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Selfdecomposable Fields

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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 L\'evy measure and the associated L\'evy-It\^o 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 functions) for a Volterra field driven by a L\'evy 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 L\'evy processes, give the L\'evy-It\^o representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of L\'evy semistationary processes with a Gamma kernel and Ornstein-Uhlenbeck processes.
Original languageEnglish
Number of pages33
Publication statusPublished - 5 Feb 2015

    Research areas

  • math.PR, 60E07, 60G51, 60G60

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