  In this paper, we focus on discrete-time stochastic systems modelled by
nonlinear stochastic difference equations and propose robust abstractions for
verifying probabilistic linear temporal specifications. The current literature
focuses on developing sound abstraction techniques for stochastic dynamics
without perturbations. However, soundness thus far has only been shown for
preserving the satisfaction probability of certain types of temporal-logic
specification. We present constructive finite-state abstractions for verifying
probabilistic satisfaction of general {\omega}-regular linear-time properties
of more general nonlinear stochastic systems. Instead of imposing stability
assumptions, we analyze the probabilistic properties from the topological view
of metrizable space of probability measures. Such abstractions are both sound
and approximately complete. That is, given a concrete discrete-time stochastic
system and an arbitrarily small L1-perturbation of this system, there exists a
family of finite-state Markov chains whose set of satisfaction probabilities
contains that of the original system and meanwhile is contained by that of the
slightly perturbed system. A direct consequence is that, given a probabilistic
linear-time specification, initializing within the winning/losing region of the
abstraction system can guarantee a satisfaction/dissatisfaction for the
original system. We make an interesting observation that, unlike the
deterministic case, point-mass (Dirac) perturbations cannot fulfill the purpose
of robust completeness.
