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Geometric stable processes and related fractional differential equations. (English) Zbl 1321.60101
Summary: We are interested in the differential equations satisfied by the density of the geometric stable processes \(\{\mathcal{G}_{\alpha }^{\beta}(t):t\geq 0\} \), with stability index \(\alpha \in (0,2]\) and symmetry parameter \(\beta \in [-1,1]\), 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 \(\mathcal{G}_{\alpha }^{\beta }(t)\). 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.

60G52 Stable stochastic processes
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
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