Tighter bounds on transient moments of stochastic chemical systems

The use of approximate solution techniques for the Chemical Master Equation is common practice for the analysis of stochastic chemical systems. Despite their widespread use, however, many such techniques rely on unverifiable assumptions and only few provide mechanisms to control the approximation error quantitatively. Addressing this gap, Dowdy and Barton [The Journal of Chemical Physics, 149(7), 074103 (2018)] proposed a method for the computation of guaranteed bounds on the moment trajectories associated with stochastic chemical systems described by the Chemical Master Equation, thereby providing a general framework for error quantification. Here, we present an extension of this method. The key contribution is a new hierarchy of convex necessary moment conditions crucially reflecting the temporal causality and other regularity conditions that are inherent to the moment trajectories associated with stochastic processes described by the Chemical Master Equation. Analogous to the original method, these conditions generate a hierarchy of semidefinite programs that furnishes monotonically improving bounds on the trajectories of the moments and related statistics. Compared to its predecessor, the presented hierarchy produces bounds that are at least as tight and it often enables the computation of dramatically tighter bounds as it enjoys superior scaling properties and the generated semidefinite programs are highly structured. We analyze the properties of the presented hierarchy, discuss some aspects of its practical implementation and demonstrate its merits with several examples.