Geometric stable processes and related fractional differential equations

Luisa Beghin

doi:10.1214/ecp.v19-2771

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 symmetry 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.

Highlights

Introduction and notationThe Geometric Stable random variable (r.v.) is usually defined through its characteristic function: let Gαβ be a GS r.v. with stability index α ∈ (0, 2], symmetry parameter β ∈ [−1, 1], position parameter μ ∈ R, scale parameter σ > 0, ΦGαβ (θ) := EeiθGαβ = +1 σα |θ|α ωα,β (θ) −, iμθ θ ∈ R, (1.1) where ωα,β(θ) :=
We are interested in the differential equations satisfied by the density of the Geometric Stable processes Gαβ(t); t ≥ 0, with stability index α ∈
We resort to their representation as compositions of stable processes with an independent Gamma subordinator

Summary

Introduction and notation

The Geometric Stable (hereafter GS) random variable (r.v.) is usually defined through its characteristic function: let Gαβ be a GS r.v. with stability index α ∈ In the positively asymmetric case, Gαβ(t) reduces to a GS subordinator, which is used in particular as random time argument of the subordinated Brownian motion, via successive iterations (see [6], [27]) for α = 1/2, we can obtain, as a corollary, the fractional equation satisfied by the density g1/2(x, t) of the first-passage time of a standard Brownian motion B(t) through a Gamma distributed random barrier, i.e. We prove in Proposition 7 below that, in the special case where L(t) coincides with a symmetric α-stable process, the generator AY is given by the following fractional operator: Pkα,xu(x) :=. The study of the generators A+Y , A−Y (which, in the stable case, should be fractional as well) is left as an important open issue for future research

Preliminary results

Univariate GS process

Symmetric GS process

GS subordinator

Multivariate GS process