Itô's Formula for Itô processes defined with respect to a cylindrical-martingale valued measure

Abstract

Using the authors' recently developed stochastic integration [Stoch PDE: Anal Comp, 2024], we prove an Itô formula for Hilbert space-valued Itô processes defined with respect to a cylindrical martingale-valued measure. We develop some tools from stochastic analysis, as are the predictable and optional quadratic variation of a stochastic integral, the continuous and purely discontinuous parts of an integral process, and a Riemann representation formula. As an application of our Itô formula, we prove a Burkholder inequality for the stochastic integral defined with respect to a cylindrical martingale-valued measure. Finally, we derive Itô formulas for Hilbert space-valued martingale-valued measures and for cylindrical square integrable martingales.