The Itô Integral for a Certain Class of Lévy Processes and Its Application to Stochastic Partial Differential Equations

Stochastic integration with respect to a Wiener process in Banach spaces have been considered by several authors (Brzeźniak [5], Dettweiler [13], Neidhart [22] and Van Neerven, Veraar and Weiss [30]). Similarly, stochastic integration with respect to Lévy processes in Banach spaces is of increasing interest. So, these articles [2, 3, 7, 15, 24, 25] are devoted to this topic. Nevertheless, in the articles above the focus was on Lévy processes of finite p-variation, where p ∈ (1, 2]. In this paper, our focus will be on the Itô integral driven by Lévy processes of finite p-variation, p ∈ (0, 1] an issue in which Laurent Schwartz [27] was interested. We will show under which conditions the Itô integral is well defined even for 0 < p ≤ 1 and will present some inequalities satisfied by the Itô integral. Additionally, we apply our result to SPDEs. To illustrate the consequences of our results, let us state the following example. Let O be a bounded domain in R with smooth boundary and A be an infinitesimal generator of an analytic semigroup on L(O), 1 ≤ q < ∞. Let (Z,Z) be a measurable space and η be a time homogeneous Poisson random measure defined on Z having as intensity measure a finite Lévy measure ν on Z. Then it is a sufficient condition for the equation