Generalization of Doob's Inequality and A Tighter Estimate on Look-back Option Price

TL;DR

This paper strengthens Doob's $L^p$ inequality for sub-martingales, providing a more precise estimate that could enhance various applications in stochastic process theory.

Contribution

It introduces a generalized version of Doob's inequality, offering a tighter bound for sub-martingale processes, which is a novel theoretical advancement.

Findings

Strengthened Doob's $L^p$ inequality for sub-martingales

Potential applications in stochastic process analysis

Improved bounds for look-back option pricing

Abstract

In this short note, we will strengthen the classic Doob's $L^p$ inequality for sub-martingale processes. Because this inequality is of fundamental importance to the theory of stochastic process, we believe this generalization will find many interesting applications.