  In the theory of Partially Observed Markov Decision Processes (POMDPs),
existence of optimal policies have in general been established via converting
the original partially observed stochastic control problem to a fully observed
one on the belief space, leading to a belief-MDP. However, computing an optimal
policy for this fully observed model, and so for the original POMDP, using
classical dynamic or linear programming methods is challenging even if the
original system has finite state and action spaces, since the state space of
the fully observed belief-MDP model is always uncountable. Furthermore, there
exist very few rigorous value function approximation and optimal policy
approximation results, as regularity conditions needed often require a tedious
study involving the spaces of probability measures leading to properties such
as Feller continuity. In this paper, we study a planning problem for POMDPs
where the system dynamics and measurement channel model are assumed to be
known. We construct an approximate belief model by discretizing the belief
space using only finite window information variables. We then find optimal
policies for the approximate model and we rigorously establish near optimality
of the constructed finite window control policies in POMDPs under mild
non-linear filter stability conditions and the assumption that the measurement
and action sets are finite (and the state space is real vector valued). We also
establish a rate of convergence result which relates the finite window memory
size and the approximation error bound, where the rate of convergence is
exponential under explicit and testable exponential filter stability
conditions. While there exist many experimental results and few rigorous
asymptotic convergence results, an explicit rate of convergence result is new
in the literature, to our knowledge.
