Paul Kruehner: Subordination of Hilbert space valued Lévy processes

We generalise multivariate subordination of Lévy processes as introduced by Barndorff-Nielsen, Pedersen, and Sato [1] to Hilbert space valued Lévy processes. The processes are explicitly characterised and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space as a subordination of a Hilbert space valued Wiener process by an inverse Gaussian Lévy 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 [2].

References
[1] O. Barndorff-Nielsen, J. Pedersen, and K. Sato. Multivariate subordination, self-decomposability and stability. Advances in Applied Probability, 33(1):160-187, 2001

[2] T. Rydberg. The normal inverse Gaussian Levy process: simulation and approximation. Communications in Statistics. Stochastic Models, 13:887-910, 1997