Selfdecomposable Fields

Abstract

In the present paper, we study selfdecomposability of random fields, as defined directly rather than in terms of finite-dimensional distributions. The main tools in our analysis are the master Lévy measure and the associated Lévy-Itô representation. We give the dilation criterion for selfdecomposability analogous to the classical one. Next, we give necessary and sufficient conditions (in terms of the kernel function) for a Volterra field driven by a Lévy basis to be selfdecomposable. In this context, we also study the so-called Urbanik classes of random fields. We follow this with the study of existence and selfdecomposability of integrated Volterra fields. Finally, we introduce infinitely divisible field-valued Lévy processes, give the Lévy-Itô representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of Lévy semistationary processes with a Gamma kernel and Ornstein-Uhlenbeck processes.

On the class of distributions of subordinated Lévy processes and bases

“…See also Brockwell et al (2013). In the absence of a drift and a stochastic volatility component, an LSS process is strictly stationary and infinitely divisible in the sense of BarndorffNielsen et al (2006) and Barndorff-Nielsen et al (2015).…”

Some Recent Developments in Ambit Stochastics

“…In particular, we have discussed the existence of the ambit fields driven by metatime changed Lévy bases, selfdecomposability of random fields [13], applications of BSS processes in the modelling of turbulent time series [35] and new results on the distributional collapse in financial data. Some of the topics not mentioned here but also under development are the integration theory with respect to time-changed volatility modulated Lévy bases [7]; integration with respect to volatility Gaussian processes in the White Noise Analysis setting in the spirit of [34] and extending [6]; modelling of multidimensional turbulence based on ambit fields; and in-depth study of parsimony and universality in BSS and LSS processes motivated by some of the discussions in the present paper.…”

“…Therefore, if the Lévy basis L in (17) is chosen so that L(A) follows a Gumbel distribution with b = 2, then exp(L(A(t))) will be infinitely divisible. For a general discussion of selfdecomposable fields we refer to [13]. See also [32] which provides a survey of when a selfdecomposable random variable can be represented as a stochastic integral, like in (12).…”

“…ω. If the temporal process is selfdecomposable then, subject to a further weak condition (see [13]), such a field can be constructed.…”

“…It has been shown in [13] that, in this case, provided g is integrable with respect to the Lebesgue measure, as well as to L, and if the Fourier transform of g is nonvanishing, then X , as a process, is selfdecomposable if and only if L is selfdecomposable. When that holds we may, as above, construct a selfdecomposable field X (x, t) with X (x, · ) ∼ X ( · ) for every x ∈ R and X ( · , t) of OU type for every t ∈ R.…”