  For classical finite time horizon stopping problems driven by a Brownian
motion
  \[V(t,x) = \sup_{t\leq\tau\leq0}E_{(t,x)}[g(\tau,W_{\tau})],\] we derive a
new class of Fredholm type integral equations for the stopping set. For large
problem classes of interest, we show by analytical arguments that the equation
uniquely characterizes the stopping boundary of the problem. Regardless of the
uniqueness, we use the representation to rigorously find the limit behavior of
the stopping boundary close to the terminal time. Interestingly, it turns out
that the leading-order coefficient is universal for wide classes of problems.
We also discuss how the representation can be used for numerical purposes.
