Abstract
In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on a locally convex space Φ. Our construction of the stochastic integral is based on the theory of tensor products of topological vector spaces and the property of good integrators of real-valued semimartingales. This theory is further developed in the case where Φ is a complete, barrelled, nuclear space, where we obtain a complete description of the class of integrands as Φ-valued locally bounded and weakly predictable processes. Several other properties of the stochastic integral are proven, including a Riemann representation, a stochastic integration by parts formula and a stochastic Fubini theorem. Our theory is then applied to provide sufficient and necessary conditions for existence and uniqueness of solutions to linear stochastic evolution equations driven by semimartingale noise taking values in the strong dual of Φ. In the last part of this article we apply our theory to define stochastic integrals with respect to a sequence of real-valued semimartingales.
Funding Statement
The author acknowledges The University of Costa Rica for providing financial support through the grant “B9131-Análisis Estocástico con Procesos Cilíndricos”.
Acknowledgments
The author thank an anonymous referee for valuable comments and suggestions that contributed greatly to improve the presentation of this article. Thanks also to Darío Mena-Arias and Ronald Zuñiga-Rojas for useful discussions.
Citation
Christian A. Fonseca-Mora. "Stochastic integration with respect to cylindrical semimartingales." Electron. J. Probab. 26 1 - 48, 2021. https://doi.org/10.1214/21-EJP718
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