Stochastic integration with respect to cylindrical semimartingales

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 ${\Phi }^{\prime }$ 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.

References in corpus (6)

• Cylindrical continuous martingales and stochastic integration in infinite dimensions
• Stochastic Integration and Stochastic PDEs Driven by Jumps on the Dual of a Nuclear Space
• Existence of Continuous and Càdlàg Versions for Cylindrical Processes in the Dual of a Nuclear Space
• Tensor products in the category of topological vector spaces are not associative
• Tightness and Weak Convergence of Probabilities on the Skorokhod Space on the Dual of a Nuclear Space and Applications
• Semimartingales on Duals of Nuclear Spaces

Cited by in corpus (2)

• Convergence Uniform on Compacts in Probability with Applications to Stochastic Analysis in Duals of Nuclear Spaces
• Tightness and weak convergence in the topology of local uniform convergence for stochastic processes in the dual of a nuclear space