  We consider a Markov process in continuous time with a finite number of
discrete states. The time-dependent probabilities of being in any state of the
Markov chain are governed by a set of ordinary differential equations, whose
dimension might be large even for trivial systems. Here, we derive a reduced
ODE set that accurately approximates the probabilities of subspaces of interest
with a known error bound. Our methodology is based on model reduction by
balanced truncation and can be considerably more computationally efficient than
the Finite State Projection Algorithm (FSP) when used for obtaining transient
responses. We show the applicability of our method by analysing stochastic
chemical reactions. First, we obtain a reduced order model for the
infinitesimal generator of a Markov chain that models a reversible,
monomolecular reaction. In such an example, we obtain an approximation of the
output of a model with 301 states by a reduced model with 10 states. Later, we
obtain a reduced order model for a catalytic conversion of substrate to a
product; and compare its dynamics with a stochastic Michaelis-Menten
representation. For this example, we highlight the savings on the computational
load obtained by means of the reduced-order model. Finally, we revisit the
substrate catalytic conversion by obtaining a lower-order model that
approximates the probability of having predefined ranges of product molecules.
