Abstract
We introduce a stochastic integral with respect to cylindrical Lévy processes with finite p-th weak moment for pin [1,2]. The space of integrands consists of p-summing operators between Banach spaces of martingale type p. We apply the developed integration theory to establish the existence of a solution for a stochastic evolution equation driven by a cylindrical Lévy process.
Highlights
Cylindrical Lévy processes are a natural generalisation of cylindrical Brownian motions to the non-Gaussian setting, and they can serve as a model of random perturbation of partial differential equations or other dynamical systems
Since conventional approaches to stochastic integration rely on either stopping time arguments or a semimartingale decomposition in the one or other form, a completely novel method for stochastic integration has been introduced in the work [12] by one of us with Jakubowski
In order to provide a control of the stochastic integral, we develop a theory of stochastic integration for random integrands with respect to cylindrical Lévy processes with finite p-th weak moments for p ∈ [1, 2] in this work
Summary
Cylindrical Lévy processes are a natural generalisation of cylindrical Brownian motions to the non-Gaussian setting, and they can serve as a model of random perturbation of partial differential equations or other dynamical systems. In order to provide a control of the stochastic integral, we develop a theory of stochastic integration for random integrands with respect to cylindrical Lévy processes with finite p-th weak moments for p ∈ [1, 2] in this work. The class of p-summing operators coincides with the class of Hilbert–Schmidt operators in Hilbert spaces, and as such, the aforementioned space of admissible integrands is a natural generalisation of the integration theory in Hilbert spaces with respect to genuine Lévy process in, for example, [18] In typical applications such as the heat equation, the p-summing norm of the operators appearing in the equation can be explicitly estimated, see [3].
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call