  The predictive ability of stochastic chemical reactions is currently limited
by the lack of closed form solutions to the governing chemical master equation.
To overcome this limitation, this paper proposes a computational method capable
of predicting mathematically rigorous upper and lower bounds of transient
moments for reactions governed by the law of mass action. We first derive an
equation that transient moments must satisfy based on the moment equation.
Although this equation is underdetermined, we introduce a set of semidefinite
constraints known as moment condition to narrow the feasible set of the
variables in the equation. Using these conditions, we formulate a semidefinite
program that efficiently and rigorously computes the bounds of transient moment
dynamics. The proposed method is demonstrated with illustrative numerical
examples and is compared with related works to discuss advantages and
limitations.
