A Doob-type maximal inequality and its applications to various stochastic processes

TL;DR

This paper extends Doob's maximal inequality to a broader class of stochastic processes, providing new bounds and inequalities applicable to diverse processes like Levy processes, branching processes, and Markov processes.

Contribution

It introduces a generalized submartingale concept and derives new maximal inequalities, expanding the applicability of Doob's inequality to various stochastic processes.

Findings

Derived new inequalities for Levy processes and random walks

Extended maximal inequalities to branching and superdiffusion processes

Provided bounds for geometric Brownian motion and Markov processes

Abstract

We generalize the notion of the submartingale property and Doob's inequality. Furthermore, we show how the latter leads to new inequalities for several stochastic processes: certain time series, Levy processes, random walks, processes with independent increments, branching processes and continuous state branching processes, branching diffusions and superdiffusions, as well as some Markov processes, including geometric Brownian motion.