Additive Geometric Stable Processes and Related Pseudo-Differential Operators

L. Beghin, C. Ricciuti

2019, v.25, Issue 3

ABSTRACT

Additive processes are obtained from L\'{e}vy
ones by relaxing the condition of stationary increments, hence they are spatially (but not
temporally) homogeneous. By analogy with the case of time-homogeneous Markov processes,
one can define an infinitesimal generator, which is, of course,
a time-dependent operator. Additive versions of stable and Gamma processes
have been considered in the literature. We
introduce here time-inhomogeneous generalizations of the well-known geometric
stable process, defined by means of time-dependent versions of fractional
pseudo-differential operators of logarithmic type. The local L\'{e}vy measures are
expressed in terms of Mittag\tire Leffler functions or $H$-functions with time-dependent
parameters. This article also presents some results
about propagators related to additive processes.

Keywords: time-inhomogeneous processes; geometric stable distributions; fractional logarithmic operator; additive processes; variance gamma process

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