Stochastic Integration with Respect to Cylindrical Lévy Processes by p-Summing Operators

Abstract

We introduce a stochastic integral with respect to cylindrical Lévy processes with finite p-th weak moment for p ∈ [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.

Stochastic wave equation with Lévy white noise

“…We refer the reader to the excellent monograph Peszat and Zabczyk (2007) and the references therein. The recent papers Jakubowski and Riedle (2017); Kosmala and Riedle (2021) contain significant advances on stochastic integration with respect to cylindrical Lévy noise, while Griffiths and Riedle (2021) gives a comparison between the cylindrical approach and the random field approach, which complements the similar comparison that was done by Dalang and Quer-Sardanyons (2011) in the Gaussian case.…”

Stochastic integration with respect to cylindrical semimartingales

“…In the context of Banach spaces, there are several theories of (vector-valued) stochastic integration with respect to specific classes of cylindrical semimartingales. For example, in Hilbert spaces we have stochastic integration with respect to cylindrical square integrable martingales and cylindrical Lévy processes [6,21,31], in separable Banach spaces we have stochastic integration with respect to cylindrical Brownian motion [23], in Banach spaces of martingale type p ∈ [1,2] with respect to cylindrical Lévy processes of order p [26], and in UMD Banach spaces we have stochastic integration with respect to cylindrical Brownian motion [53] and with respect to continuous cylindrical local martingales [54].…”

“…[5,6,23,53]). However, in recent years there has been an increasing interest in the usage of other classes of cylindrical semimartingales as such driving noise; we can cite, for example, the cylindrical Lévy processes [21,26,28,38,43], cylindrical martingalevalued measures [13] and cylindrical continuous local martingales [33,34,54]. To the extent of our knowledge, none of these works considers stochastic integration with respect to general cylindrical semimartingales in a general locally convex space.…”