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    题名: 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
    作者: 許慧儀
    Hsu, Hui-Yi
    贡献者: 許順吉
    姜祖恕
    洪芷漪

    Sheu, Shuenn-Jyi
    Chiang, Tzuu-Shuh
    Hong, Jyy-I

    許慧儀
    Hsu, Hui-Yi
    关键词: 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
    日期: 2024
    上传时间: 2024-07-01 12:56:06 (UTC+8)
    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。
    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.
    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.
    描述: 碩士
    國立政治大學
    應用數學系
    110751003
    資料來源: http://thesis.lib.nccu.edu.tw/record/#G0110751003
    数据类型: thesis
    显示于类别:[應用數學系] 學位論文

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