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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.

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