Ornstein-Uhlenbeck Processes Driven by Cylindrical Lévy Processes

Abstract

In this article we introduce a theory of integration for deterministic, operator-valued integrands with respect to cylindrical Lévy processes in separable Banach spaces. Here, a cylindrical Lévy process is understood in the classical framework of cylindrical random variables and cylindrical measures, and thus, it can be considered as a natural generalisation of cylindrical Wiener processes or white noises. Depending on the underlying Banach space, we provide necessary and/or sufficient conditions for a function to be integrable. In the last part, the developed theory is applied to define Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes and several examples are considered.

Stochastic integration with respect to cylindrical Lévy processes

“…Furthermore, Lemma 4.3 implies for every β ∈ Ê and v ∈ V that E exp(iβ ∆ n (t), v )| F n,kn = exp (t − t n,kn )S(βΦ * n,kn v) P -a.s., (5.52) where S : U → denotes the cylindrical Lévy symbol of L. The set {Φ * n,kn (ω)v : n ∈ AE} is uniformly bounded as Φ n,kn (ω) is in the closure of {Ψ(t)(ω) : t ∈ [0, T ]} for every ω ∈ Ω and the closure of the latter is compact by Proposition 1.1 in [9]. As S maps bounded sets to bounded sets by Lemma 3.2 in [24], we conclude from (5.51) and (5.52) that lim n→∞ E exp(iβ ∆ n (t), v )| F n,kn = 0 P -a.s. for every β ∈ Ê.…”

“…Some specific examples and their constructions of cylindrical Lévy processes are presented in the work [24] by Riedle. Linear and semi-linear stochastic partial differential equations perturbed by an additive noise which is modelled by various but specific examples of cylindrical Lévy processes can be found for example in the works Brzeźniak and Zabczyk [3], Peszat and Zabczyk [20], and Priola and Zabczyk [21]. However, modelling an arbitrary perturbation of a general stochastic partial differential equations beyond the purely additive case requires a theory of stochastic integration of random integrands with respect to cylindrical Lévy processes.…”

Time Regularity of Generalized Ornstein–Uhlenbeck Processes with Lévy Noises in Hilbert Spaces

“…In the very recent paper [19], Riedle systemically researched OU processes driven by cylindrical Lévy processes in infinite dimensional space. His Theorem 8.4 obtains a result similar to Theorem 2.3 (2).…”

Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

“…This refers to finding a mapping into a possibly larger space that transforms the cylindrical object into a bona fide one. A important theorem [26] states that if (M(t), t ≥ 0) is a cylindrical semimartingale in H, then there exists a Hilbert-Schmidt operator T on H and a semimartingale (N(t), t ≥ 0) so that the real-valued processes (M(t)(T * x), t ≥ 0) and ( N(t), x , t ≥ 0) are indistinguishable, for all x ∈ H. In Theorem 5.10 of [47], conditions are found for a suitable deterministic function f so that its cylindrical stochastic integral t 0 f (s)dL(s) has the property of stochastic integrability in that there exists a random variable I t such that for all u ∈ H, I t , u = t 0 f (s)dL(s) (u) (see also Corollary 4.4 in [46]). An alternative approach has been developed in a series of papers that focus on the specific class of cylindrical Lévy processes defined by (7.23), with the assumption that the L n 's are i.i.d.…”

“…, X(t)a n ) is a Lévy process in R n . As shown in Lemma 4.2 of [47], the representation (7.23) gives a specific class of examples of this more general notion when we identify L(t) therein with the mapping which sends each a ∈ H to the random variable ∞ n=1 β n L n (t) e n , a , where the β n 's are chosen to ensure the series converges for all t ≥ 0. In particular, we obtain a cylindrical Lévy process in the case where the L n 's are i.i.d.…”