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研究生:林奕伸
研究生(外文):Lin, Yi-Shen
論文名稱:最佳控制及停止問題
論文名稱(外文):Optimal Control and Stopping Problems
指導教授:蕭守仁蕭守仁引用關係
指導教授(外文):Hsiau, Shoou-Ren
口試委員:蕭守仁姚怡慶符麥克鄭宗琳李信宏
口試日期:2016-06-14
學位類別:博士
校院名稱:國立彰化師範大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2016
畢業學年度:104
語文別:英文
論文頁數:75
中文關鍵詞:最佳控制最佳停止秘書問題後推歸納法一致化間隔馬比諾吉昂
外文關鍵詞:optimal controloptimal stoppingsecretary problembackward inductionuniform spacingsMabinogion
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本論文主要研究幾個最佳控制及停止問題。第一及第二章探討主題為最佳停止問題。我們在第一章考慮最佳停止問題裡的秘書問題。我們想要分別最大化選取第二名與第三名應徵者的機率。經由使用後推歸納法(backward induction) 得到如同古典秘書問題答案的門檻型最佳停止策略。
接著,我們在第二章考慮一致化間隔的最佳停止問題。單位區間[0, 1] 被隨機地切割成n 段(這些碎段被稱為一致化間隔)。假設有一位決策者按順序觀察這些間隔的長度,並且每當看完一個間隔就必須立刻決定是否選取該間隔或是觀察下一個。我們想要最大化所選間隔的期望長度,以及最大化選到最長間隔的機率。
最後,在第三章我們考慮的最佳控制問題來自David Williams 的Mabinogion sheep模型。我們提出更一般的模型 M(p, q) 及其對應的策略π(p,q),其中0 <= p, q<= 1;M(1,1) 與 π(1,1) 即為Williams 所提出的原始模型及其最佳策略。當(p, q) = (1,1/2) 及(1/2,1) 時,我們依循David Williams 的論証方法可以證明 π(p,q) 是 M(p,q) 模型的最佳策略。
This thesis is concerned with several optimal control and stopping problems. Chapters 1 and 2 are on the topic of optimal stopping problems. In Chapter 1 we generalize the classical secretary problem. We want to find a stopping rule to maximize the probability of selecting the 2nd (3rd, resp.) best applicant. By using the method of backward induction, we get an optimal stopping rule, which is of threshold type as the solution of the classical secretary problem.
Next, we consider an optimal stopping problem for the uniform spacings in Chapter 2. Suppose the decision-maker observes the lengths of the uniform spacings in [0, 1] sequentially one by one. The decision-maker must decide whether to select the present spacing or to reject it and continue to observe the next one if any. We first find a strategy to maximize the expected length of the selected spacing and then a strategy to maximize the probability
of selecting the largest spacing.
Finally, in Chapter 3 we consider an optimal control problem for the Mabinogion sheep model, which is originally proposed by D. Williams. We propose a more general model M(p,q) and corresponding control policy π(p,q). We show that π(p,q) is optimal for M(p,q) in the two cases (p,q) = (1,1/2) and (1/2,1).

Overview of the Thesis 1

1 Optimal stopping for the secretary problem 4
1.1 Introduction. . . . . . . . . . . . . . . 4
1.2 The method of backward induction . . . . .7
1.3 Maximizing the probability of selecting the 2nd best applicant . . . . . . . . . 8
1.4 Maximizing the probability of selecting the 3rd best applicant . . . . . . . . . 13
1.5 Proofs of the key lemmas in Section 1.4. . . . 18

2 Optimal stopping for the uniform spacings 25
2.1 Introduction . . . . . . . . . . . .. . 25
2.2 Maximizing the expected length of the selected spacing . . . . . . . . . . . . . 28
2.3 Maximizing the probability of selecting the largest spacing . . . . . . . . . . . 35

3 Optimal control for the Mabinogion sheep model 39
3.1 Introduction . . . . . . . . . . . . . . . . 39
3.2 Main results . . . . . . . . . . . . . . . . 42
3.3 Proofs of (S1) and (S2) . . . . . . . . . . 43
3.4 Proofs of (S3) and (S4) . . . . . . . . .. . 50
3.5 Proofs of the key lemmas in Section 3.3 . . 54
3.6 Proofs of the key lemmas in Section 3.4 . . 61
3.7 Asymptotics of the value functions . . . . . 66
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