A Geometric Approach for Bounding Average Stopping Time

TL;DR

This paper introduces a geometric method using convex analysis to bound average stopping times in stochastic processes, providing explicit formulas and broad applicability to complex stopping boundaries.

Contribution

It presents a novel geometric framework that generalizes classical inequalities and offers efficient bounds for average stopping times involving complex conditions.

Findings

Explicit bounds for average stopping times derived

Generalization of Jensen's inequality, Wald's, and Lorden's inequalities

Applicable to nonlinear stopping boundaries and random vectors

Abstract

We propose a geometric approach for bounding average stopping times for stopped random walks in discrete and continuous time. We consider stopping times in the hyperspace of time indexes and stochastic processes. Our techniques relies on exploring geometric properties of continuity or stopping regions. Especially, we make use of the concepts of convex sets and supporting hyperplane. Explicit formulae and efficiently computable bounds are obtained for average stopping times. Our techniques can be applied to bound average stopping times involving random vectors, nonlinear stopping boundary, and constraints of time indexes. Moreover, we establish a stochastic characteristic of convex sets and generalize Jensen's inequality, Wald's equations and Lorden's inequality, which are useful for investigating average stopping times.