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| 研究生： |
許慧儀 Hsu, Hui-Yi |
|---|---|
| 論文名稱： |
具有狀態轉換的擴散過程及其大離差行為之研究 A study of large deviations of diffusion processes with regime switching |
| 指導教授： |
許順吉
Sheu, Shuenn-Jyi 姜祖恕 Chiang, Tzuu-Shuh 洪芷漪 Hong, Jyy-I |
| 口試委員: |
許順吉
Sheu, Shuenn-Jyi 姜祖恕 Chiang, Tzuu-Shuh 洪芷漪 Hong, Jyy-I 須上苑 Shiu, Shang-Yuan |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2024 |
| 畢業學年度： | 112 |
| 語文別： | 英文 |
| 論文頁數： | 61 |
| 中文關鍵詞： | 大離差原則 、收縮原則 、Schilder定理 、Wentzell-Freidlin定理 、擴散過程 、Markov跳躍過程 |
| 外文關鍵詞： | Large Deviation Principle, Contraction Principle, Schilder's Theorem, Wentzell-Freidlin Theorem, Diffusion Processes, Markov Jump Process |
| 相關次數： | 點閱：103 下載：66 |
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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。
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.
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
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[11] Jean-François Le Gall. Brownian motion, martingales, and stochastic calculus. Springer, 2016.
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[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.
