  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.
