论文标题
解决生育链的泊松方程：结构，不稳定和准确的近似
Solving Poisson's equation for birth-death chains: Structure, instability, and accurate approximation
论文作者
论文摘要
Poisson的方程式是马尔可夫链性能评估和优化的工具的基本角色。对于本文所述的连续时间出生死亡链，可能是无界的过渡和成本率，当分析解决方案不可用时，可以通过简单的远期复发来获得其数值溶液。然而，这可能会遭受数值不稳定的困扰，可以隐藏精确解决方案的结构。本文提出了三个主要贡献：（1）在轻度条件和成本率的轻度条件下，它建立了结构性结果（相对成本函数的凸）（相对成本函数的凸），这与马尔可夫决策模型中最佳政策的结构特性有关； （2）它通过标准的正向复发阐明了数值溶液中不稳定性的根本原因，程度和流行率； （3）它提出了一种新型的前向后复发方案，以计算准确的数值解决方案。结果应用于对偏差和渐近方差的准确评估，并在一个示例中进行了说明。
Poisson's equation plays a fundamental role as a tool for performance evaluation and optimization of Markov chains. For continuous-time birth-death chains with possibly unbounded transition and cost rates as addressed herein, when analytical solutions are unavailable its numerical solution can in theory be obtained by a simple forward recurrence. Yet, this may suffer from numerical instability, which can hide the structure of exact solutions. This paper presents three main contributions: (1) it establishes a structural result (convexity of the relative cost function) under mild conditions on transition and cost rates, which is relevant for proving structural properties of optimal policies in Markov decision models; (2) it elucidates the root cause, extent and prevalence of instability in numerical solutions by standard forward recurrence; and (3) it presents a novel forward-backward recurrence scheme to compute accurate numerical solutions. The results are applied to the accurate evaluation of the bias and the asymptotic variance, and are illustrated in an example.
