一類具有非利普希茨漂移項的隨機微分方程之相變

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楊氏三妹(DUONG THI BE BA) 查詢紙本館藏

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論文名稱

一類具有非利普希茨漂移項的隨機微分方程之相變
(THE PHASE TRANSITION IN A FAMILY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH NON-LIPSCHITZ DRIFT TERM)

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摘要(中)

我們考慮下列一維的隨機微分方程：
dXt = (Xt −Xt^3)dt +λXtdBt, λ > 0,
對應到的起始值為正數。這方程的係數函數並不滿足利普希茨連續，所以不能用一般已知的定理得到強解的存在唯一性，但我們可以將係數函數截斷使其滿足利普希茨連續，得到截斷後方程強解的存在與唯一性，再取極限得到原方程強解的存在唯一性，而這解為一強馬氏過程。這篇論文主要想要研究解的長時間行為，而這會與解中的λ有關。利用解的比較定理，我們證明當λ大於根號2的時候，只有一個顯然的不變測度；當λ小於2的時候，存在了一個非顯然的不變測度。利用耦合的方法，我們得到藉由遍歷性可以得到全變差距離收斂。另外，我們也得到強範數為指數型收斂。

摘要(英)

We consider the one-dimensional stochastic differential equation (SDE) of the form:
dXt = (Xt −Xt^3)dt +λXtdBt, λ > 0,
subject to a positive initial value. \
It should be emphasized that because the Lipschitz conditions are not satisfied, the existence of a solution to the SDE is not assured. However, we can establish the existence and uniqueness of a strong solution by truncating the drift term and taking the limit as the truncation is removed. As an application of martingale problems, the solution is a strong Markov process. Our primary goal is to investigate the large-time behavior of this process. The conclusions will depend on the value of λ. By utilizing comparison theorems, we demonstrate that the process has a unique trivial invariant measure when λ is large enough. In addition, we prove that a unique invariant probability measure exists when λ is small, which is why we are more interested in this case. We discover, in particular, that the critical point exists and is equal to the square root of 2. Furthermore, using the coupling method, we establish ergodicity, which ensures convergence in the total variation distance, is established. We also obtain an exponential rate of convergence in the strong norm based on results from the application of "drift criteria" for general state space Markov processes.

關鍵字(中)

★ 隨機微分方程

關鍵字(英)

★ Stochastic differential equation

論文目次

Contents
摘要 i
Abstract ii
Acknowledgement iii
Explanation of Symbols iv
1 STOCHASTIC INTEGRATION AND ITÔ’S FORMULA 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Stochastic Integration with Respect to Semimartingales . . . . . . . . . . . . . 9
1.2.1 Integration with Respect to Bounded Martingales . . . . . . . . . . . . 10
1.2.2 Integration with Respect to Local Martingales . . . . . . . . . . . . . . 12
1.2.3 Integration with Respect to Semimartingales . . . . . . . . . . . . . . 14
1.2.4 Itô′s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Brownian Motion and Fractional Brownian Motion . . . . . . . . . . . . . . . 18
1.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.3 Representation of Fractional Brownian Motion . . . . . . . . . . . . . 21
1.3.4 The Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Stochastic Integration with Respect to Fractional Brownian Motions . . . . . . 27
1.4.1 Forward, Backward and Symmetric Integrals . . . . . . . . . . . . . . 28
1.4.2 m-order Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4.3 Newton-Côtes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.4 Applications to Fractional Brownian Motions . . . . . . . . . . . . . . 35
2 Markov Processes 36
2.1 Basic Introductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Foster-Lyapunov Criteria for Stability of Markovian Processes . . . . . . . . . 45
2.2.1 Criteria for Finite Escape Time . . . . . . . . . . . . . . . . . . . . . . 48
2.2.2 Criteria for Harris Recurrence . . . . . . . . . . . . . . . . . . . . . . 50
2.2.3 Criteria for Positive Harris Recurrence . . . . . . . . . . . . . . . . . . 50
2.2.4 Criteria for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Stochastic Differential Equations 54
3.1 Stochastic Differential Equations Driven by Standard Brownian Motion . . . . 54
3.1.1 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . 55
3.1.2 The Comparison Theorem . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.3 Markov Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1.4 Feller Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Stochastic Differential Equations Driven by a Fractional Brownian Motion . . . 69
3.2.1 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . 69
3.2.2 Positivity of the Density for Solutions . . . . . . . . . . . . . . . . . . 73
4 Existence and Stability of Solutions to Stochastic Differential Equations with Non-Lipschitz Coefficients 76
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Bounded Moments and Tightness . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Feller Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6.1 Existence and Uniqueness of Invariant Measure . . . . . . . . . . . . . 87
4.6.2 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.7 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A On Martingale Measures 109
B On a Class of Stochastic Differential Inequalities 110
C On Deterministic Integrals and Differential Equations 113
D Martingale Problems 115

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指導教授

須上苑(Shang-Yuan Shiu)

審核日期

2023-7-14

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