Stochastic integration with respect to cylindrical semimartingales

C. A. Fonseca-Mora

Published 2020-10-11Version 1

In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales in the dual $\Phi'$ of a locally convex space $\Phi$. 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 $\Phi$ is a complete, barrelled, nuclear space, where we obtain a complete description of the class of integrands as $\Phi$-valued locally bounded and weakly predictable processes. Several other properties of the stochastic integral are proven, including a Riemann representation and a integration by parts formula. Finally, as an application to our theory we define stochastic integrals with respect to a sequence of real-valued semimartingales.