Convergence Uniform on Compacts in Probability with Applications to Stochastic Analysis in Duals of Nuclear Spaces

Abstract

Let ${\Phi }^{\prime }$ denote the strong dual of a nuclear space $\Phi$. In this paper we introduce sufficient conditions for the convergence uniform on compacts in probability for a sequence of ${\Phi }^{\prime }$-valued processes with continuous or càdlàg paths. We illustrate the usefulness of our results by considering two applications to stochastic analysis. First, we introduce a topology on the space of ${\Phi }^{\prime }$-valued semimartingales which are good integrators and show that this topology is complete and that the stochastic integral mapping is continuous on the integrators. Second, we introduce sufficient conditions for the convergence uniform on compacts in probability of the solutions to a sequence of linear stochastic evolution equations driven by semimartingale noise.

The original submission has been split into two separated works. UCP convergence covered in this work, and weak uniform convergence is studied in a separated work

References in corpus (7)

• 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
• Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space
• Semimartingales on Duals of Nuclear Spaces
• Stochastic integration with respect to cylindrical semimartingales
• Time regularity of stochastic convolutions and stochastic evolution equations in duals of nuclear spaces
• Almost Sure Uniform Convergence of Stochastic Processes in the Dual of a Nuclear Space

Cited by in corpus (1)

• Tightness and weak convergence in the topology of local uniform convergence for stochastic processes in the dual of a nuclear space