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| Title: | 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 |
| Authors: | 許慧儀
Hsu, Hui-Yi |
| Contributors: | 許順吉
姜祖恕
洪芷漪
Sheu, Shuenn-Jyi
Chiang, Tzuu-Shuh
Hong, Jyy-I
許慧儀
Hsu, Hui-Yi |
| Keywords: | 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
Large Deviation Principle
Contraction Principle
Schilder's Theorem
Wentzell-Freidlin Theorem
Diffusion Processes
Markov Jump Process |
| Date: | 2024 |
| Issue Date: | 2024-07-01 12:56:06 (UTC+8) |
| Abstract: | 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. |
| Reference: | [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. |
| Description: | 碩士
國立政治大學
應用數學系
110751003 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0110751003 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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