This paper investigates the random horizon optimal stopping problem for measure-valued piecewise deterministic Markov processes (PDMPs). This is motivated by population dynamics applications, when one wants to monitor some characteristics of the individuals in a small population. The population and its individual characteristics can be represented by a point measure. We first define a PDMP on a space of locally finite measures. Then we define a sequence of random horizon optimal stopping problems for such processes. We prove that the value function of the problems can be obtained by iterating some dynamic programming operator. Finally we prove via a simple counter-example that controlling the whole population is not equivalent to controlling a random lineage.

中文翻译：

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测量值分段确定性马尔可夫过程的最优停止

本文研究了测量值分段确定性马尔可夫过程 (PDMP) 的随机水平最优停止问题。这是由人口动态应用程序推动的，当人们想要监控小群体中个体的某些特征时。人口及其个体特征可以用点测度来表示。我们首先在局部有限度量空间上定义一个 PDMP。然后我们为这些过程定义了一系列随机水平最优停止问题。我们证明了问题的价值函数可以通过迭代一些动态规划算子来获得。最后我们通过一个简单的反例证明，控制整个种群并不等同于控制一个随机谱系。