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One problem of wide interest involves estimating expected crossing-times.Several tools have been developed to solve this problem beginning with theworks of Wald and the theory of sequential analysis. An extension of hisapproach is provided by the optional sampling theorem in conjunction withmartingale inequalities. Deriving the explicit close form solution for theexpected crossing times may be difficult. In this paper, we provide a frameworkthat can be used to estimate expected crossing times of arbitrary stochasticprocesses. Our key assumption is the knowledge of the average behavior of thesupremum of the process. Our results include a universal sharp lower bound onthe expected crossing times.

From Boundary Crossing of Non-Random Functions to Boundary Crossing of Stochastic Processes