Pseudo-differential operators and related additive geometric stable processes

TL;DR

This paper introduces time-inhomogeneous geometric stable processes using fractional pseudo-differential operators with time-dependent parameters, extending classical stable processes to a more general, non-stationary setting.

Contribution

It develops a novel framework for time-inhomogeneous geometric stable processes via fractional pseudo-differential operators with explicit time-dependent Lévy measures.

Findings

Defined new class of time-inhomogeneous processes

Expressed Lévy measures using Mittag-Leffler and H-functions

Extended stable process theory to non-stationary case

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-Leffler functions or $H$-functions with time-dependent parameters.