  Providing safety guarantees for stochastic dynamical systems has become a
central problem in many fields, including control theory, machine learning, and
robotics. Existing methods either employ Stochastic Barrier Functions (SBFs) or
rely on numerical approaches based on abstractions. While SBFs are analogous to
Lyapunov functions to prove (probabilistic) set invariance, abstraction-based
approaches approximate the stochastic system into a finite model for the
computation of safety probability bounds. In this paper, we offer a new
perspective on these seemingly different methods. We show that both these
approaches arise as approximations of a stochastic dynamic programming problem.
Such a new and unifying perspective allows us to formally show the correctness
of both approaches, characterize their convergence and optimality properties,
and examine advantages and disadvantages of each. Our analysis reveals that
abstraction-based methods can provide more accurate certificates of safety, but
generally at a higher computational cost. We conclude the article with a
discussion that highlights possible future directions.
