條件期望值(Conditional Expectation)源於財務金融領域之條件風險值(Conditional Value at Risk)，為一個被廣泛使用的風險管理指標。本研究欲求解其無限制式之最佳化問題，改善其估計方法以提升計算效率，並拓展其應用層面。本論文以SNM (Stochastic Nelder-Mead Simplex Method)演算法為基礎，提出一個名為SNM-CE (Stochastic Nelder-Mead Simplex Method for Conditional Expectation)之無微分最佳化演算法。因條件期望值具有隨機性，故需利用隨機模擬以進行估計，本研究基於重要性抽樣(Importance Sampling)與最佳資源分配法(Optimal Computing Budget Allocation, OCBA)，在較少之模擬資源下達到一定之估計精確度，且經由修正問題模型之定義與流程，使得上述方法更適合融入本演算架構，提升演算效率。針對SNM演算法本身，本研究進行細部流程上之修正，並提出新的適應性隨機搜尋法(Adaptive Random Search)，以提升此演算法之效率。本研究之數值實驗也顯示本演算架構之實用及有效性，值得後續深入研究。

Conditional Expectation(CE) origins from the term “Conditional Value at Risk” in the financial field, which is one of the widely used risk measurement in the practice of risk management. Consider the unconstrained optimization problem of CE, this research aims to improve the computational efficiency with a different estimation method and expand the area of its application. We propose a gradient free optimization method, called Stochastic Nelder-Mead Simplex Method for Conditional Expectation, which is based on the Stochastic Nelder-Mead Simplex Method, to solve the optimization problem of CE. Because of the randomness and complexities, Monte Carlo simulation is applied to estimate the conditional expectation. We use Importance Sampling as variance reduction technique and Optimal Computing Budget Allocation to utilize the simulation resources more efficiently. By revising the problem definition and process of the model, we integrate these above methods into the proposed framework to increase the computational efficiency. As for the original SNM algorithm, this research not only makes revisions on the process details, but also develop a new Adaptive Random Search method to improve the computational efficiency. In the end, a series of numerical experiments was conducted to verify the performance and applicability in real problem. The results show the efficiency and efficacy of our proposed method, which is worth for further investigation.

摘要.........................................................I
Abstract....................................................II
圖目錄.......................................................IV
表目錄........................................................V
第一章 緒論...................................................1
1.1研究背景與動機..............................................1
1.2研究目的...................................................2
1.3論文架構...................................................3
第二章　文獻回顧...............................................4
2.1風險值與條件風險值..........................................4
2.1.1條件期望值之估計..........................................6
2.1.2條件期望值之最佳化問題.....................................8
2.2模擬最佳化..................................................9
2.3最佳資源分配法(Optimal Computing Budget Allocation, OCBA)...11
第三章　數學模型...............................................15
第四章　求解方法...............................................16
4.1條件期望值的估計............................................17
4.1.1原始蒙地卡羅模擬法(Crude Monte Carlo Simulation) ........ 17
4.1.2 重要性抽樣(Importance Sampling).........................18
4.2 最佳資源分配法(OCBA) ......................................19
4.2.1 適用條件期望值之OCBA方法 (OCBA-CE) ......................20
4.2.2 適合SNM演算法之OCBA方法 (SOCBA-mn) ......................21
4.2.3 SOCBA-mn流程詳述........................................22
4.3 SNM-CE (Stochastic Nelder-Mead Simplex Method for Conditional Expectation) ..24
4.3.1 SNM-CE之架構與步驟說明..................................24
4.3.2 收縮失敗時之取代修正....................................28
4.3.3 新適應性隨機搜尋(New Adaptive Random Search) ...........30
第五章　數值實驗..............................................35
5.1實驗設計..................................................35
5.2比較指標..................................................37
5.3數值結果..................................................38
第六章 例題驗證.............................................44
6.1數學模型..................................................44
6.2例題結果..................................................46
第七章 結論與未來研究........................................47
參考文獻.....................................................48

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