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arXiv · 2407.16086

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

Santiago Cambronero·David Campos·C. A. Fonseca-Mora·Darío Mena
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Abstract

Using the authors' recently developed stochastic integration [Stoch PDE: Anal Comp, 2024], we prove an It\^{o} formula for Hilbert space-valued It\^{o} 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 It\^{o} formula, we prove a Burkholder inequality for the stochastic integral defined with respect to a cylindrical martingale-valued measure. Finally, we derive It\^{o} formulas for Hilbert space-valued martingale-valued measures and for cylindrical square integrable martingales.

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Santiago Cambronero, David Campos, C. A. Fonseca-Mora, Darío Mena. 2024-07-22. It\^o's Formula for It\^{o} processes defined with respect to a cylindrical-martingale valued measure. https://arxiv.org/abs/2407.16086

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