Vector-Valued Stochastic Integration With Respect to Semimartingales in the Dual of Nuclear Space

Abstract: In this work, we introduce a theory of stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. These semimartingales are required to have the good integrator property, which is a property that we explore in detail and provide several examples. Our construction of the stochastic integral uses a regularization argument for cylindrical semimartingales and the theory of real-valued stochastic integration introduced by the author in a previous work [Electron. J. Probab., Volume 26, paper no. 147, 2021]. We show various properties of the stochastic integral; in particular we study continuity of the integral mapping for integrands and for integrators, we prove a Riemman representation formula, and we introduce sufficient conditions for the stochastic integral to be a good integrator. Finally, we apply our theory to show an extension of \"{U}st\"{u}nel's version of It^{o}'s formula in the spaces of distributions and of tempered distributions.