Some Martingale Properties of Simple Random Walk and Its Maximum Process

TL;DR

This paper explores martingale properties related to simple random walks and their maximum processes, providing new characterizations, alternative derivations, and applications including proofs of inequalities and solutions to embedding problems.

Contribution

It offers a complete characterization of martingales involving the walk and its maximum, and introduces a discrete Azéma–Yor martingale with applications.

Findings

Derived a sufficient condition for a three-argument function to be a martingale.

Provided an alternative derivation of the Kennedy martingale.

Formulated a discrete Azéma–Yor solution for the Skorokhod embedding problem.

Abstract

In this paper, martingales related to simple random walks and their maximum process are investigated. First, a sufficient condition under which a function with three arguments, time, the random walk, and its maximum process becomes a martingale is presented, and as an application, an alternative way of deriving the Kennedy martingale is provided. Then, a complete characterization of a function with two arguments, the random walk and its maximum, being a martingale is presented. This martingale can be regarded as a discrete version of the Az\'ema--Yor martingale. As applications of discrete Az\'ema--Yor martingale, a proof of the Doob's inequalities is provided and a discrete Az\'ema--Yor solution for the Skorokhod embedding problem for the simple random walk is formulated and examined in detail.