an:06467396
Zbl 1327.60103
Benth, Fred Espen; Kr??hner, Paul
Integrability of multivariate subordinated L??vy processes in Hilbert space
EN
Stochastics 87, No. 3, 458-476 (2015).
00344969
2015
j
60G51 60G15 60G52
L??vy processes; multivariate subordination; Hilbert space; integrability; infinite variate normal inverse Gaussian process
Summary: We investigate multivariate subordination of L??vy processes, which was first introduced by \textit{O.E. Barndorff-Nielsen} et al. [Adv. Appl. Probab. 46, No. 3, 719--745 (2014; Zbl 1304.91213)], in a Hilbert space valued setting, which has been introduced in [\textit{V. P??rez-Abreu} and \textit{A. Rocha-Arteaga}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8, No. 1, 32--54 (2005; Zbl 1067.60052)]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions on the L??vy process and the subordinator. As an application of our theory, we construct explicitly some Hilbert space valued versions of L??vy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian L??vy process in a Hilbert space. The resulting process has the property that at each time all of its finite-dimensional projections are multivariate normally inverse Gaussian distributed as introduced in [\textit{T. H. Rydberg}, Commun. Stat., Stochastic Models 13, No. 4, 887--910 (1997; Zbl 0899.60036)].
Zbl 1304.91213; Zbl 1067.60052; Zbl 0899.60036