It\^{o} isomorphisms for $L^{p}$-valued Poisson stochastic integrals

TL;DR

This paper establishes Itô isomorphisms for $L^p$-valued Poisson stochastic integrals, providing new inequalities and moment estimates relevant for stochastic PDEs with jump noise.

Contribution

It introduces novel Rosenthal type inequalities for $L^p$-valued random variables and proves Itô isomorphisms for Poisson stochastic integrals, advancing the analysis of jump noise in stochastic PDEs.

Findings

Proved Itô isomorphisms for $L^p$-valued Poisson integrals.

Developed Rosenthal type inequalities for noncommutative $L^p$-spaces.

Derived moment estimates for sums of independent random matrices.

Abstract

Motivated by the study of existence, uniqueness and regularity of solutions to stochastic partial differential equations driven by jump noise, we prove It\^{o} isomorphisms for $L^p$-valued stochastic integrals with respect to a compensated Poisson random measure. The principal ingredients for the proof are novel Rosenthal type inequalities for independent random variables taking values in a (noncommutative) $L^p$-space, which may be of independent interest. As a by-product of our proof, we observe some moment estimates for the operator norm of a sum of independent random matrices.