| 摘要: | 在第一部分，我們主要介紹不同作者所定義的 Markov 過程，並且給出它們的強弱關係。在章節1.1, 我們介紹 Markov 過程最直觀的定義，由 Cinlar和鍾開萊的定義統整而來。 在章節1.2, 我們介紹 transition kernels 的原始定義，並且在章節 1.3 討論 Markov 過程的 transition function 的存在性，其中涉及 “Regular Conditional Distribtution (RCD)” 的概念. 在章節 1.3,我們介紹由 Morters-Peres和 Revuz-Yor 所定義的 time-homogeneous (時間齊次) Markov 過程。我們給出一個隨機過程是時間齊次 Markov 過程的充分條件 (Corollary 1.46)。在章節 1.5, 我們介紹分別由 Bass， Morters-Peres，Revuz-Yor 和 Shreve-Ioannis 所定義的 Markov 過程，其定義涉及一族初始測度。我們給出了這些定義彼此間的強弱關係。在章節1.6，我們介紹 Brownian motion 的數學定義，並且證明它是一個 Markov 過程。
在第 2章，我們介紹擴展 Markov 過程的 filtration 的概念。
在章節3.1，我們介紹 strong Markov property (SMP)。與 Markov property 不同的地方是，strong Markov 把 Markov property 中固定的時間換成了 stopping time。我們證明 Brownian motion 是一個 strong Markov process，並且利用這個 strong Markov property 去證明 Brownian motions 的一些簡單的性質。在本章最後，我們給出一個有左極限右連續 (cadlag) 的路徑的 strong Markov process 擁有連續路徑的充分條件。在章節 3.2，我們介紹 Dirichlet problem, 並且利用 Brownian motion 的 strong Markov property 給出 Dirichlet problem 的一個解。
在章節 4.1，我們討論一個隨機過程 killed on leaving a set, 並且在章節 4.2證明如果原始的過程有 SMP, 那麼被切斷的過程仍然具有 SMP。在章節 4.3, 我們介紹一種 Markov 過程的變形，其又名為 Doob′s h-path transform。
最後，在章節5.1, 我們介紹 Galton-Watson branching process 和 continuous-State branching process, 並且整理 continuous-state branching processes 作為 Galton-Watson branching processes 的 scaling limits 的定理。 在章節 5.2,我們介紹 reflected Brownian motion 是一個 Markov 過程但不是 Brownian motion，並且介紹 Paul Levy 在1948年證明的定理，它給出一個 reflected Brownian motion 的例子。
在第二部分，我們主要給出 Morters-Peres的著作 Brownian motion 中前兩章節的註解，它們分別為第 6章與第 7章，其中深入討論 Brownian motion 的存在性、拓樸性質以及最重要的 strong Markov property。
在附錄中，我們列出幾個機率論、分析和隨機過程的基本結論。在章節 A,我們介紹條件機率的定義；在章節 B.2,我們介紹 Dynkin′s $\lambda-\pi$ system 並且證明 Dynkin′s Theorem；在章節B.3和B.4我們分別介紹兩個版本的 monotone class theorem； 在章節C，我們介紹 optional stopping theorem， 作為一個解決涉及 stopping times 和 martingales 的常用工具。在章節 E，我們給出一個簡單版本的 Kolmogorov extension theorem。在章節 D，我們解釋何謂隨機過程的 law，透過 Kolmogorov extension theorem，可以知道一個隨機過程的 law 可以被它自身的 f.d.d. 所決定。在章節 F，我們給出 Gaussian process 的定義，並且給出 Gaussian process 的幾個重要性質，利用 Brownian motion 作為 Gaussian process 的一個特例，我們可以得出許多有趣的結果。最後，在章節G，我們介紹 stable subordinator 的定義與一個重要的 subordinator 的 representation 定理。;In the first part, we survey several definitions of Markov properties from different authors and explain the relationship between all of them. In section 1.1, we introduce the main definition of Markov process defined by Cinlar and Chung Kai Lai, which is the most intuitive way to define a Markov process. In section 1.2, we introduce the general definition of transition kernels, and discuss the existence of transition kernels for a Markov process in the section 1.3, which includes a discussion of “Regular Conditional Distribution (RCD)”. In section 1.4, we introduce the homogeneous Markov processes as defined by Morters-Peres and Revuz-Yor. We give a sufficient condition (Corollary 1.46) for a process to be a homogeneous Markov process. In section 1.5, we introduce Markov processes with respect to a family of probability measures, as defined by Bass, Morters-Peres, Revuz-Yor and Shreve-Ioannis. We work out the relationships between them. In section 1.6, we introduce the Brownian motion and show that a Brownian motion is a Markov process.
In section 2, we introduce the idea of enlarging the filtration of a Markov process.
In section 3.1, we introduce the strong Markov property (SMP). To do this, we need to replace a deterministic time by a stopping time. We prove that the Brownian motion is a strong Markov process, and then use the strong Markov property to prove several results on Brownian motions. Finally, we give a sufficient condition for a strong Markov process with cadlag sample paths to have continuous paths. In section 3.2, we introduce the Dirichlet problem, and solve the Dirichlet problem by using the strong Markov property of Brownian Motions.
In section 4.1, we discuss processes killed on leaving a set, and show that if the original process has SMP, then so does the killed process in section 4.2. In section 4.3, we introduce a type of transformation of a Markov process which is by conditioning, known as Doob′s h-path transform.
Finally, in the section 5.1, we survey the Galton-Watson branching process and the Continuous-State Branching process, and state the theorem of Continuous-State Branching processes as the scaling limits of Galton-Watson branching processes. In the section 5.2, we introduce the idea of reflected Brownian motion, as an example of Markov process but not a Brownian motion, and introduce a well-known theorem of Paul Levy.
In the second part, we mainly discuss the Brownian motion introduced by Morters and Pere, and survey the details for each statement of section 1 and section 2 in this book. See section 6 and section 7.
In appendices, we survey several results from probability, analysis and stochastic processes. In the section A, we introduce the definition of conditional expectation; In the section B.2, we give the definition of Dynkin′s lambda-pi system and prove the Dynkin′s Theorem; We introduce two versions of Monotone Class Theorem in the section B.3 and B.4; Finally, in the section C, we introduce the Optional Stopping Theorem, which is useful for solving problems which include stopping times and martingales. In section E, we survey the Kolmogorov extension theorem. In section D we introduce the law of a stochastic process, as a consequence of Kolmogorov extension theorem, it provides that the law of a stochastic process is determined by its finite dimensional distribution. In section F, we introduce the definition of Gaussian process and survey several results of Gaussian process. In section G, we introduce the definition of stable processes, subordinators and a representation theorem. |
