On maximal inequalities for purely discontinuous martingales in infinite dimensions

TL;DR

This paper surveys maximal inequalities for purely discontinuous martingales and related stochastic integrals in infinite-dimensional spaces, highlighting their importance in stochastic PDEs with jump noise.

Contribution

It provides a comprehensive overview of maximal inequalities in infinite dimensions, emphasizing their applications in stochastic partial differential equations with jump noise.

Findings

Summarizes key maximal inequalities for jump-type martingales.

Highlights the role of these inequalities in stochastic PDE analysis.

Provides insights into stochastic integrals with respect to Poisson measures.

Abstract

The purpose of this paper is to give a survey of a class of maximal inequalities for purely discontinuous martingales, as well as for stochastic integral and convolutions with respect to Poisson measures, in infinite dimensional spaces. Such maximal inequalities are important in the study of stochastic partial differential equations with noise of jump type.