Variance of finite difference methods for reaction networks with non-Lipschitz rate functions

TL;DR

This paper proves that the variance of finite difference estimators using split coupling remains bounded for a broad class of reaction network models with non-Lipschitz rate functions, including time-dependent systems.

Contribution

It extends variance analysis of split coupling methods to non-Lipschitz and time-dependent reaction network models, broadening applicability beyond previous Lipschitz assumptions.

Findings

Variance of coupled processes scales as desired for non-Lipschitz models

Analysis includes time-dependent parameters in reaction networks

Results apply to binary systems common in literature

Abstract

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.