  This manuscript presents an innovative framework for constructing barrier
functions to bound reachability probabilities for continuous-time stochastic
systems described by stochastic differential equations (SDEs). The reachability
probabilities considered in this paper encompass two aspects: the probability
of reaching a set of specified states within a predefined finite time horizon,
and the probability of reaching a set of specified states at a particular time
instant. The barrier functions presented in this manuscript are developed
either by relaxing a parabolic partial differential equation that characterizes
the exact reachability probability or by applying the Gr\"onwall's inequality.
In comparison to the prevailing construction method, which relies on Doob's
non-negative supermartingale inequality (or Ville's inequality), the proposed
barrier functions provide stronger alternatives, complement existing methods,
or fill gaps.
