论文标题
射线骑士的出生和死亡过程
Ray-Knight compactification of birth and death processes
论文作者
论文摘要
出生和死亡过程是一个连续的马尔可夫链，其状态空间$ \ mathbb n $，其过渡矩阵是标准的，其密度矩阵是给定的出生死亡Matrix。且仅当$ \ infty $是入口或自然的情况下，出生和死亡过程是独一无二的。当$ \ infty $既不是入口也不是自然的，文献中有两种获得所有出生和死亡过程的方法。第一种是Feller于1959年提出的一种分析治疗，第二种是Wang于1958年完成的概率结构。 在本文中，我们将使用Ray-Knight紧凑型研究出生和死亡过程的另一种方法。这种方式具有上述分析和概率治疗的优势。通过应用射线骑士紧凑型，可以将每个出生和死亡过程修改为$ \ mathbb n \ cup \ {\ infty \} \ cup \ cup \ {\ partial \} $的càdlàg射线过程，这是doob过程，要么是一个doob process，要么是teller $ q $ q $ - $ q $ -process。第二类中的每个出生和死亡过程都有一个修改，这是$ \ Mathbb n \ cup \ {\ infty \} \ cup \ {\ partial \} $上的谋杀过程。我们将得出其无限发电机的表达，该发电机将其边界行为解释为$ \ infty $。此外，通过利用杀戮的转变和Ikeda-Nagasawa-Watanabe的拼凑而成，我们还将为出生和死亡过程提供概率的结构。这种构造依赖于三重确定Wang and Yang在其工作（1992'）}中引入的分解矩阵。
A birth and death process is a continuous-time Markov chain with the minimal state space $\mathbb N$, whose transition matrix is standard and whose density matrix is the given birth-death matrix. Birth and death process is unique if and only if $\infty$ is an entrance or natural. When $\infty$ is neither an entrance nor natural, there are two ways in the literature to obtain all birth and death processes. The first one is an analytic treatment proposed by Feller in 1959, and the second one is a probabilistic construction completed by Wang in 1958. In this paper we will give another way to study birth and death processes using the Ray-Knight compactification. This way has the advantage of both the analytic and probabilistic treatments above. By applying the Ray-Knight compactification, every birth and death process can be modified into a càdlàg Ray process on $\mathbb N\cup \{\infty\}\cup\{\partial\}$, which is either a Doob processes or a Feller $Q$-process. Every birth and death process in the second class has a modification that is a Feller process on $\mathbb N\cup\{\infty\}\cup \{\partial\}$. We will derive the expression of its infinitesimal generator, which explains its boundary behaviours at $\infty$. Furthermore, by utilizing transformations of killing and Ikeda-Nagasawa-Watanabe's piecing out, we will also provide a probabilistic construction for birth and death processes. This construction relies on a triple determining the resolvent matrix introduced by Wang and Yang in their work (1992')}.
