  Poisson's equation has a lot of applications in various areas. Usually it is
hard to derive the explicit expression of the solution of Poisson's equation
for a Markov chain on an infinitely many state space. We will present a
computational framework for the solution for both discrete-time Markov chains
(DTMCs) and continuous-time Markov chains (CTMCs), by developing the technique
of augmented truncation approximations. The convergence to the solution is
investigated in terms of the assumption about the monotonicity of the first
return times, and is further established for two types of truncation
approximation schemes: the censored chain and the linear augmented truncation.
Moreover, truncation approximations to the variance constant in central limit
theorems (CLTs) are also considered. The results obtained are applied to
discrete-time single-birth processes and continuous-time single-death
processes.
