| dc.contributor.advisor | 許順吉<br>姜祖恕<br>洪芷漪 | zh_TW |
| dc.contributor.advisor | Sheu, Shuenn-Jyi<br>Chiang, Tzuu-Shuh<br>Hong, Jyy-I | en_US |
| dc.contributor.author (Authors) | 許慧儀 | zh_TW |
| dc.contributor.author (Authors) | Hsu, Hui-Yi | en_US |
| dc.creator (作者) | 許慧儀 | zh_TW |
| dc.creator (作者) | Hsu, Hui-Yi | en_US |
| dc.date (日期) | 2024 | en_US |
| dc.date.accessioned | 1-Jul-2024 12:56:06 (UTC+8) | - |
| dc.date.available | 1-Jul-2024 12:56:06 (UTC+8) | - |
| dc.date.issued (上傳時間) | 1-Jul-2024 12:56:06 (UTC+8) | - |
| dc.identifier (Other Identifiers) | G0110751003 | en_US |
| dc.identifier.uri (URI) | https://nccur.lib.nccu.edu.tw/handle/140.119/152098 | - |
| dc.description (描述) | 碩士 | zh_TW |
| dc.description (描述) | 國立政治大學 | zh_TW |
| dc.description (描述) | 應用數學系 | zh_TW |
| dc.description (描述) | 110751003 | zh_TW |
| dc.description.abstract (摘要) | 我們考慮以下的d維隨機微分方程系統：
dX^ε(t) = b(X^ε(t),Y(t))dt + √ε dW(t), t在[0,T]之間，
X^ε(0) = x在R^d中，
其中W是標準的d維布朗運動，b: R^d × {1,2,...,n} → R^d是有界的，且對於i在{1,2,...,n}中，b(·,i)在全域上滿足Lipschitz連續。狀態轉換過程Y是一個有n個狀態的連續時間馬爾可夫過程，並且與W獨立。我們將考慮當ε趨近於0時，此方程解所形成的擴散過程的大離差原則。
在1987年，Carol Bezuidenhout(參見[2])推導了{X^ε}的大離差原則，其中Y為一般的隨機過程，並將其樣本路徑視為L^1空間中的一個元素。該結果包括了Y為n個狀態馬爾可夫過程的情形。在本論文中，我們將Y的樣本路徑視為具有Skorokhod拓撲的D空間中的一個元素。 | zh_TW |
| dc.description.abstract (摘要) | We consider the following system of d-dimensional stochastic differential equations,
dX^ε(t) = b(X^ε(t),Y(t))dt + √ε dW(t), t ∈ [0,T],
X^ε(0) = x ∈ R^d,
where W is a standard d-dimensional Brownian motion, b:R^d × {1,2,...,n} → R^d is bounded and each component b(·,i) is Lipschitz continuous for i in {1,2,...,n}. Also, the switching process Y is modeled by an n-state continuous time Markov jump process and is independent of W. We shall consider the large deviation principle for the law of the solution diffusion process as ϵ → 0.
In 1987, Carol Bezuidenhout (cf. [2]) derived the large deviations principle of these processes {X^ε} for a general random process Y which is considered the sample path of Y as an element of the L^1-space . The result includes the case where Y is an n-state Markov process. In this thesis, we consider the sample path of Y as an element of the D-space with the Skorokhod topology. | en_US |
| dc.description.tableofcontents | 致謝 i
中文摘要 ii
Abstract iii
Contents iv
1 Introduction 1
2 Preliminary 4
2.1 Large Deviation Principle and Contraction Principle 4
2.2 Schilder's Theorem 11
2.3 Wentzell–Freidlin Theorem 12
3 Main Result 18
3.1 Formulation of The Problem 18
3.2 Main Theorem 19
3.3 Some Properties 21
3.4 Proof of The Bounds 27
3.5 Proof of Theorem 3.2.1 31
3.6 A Counterexample 47
Appendix A The Skorokhod Topology 51
Appendix B Arzelà–Ascoli Theorem 53
Appendix C Grönwall's Inequality 58
Bibliography 60 | zh_TW |
| dc.format.extent | 576889 bytes | - |
| dc.format.mimetype | application/pdf | - |
| dc.source.uri (資料來源) | http://thesis.lib.nccu.edu.tw/record/#G0110751003 | en_US |
| dc.subject (關鍵詞) | 大離差原則 | zh_TW |
| dc.subject (關鍵詞) | 收縮原則 | zh_TW |
| dc.subject (關鍵詞) | Schilder定理 | zh_TW |
| dc.subject (關鍵詞) | Wentzell-Freidlin定理 | zh_TW |
| dc.subject (關鍵詞) | 擴散過程 | zh_TW |
| dc.subject (關鍵詞) | Markov跳躍過程 | zh_TW |
| dc.subject (關鍵詞) | Large Deviation Principle | en_US |
| dc.subject (關鍵詞) | Contraction Principle | en_US |
| dc.subject (關鍵詞) | Schilder's Theorem | en_US |
| dc.subject (關鍵詞) | Wentzell-Freidlin Theorem | en_US |
| dc.subject (關鍵詞) | Diffusion Processes | en_US |
| dc.subject (關鍵詞) | Markov Jump Process | en_US |
| dc.title (題名) | 具有狀態轉換的擴散過程及其大離差行為之研究 | zh_TW |
| dc.title (題名) | A study of large deviations of diffusion processes with regime switching | en_US |
| dc.type (資料類型) | thesis | en_US |
| dc.relation.reference (參考文獻) | [1] Carol Bezuidenhout. Small random perturbation of stochastic systems, Thesis. University of Minnesota, 1985.
[2] Carol Bezuidenhout. A large deviations principle for small perturbations of random evolution equations. The Annals of Probability, pages 646–658, 1987.
[3] Patrick Billingsley. Convergence of probability measures. John Wiley & Sons, 2013.
[4] Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38. Springer Science & Business Media, 2009.
[5] Alexander Eizenberg and Mark Freidlin. Large deviations for markov processes corresponding to pde systems. The Annals of Probability, pages 1015–1044, 1993.
[6] MI Freidlin, AD Wentzell, et al. Random perturbations of dynamical systems [electronic resource].
[7] Qi He and G Yin. Large deviations for multi-scale markovian switching systems with a small diffusion. Asymptotic Analysis, 87(3-4):123–145, 2014.
[8] Frank Hollander. Large deviations, volume 14. American Mathematical Soc., 2000.
[9] Hu Y J. A unified approach to the large deviations for small perturbations of random evolution equations with small perturbations. Sci China Ser A, (7):302–310, 1997.
[10] Vitalii Konarovskyi. An introduction to large deviations. 2019.
[11] Jean-François Le Gall. Brownian motion, martingales, and stochastic calculus. Springer, 2016.
[12] Xiaocui Ma and Fubao Xi. Large deviations for empirical measures of switching diffusion processes with small parameters. Frontiers of Mathematics in China, 10:949–963, 2015.
[13] Halsey Lawrence Royden and Patrick Fitzpatrick. Real analysis, volume 2. Macmillan New York, 1968.
[14] Anatoly V Skorokhod. Limit theorems for stochastic processes. Theory of Probability & Its Applications, 1(3):261–290, 1956.
[15] Varadhan and SR Srinivasa. Asymptotic probabilities and differential equations. Communications on Pure and Applied Mathematics, 19(3):261–286, 1966. | zh_TW |
