A trajectorial interpretation of Doob's martingale inequalities

TL;DR

This paper introduces a unified, deterministic approach to deriving Doob's $L^p$ maximal inequalities for martingales, connecting probabilistic inequalities with robust hedging interpretations and achieving optimal bounds.

Contribution

It presents a novel deterministic framework that yields new, sharp versions of Doob's inequalities, enhancing understanding and potential applications in stochastic analysis.

Findings

Derived new versions of Doob's maximal inequalities

Established inequalities as consequences of deterministic counterparts

Achieved equality cases with optimally chosen martingales

Abstract

We present a unified approach to Doob's $L^p$ maximal inequalities for $1\leq p<\infty$. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover, our deterministic inequalities lead to new versions of Doob's maximal inequalities. These are best possible in the sense that equality is attained by properly chosen martingales.