Geometric Stable processes and related fractional differential equations

TL;DR

This paper investigates the differential equations governing the densities of Geometric Stable processes, revealing their connection to fractional differential equations and related well-known stochastic processes.

Contribution

It derives the space-fractional differential equations for Geometric Stable processes using their representation as compositions with Gamma subordinators, extending understanding of their mathematical properties.

Findings

Derived differential equations for Geometric Stable process densities.

Connected special cases to Variance Gamma and Brownian first passage processes.

Established the governing equations using shift operators.

Abstract

We are interested in the differential equations satisfied by the density of the Geometric Stable processes $\mathcal{G}_{\alpha}^{\beta}=\left\{\mathcal{G}_{\alpha}^{\beta}(t);t\geq 0\right\} $, with stability \ index $% \alpha \in (0,2]$ and asymmetry parameter $\beta \in \lbrack -1,1]$, both in the univariate and in the multivariate cases. We resort to their representation as compositions of stable processes with an independent Gamma subordinator. As a preliminary result, we prove that the latter is governed by a differential equation expressed by means of the shift operator. As a consequence, we obtain the space-fractional equation satisfied by the density of $\mathcal{G}_{\alpha}^{\beta}.$ For some particular values of $% \alpha $ and $\beta ,$ we get some interesting results linked to well-known processes, such as the Variance Gamma process and the first passage time of the Brownian motion.