It\^o's Formula for It\^{o} processes defined with respect to a cylindrical-martingale valued measure

TL;DR

This paper extends Itô's formula to Hilbert space-valued processes driven by cylindrical-martingale measures, introducing new stochastic analysis tools and applying them to establish inequalities for such stochastic integrals.

Contribution

It develops an Itô formula for Hilbert space-valued processes with respect to cylindrical-martingale measures and introduces related stochastic analysis tools.

Findings

Established an Itô formula for cylindrical-martingale driven processes.

Developed tools for quadratic variation and process decomposition.

Proved a Burkholder inequality for the stochastic integral.

Abstract

Using the theory of stochastic integration developed recently by the authors, in this paper we prove an It\^{o} formula for Hilbert space-valued It\^{o} processes defined with respect to a cylindrical-martingale valued measure. As part of our study, we develop some tools from stochastic analysis as are the predictable and optional quadratic variation of the stochastic integral, the continuous and purely discontinuous parts of the integral process, and a Riemann representation formula. Finally, as an application of It\^{o}'s formula we prove a Burkholder inequality for the stochastic integral defined with respect to a cylindrical-martingale valued measure.