多維非線性跨界問題暨相關應用

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姓名

高竹嵐(Chu-Lan Kao) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

多維非線性跨界問題暨相關應用
(Multidimensional Nonlinear Boundary Crossing Problems with Applications)

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摘要(中)

本論文研究一高維簡單隨機漫步跨越一高維度曲面的首次穿越時間。此類問題在實務上有相當廣泛的運用，諸如多重CUSUM、公司違約相關性，以及多重偵測問題等。在若干條件下，我推導出首次穿越時間機率分佈與期望值的漸近行為。其中期望值漸近行為的推導，乃奠基於本論文率先提出的新方法：我首先將問題轉化為一個一維馬可夫隨機漫步跨越一直線的首次穿越時間，接著再以一序列具有遍歷性的馬可夫隨機漫步來逼近之。最後，本文將既有的馬可夫隨機漫步更新理論，推廣至一序列之馬可夫隨機漫步並應用之，終得到原期望值之漸近行為。本論文同時呈現相關數值結果，討論相關應用、推廣，以及此『馬可夫化』手法之未來可能應用。

摘要(英)

In this dissertation, I study the first passage time of a
multidimensional simple random walk crosses a certain type of
nonlinear boundary, which is motivated by a wide class of
applications, including MCUSUM, correlated defaults, and
multi-sensor problem. Under some regularity conditions, I derive
asymptotic expansions for the ruin probability and the expected
value. The evaluation of the expected value is through an
innovative device that first rewrite the problem as a one
dimensional Markov random walk crossing a linear boundary, and then approximate this Markov random walk by a sequence of uniformly ergodic Markov random walks. For this purpose, I also study renewal theory for a sequence of Markov random walks. Numerical simulations are given for illustration. Applications and further extensions are presented, along with the discussion of possible future usage of this Markovianlize device.

關鍵字(中)

★ 首次通過機率
★ 馬可夫更新理論
★ 收斂速度

關鍵字(英)

★ first-passage probabilities
★ Markov renewal theory
★ rate of convergence

論文目次

中文提要 ………………………………………………………………………… i
英文提要 ………………………………………………………………………… ii
誌謝 ………………………………………………………………………… iii
目錄 ………………………………………………………………………… iv
表目錄 ………………………………………………………………………… v
1. Introduction …………………………………………………………… 1
2. Review of Simple Random Walks…………………………………… 4
2.1 Inequalities for Maximum…………………………………………… 4
2.2 Berry-Esseen Bounds………………………………………………… 4
2.3 Anscombe-Doeblin Theorem………………………………………… 5
2.4 Renewal Measures and Ladder Epochs……………………………… 5
2.5 Wald’s Identity……………………………………………………… 7
2.6 Nonlinear Renewal Theory…………………………………………… 7
3. Review of Markov Chains…………………………………………… 9
3.1 Irreducibility………………………………………………………… 9
3.2 Aperiodicity…………………………………………………………… 10
3.3 Atoms, Minorization, and Small Sets………………………………… 12
3.4 Invariant Measure and Uniform Ergodicity…………………………… 13
3.5 Connection with Simple Random Walks……………………………… 14
4. Main Results……………………………………………… 15
5. Proofs of Theorems 1 and 2…………………………………………… 21
5.1. Proof of Theorem 1…………………………………………………… 22
5.2. Proof of Theorem 2…………………………………………………… 24
6. Numerical Simulations……………………………………………… 37
7. Applications…………………………………………………………… 43
7.1. Correlated Defaults …………………………………………………… 43
7.2. MCUSUM and Multi-Sensors..……………………………………… 45
7.3. Special Case: Periodic Boundary……………………………………… 47
8. Future Works ………………………………………………………… 49
8.1. Boundary Conditions ………………………………………………… 49
8.2. Negative, Positive and Zero Drifts…………………………………… 50
8.3. Variance ……………………………………………………………… 51
8.4. Practical Applications………………………………………………… 51
8.5. Further Application of Markovianlization …………………………52
9. Conclusion…………………………………………………………… 53
Reference ………………………………………………………………………… 54
Appendix 1. Markov Renewal Theory for a sequence of Markov random walks… 57
Appendix 2. Theorems 1 and 2 with slowly changing term ………… 60
Appendix 3. Derivation of (57)……………………………………… 64

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指導教授

傅承德(Cheng-Der Fuh)

審核日期

2014-6-27

推文