New Optional Stopping Theorems and Maximal Inequalities on Stochastic Processes

TL;DR

This paper introduces novel optional stopping theorems for bounded regions and derives inequalities for the extremal values of stochastic processes over all time points.

Contribution

It presents new theoretical results on optional stopping and maximal inequalities for stochastic processes with bounded stopping regions.

Findings

New optional stopping theorems for bounded continuity regions

A broad class of inequalities for supremums and infimums of stochastic processes

Theoretical framework applicable to various stochastic process analyses

Abstract

In this paper, we develop new optional stopping theorems for scenarios where the stopping rules are defined by bounded continuity regions. Moreover, we establish a wide variety of inequalities on the supremums and infimums of functions of stochastic processes over the whole range of time indexes.