Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces: An L² approach

Abstract

In this work stochastic integration with respect to cylindrical Lévy processes with weak second moments is introduced. It is well known that a deterministic Hilbert-Schmidt operator radonifies a cylindrical random variable, i.e. it maps a cylindrical random variable to a classical Hilbert space valued random variable. Our approach is based on a generalisation of this result to the radonification of the cylindrical increments of a cylindrical Lévy process by random Hilbert-Schmidt operators. This generalisation enables us to introduce a Hilbert space valued random variable as the stochastic integral of a predictable stochastic process with respect to a cylindrical Lévy process. We finish this work by deriving an Itô isometry and by considering shortly stochastic partial differential equations driven by cylindrical Lévy processes.

Weak Convergence of Finite Element Approximations of Linear Stochastic Evolution Equations with Additive Lévy Noise

“…As a consequence of Douglas' theorem as stated in [38,Appendix A.4], compare also [41, Corollary C.0.6], the reproducing kernel Hilbert space of L has the alternative representation [38,Proposition 7.7]. The unique continuous extensions of the linear mappings U * 1 ∋ x → L(t), x ∈ L 2 (P), t 0, to the larger space U * determine a 2-cylindrical U-process in the sense of [35], compare also [1], [43], [44]. Remark 2.2.…”

Stochastic integration with respect to cylindrical Lévy processes

“…In Métivier and Pellaumail [15], the construction is extended to cylindrical local martingales. For the special case of a cylindrical Lévy process with finite weak second moments one can follow a classical Itô aproach to define the stochastic integral for random integrands; see Riedle [23].…”

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.…”