Parametric sensitivity analysis is a critical component in the study of mathematical models of physical systems. Due to its simplicity, finite difference methods are used extensively for this analysis in the study of stochastically modeled reaction networks. Different coupling methods have been proposed to build finite difference estimators, with the "split coupling," also termed the "stacked coupling," yielding the lowest variance in the vast majority of cases. Analytical results related to this coupling are sparse, and include an analysis of the variance of the coupled processes under the assumption of globally Lipschitz intensity functions [Anderson, SIAM Numerical Analysis, Vol. 50, 2012]. Because of the global Lipschitz assumption utilized in [Anderson, SIAM Numerical Analysis, Vol. 50, 2012], the main result there is only applicable to a small percentage of the models found in the literature, and it was conjectured that similar results should hold for a much wider class of models. In this paper we demonstrate this conjecture to be true by proving the variance of the coupled processes scales in the desired manner for a large class of non-Lipschitz models. We further extend the analysis to allow for time dependence in the parameters. In particular, binary systems with or without time-dependent rate parameters, a class of models that accounts for the vast majority of systems considered in the literature, satisfy the assumptions of our theory.

中文翻译：

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具有非 Lipschitz 速率函数的反应网络的有限差分方法的方差

参数灵敏度分析是研究物理系统数学模型的关键组成部分。由于其简单性，有限差分方法在随机建模反应网络的研究中被广泛用于这种分析。已经提出了不同的耦合方法来构建有限差分估计器，其中“分裂耦合”也称为“堆叠耦合”，在绝大多数情况下产生最低的方差。与这种耦合相关的分析结果是稀疏的，包括在全局 Lipschitz 强度函数的假设下对耦合过程的方差进行分析 [Anderson，SIAM 数值分析，卷。50, 2012]。由于 [Anderson, SIAM Numerical Analysis, Vol. 50, 2012], 那里的主要结果仅适用于文献中发现的一小部分模型，据推测，类似的结果应该适用于更广泛的模型类别。在本文中，我们通过为一大类非 Lipschitz 模型以所需方式证明耦合过程尺度的方差来证明这一猜想是正确的。我们进一步扩展分析以允许参数的时间依赖性。特别是，带有或不带有时间相关速率参数的二元系统（一类占文献中考虑的绝大多数系统的模型）满足我们理论的假设。在本文中，我们通过为一大类非 Lipschitz 模型以所需方式证明耦合过程尺度的方差来证明这一猜想是正确的。我们进一步扩展分析以允许参数的时间依赖性。特别是，带有或不带有时间相关速率参数的二元系统（一类占文献中考虑的绝大多数系统的模型）满足我们理论的假设。在本文中，我们通过为一大类非 Lipschitz 模型以所需方式证明耦合过程尺度的方差来证明这一猜想是正确的。我们进一步扩展分析以允许参数的时间依赖性。特别是，带有或不带有时间相关速率参数的二元系统（一类占文献中考虑的绝大多数系统的模型）满足我们理论的假设。