Subordination of Hilbert space valued L\'evy processes

TL;DR

This paper extends the concept of subordination of Le9vy processes to Hilbert space valued processes, providing explicit characterizations and constructing examples like the Hilbert space normal inverse Gaussian process.

Contribution

It generalizes multivariate subordination to infinite-dimensional Hilbert spaces and constructs new Hilbert space valued Le9vy processes, including a Hilbert space normal inverse Gaussian process.

Findings

Explicit characterization of Hilbert space valued Le9vy processes

Conditions for integrability and martingale properties

Construction of Hilbert space valued normal inverse Gaussian process

Abstract

We generalise multivariate subordination of L\'evy processes as introduced by Barndorff-Nielsen, Pedersen, and Sato to Hilbert space valued L\'evy processes. The processes are explicitly characterised and conditions for integrability and martingale properties are derived under various assumptions of the L\'evy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of L\'evy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian L\'evy process in Hilbert space as a subordination of a Hilbert space valued Wiener process by an inverse Gaussian L\'evy process. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg(97).