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arXiv · 2211.05745

Some Martingale Properties of Simple Random Walk and Its Maximum Process

Takahiko Fujita·Shotaro Yagishita·Naohiro Yoshida
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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 Azéma--Yor martingale. As applications of discrete Azéma--Yor martingale, a proof of the Doob's inequalities is provided and a discrete Azéma--Yor solution for the Skorokhod embedding problem for the simple random walk is formulated and examined in detail.

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Takahiko Fujita, Shotaro Yagishita, Naohiro Yoshida. 2022-11-10. Some Martingale Properties of Simple Random Walk and Its Maximum Process. https://arxiv.org/abs/2211.05745

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