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

TL;DR

This paper introduces a framework for estimating expected crossing times of stochastic processes using supremum behavior, providing universal bounds and extending classical boundary crossing theories.

Contribution

It extends boundary crossing analysis from non-random functions to stochastic processes, offering a new framework based on supremum behavior and universal bounds.

Findings

Universal sharp lower bound on expected crossing times

Framework applicable to arbitrary stochastic processes

Extension of classical boundary crossing theories

Abstract

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.