  One problem of wide interest involves estimating expected crossing-times.
Several tools have been developed to solve this problem beginning with the
works of Wald and the theory of sequential analysis. An extension of his
approach is provided by the optional sampling theorem in conjunction with
martingale inequalities. Deriving the explicit close form solution for the
expected crossing times may be difficult. In this paper, we provide a framework
that can be used to estimate expected crossing times of arbitrary stochastic
processes. Our key assumption is the knowledge of the average behavior of the
supremum of the process. Our results include a universal sharp lower bound on
the expected crossing times.
