Stochastic integration for Lévy processes with values in Banach spaces

Abstract

A stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces nor on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated to the integrand. The integral is used to prove a Lévy-Itô decomposition for Banach space valued Lévy processes and to study existence and uniqueness of solutions of stochastic Cauchy problems driven by Lévy processes.

Cylindrical Lévy processes in Banach spaces

“…1 C (ΔX(s)) for C ∈ B(U ); see [18]. Obviously, the cylindrical Lévy process L is decomposed according to…”

“…, C * a n ; B)). Definition 5.2 extends the concept of a solution of stochastic Cauchy problems on a Hilbert space or a Banach space driven by a Lévy process to the cylindrical situation; see [6] for the case of a Hilbert space and [18] for the case of a Banach space. The following example illustrates this generalisation.…”

“…for all a ∈ dom(A * ) and t 0. These kinds of equations in Hilbert spaces are considered in [2,15] and in Banach spaces in [18]. If we define a cylindrical Lévy process (M 2 (t) : t 0) by…”

Maximal Inequalities of the Itô Integral with Respect to Poisson Random Measures or Lévy Processes on Banach Spaces

“…Stochastic integration in Banach spaces with respect to Lévy processes is not that well established and of increasing interest. For example, the works [2,4,8,24,28] are about stochastic integration in Banach spaces. In this paper, our focus will be on maximal inequalities satisfied by the stochastic integral driven by Lévy processes respective Poisson random measures.…”

Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

“…Now, we will show that for each φ ∈ Φ, the real-valued process X[φ] = {X t [φ]} t≥0 is a Wiener process. We proceed in a similar way as in the proof of Proposition 6.2 in [25], where a similar result for the separable Banach space case is considered.…”