Poisson stochastic integration in Banach spaces

TL;DR

This paper establishes new bounds for Banach space-valued Poisson stochastic integrals, develops a Malliavin calculus framework, and derives a Clark-Ocone formula, advancing the theoretical understanding of stochastic integration in Banach spaces.

Contribution

It introduces novel bounds for Poisson stochastic integrals in Banach spaces and extends Malliavin calculus tools to this setting, including a Clark-Ocone representation.

Findings

New upper and lower bounds for Banach space-valued stochastic integrals

Extension of Malliavin calculus to Poisson integrals in Banach spaces

Derivation of a Clark-Ocone representation formula

Abstract

We prove new upper and lower bounds for Banach space-valued stochastic integrals with respect to a compensated Poisson random measure. Our estimates apply to Banach spaces with non-trivial martingale (co)type and extend various results in the literature. We also develop a Malliavin framework to interpret Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a Clark-Ocone representation formula.