Abstract 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.

中文翻译：

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求解生死链的泊松方程：结构、不稳定性和精确近似

摘要 泊松方程作为马尔可夫链的性能评估和优化工具发挥着重要的作用。对于如本文所述的可能具有无限过渡和成本率的连续时间生死链，当解析解不可用时，理论上可以通过简单的前向递归获得其数值解。然而，这可能会受到数值不稳定性的影响，这会隐藏精确解的结构。本文提出了三个主要贡献：（1）在过渡和成本率的温和条件下建立了结构结果（相对成本函数的凸性），这与证明马尔可夫决策模型中最优策略的结构特性有关；(2) 它通过标准正向递归阐明数值解中不稳定性的根本原因、程度和普遍性；(3) 它提出了一种新的前后递归方案来计算精确的数值解。将结果应用于偏差和渐近方差的准确评估，并在示例中进行说明。