  Consider the standard stochastic reaction network model where the dynamics is
given by a continuous-time Markov chain over a discrete lattice. For such
models, estimation of parameter sensitivities is an important problem, but the
existing computational approaches to solve this problem usually require
time-consuming Monte Carlo simulations of the reaction dynamics. Therefore
these simulation-based approaches can only be expected to work over finite
time-intervals, while it is often of interest in applications to examine the
sensitivity values at the steady-state after the Markov chain has relaxed to
its stationary distribution. The aim of this paper is to present a
computational method for the estimation of steady-state parameter
sensitivities, which instead of using simulations, relies on the recently
developed stationary Finite State Projection (sFSP) algorithm [J. Chem. Phys.
147, 154101 (2017)] that provides an accurate estimate of the stationary
distribution at a fixed set of parameters. We show that sensitivity values at
these parameters can be estimated from the solution of a Poisson equation
associated with the infinitesimal generator of the Markov chain. We develop an
approach to numerically solve the Poisson equation and this yields an efficient
estimator for steady-state parameter sensitivities. We illustrate this method
using several examples.
