查詢紙本館藏   畢業系所 數學系
論文名稱 兩個獨立的基本Lévy隨機過程之極值過程
(On Maximum and Minimum of Two Independent Simple Lévy Processes)
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摘要(中) 在這本論文中,我們研究兩個獨立的基本Lévy隨機過程之極值過程。我們考慮的Lévy過程有布朗運動、卜松過程,以及此兩種隨機過程的和。
在第二章,我們對兩個獨立的布朗運動,找出其極值過程的期望值、平方期望值,進一步討論其機率性質:獨立增量、平穩增量、到達時間的分布和隨機積分等。統計分析則有似然比值、最大似然估計及漸近的不偏估計。
第三章探討的是兩個獨立的卜松過程,我們找出其極值過程的期望值、平方期望值、邊際分布及條件分布。機率性質的討論包含獨立增量、平穩增量、跳躍時間的分布和間隔時間的分布。統計分析則是同第二章,有似然比值、最大似然估計及近似的不偏估計。
第四章則是將布朗運動與卜松過程結合,我們對兩者之和所形成新的兩獨立過程,寫出其極值過程的期望值、平方期望值的表示式,討論其獨立增量、平穩增量、到達時間的分布等機率性質,統計分析同前兩章。
第五章,我們將對第二章到第四章的結論作整理與解析。
摘要(英) In this dissertation, we study the maximum and minimum of two independent
simple L$acute{mbox e}$vy processes. L$acute{mbox e}$vy processes considered are
Brownian motion with drift, Poisson process and their sum.
Mathematical or probabilistic properties discussed include stationary, martingale, diffusion,
stochastic integral, interarrival time, jumping time, first passage time, etc.
Statistical properties investigated cover likelihood ratio, maximum likelihood estimator and
asymptotically unbiased estimator.
This dissertation is structured as follows. In Chapter 2, we deal with the maximum and minimum of two independent
Brownian motions with drifts. Chapter 3 contains the works about the maximum and minimum of two independent
Poisson processes. Chapter 4 is covered by the results about the maximum and minimum of two independent
sums of Brownian motion with drift and Poisson process. In Chapter 5, we provide conclusions and comparisons.
關鍵字(中) ★ 極值過程
★ Lévy隨機過程
關鍵字(英) ★ Lévy Processes
★ Maximum and Minimum
論文目次 List of Notations III
List of Figures VII
List of Tables VIII
Chapter 1 Introduction 1
Chapter 2 Maximum and Minimum of Two Independent
Brownian Motions with Drifts 4
2.1 Moments and Probabilistic Properties 7
2.2 First Passage Times 20
2.3 Stochastic Analysis 26
2.4 Likelihood Ratio and Maximum Likelihood Estimator 31
Chapter 3 Maximum and Minimum of Two Independent
Poisson Processes 42
3.1 Marginal and Conditional Distributions 45
3.2 First Passage Times 55
3.3 Interarrival Times and Conditional Jumping Times 65
3.4 Likelihood Ratio and Maximum Likelihood Estimator 114
Chapter 4 Maximum and Minimum of Two Independent
Sums of Brownian Motion with Drift and
Poisson Process 130
4.1 Moments and Probabilistic Properties 133
4.2 First Passage Times 144
4.3 Likelihood Ratio and Maximum Likelihood Estimator 148
Chapter 5 Conclusions 166
5.1 Similarities and Differences 167
5.2 Expectations 170
5.3 Second Moments 174
5.4 Interarrival times and Conditional Jumpings 176
5.5 Likelihood Ratio 179
References 181
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指導教授 許玉生(Yu-Sheng Hsu) 審核日期 2005-6-22
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