Stochastic integration with respect to cylindrical semimartingales

TL;DR

This paper develops a comprehensive theory of stochastic integration for cylindrical semimartingales in locally convex spaces, including properties, equations, and applications to stochastic evolution equations and real-valued semimartingales.

Contribution

It introduces a novel stochastic integral framework for cylindrical semimartingales in locally convex spaces, expanding existing theories and providing tools for stochastic evolution equations.

Findings

Established a stochastic integral with cylindrical semimartingales.

Proved properties like integration by parts and Fubini theorem.

Derived conditions for solutions to stochastic evolution equations.

Abstract

In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on 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, 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 $\Phi'$ of $\Phi$. In the last part of this article we apply our theory to define stochastic integrals with respect to a sequence of real-valued semimartingales.