### On distributions of exponential functionals of the processes with independent increments

Published: 8 September 2020
by  VTeX
Modern Stochastics: Theory and Applications pp 291-313; doi:10.15559/20-vmsta159

Abstract: Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: On distributions of exponential functionals of the processes with independent increments, Authors: Lioudmila Vostrikova , The aim of this paper is to study the laws of exponential functionals of the processes $X={({X_{s}})_{s\ge 0}}$ with independent increments, namely ${I_{t}}={\int _{0}^{t}}\exp (-{X_{s}})ds,\hspace{0.1667em}\hspace{0.1667em}t\ge 0,$ and also ${I_{\infty }}={\int _{0}^{\infty }}\exp (-{X_{s}})ds.$ Under suitable conditions, the integro-differential equations for the density of ${I_{t}}$ and ${I_{\infty }}$ are derived. Sufficient conditions are derived for the existence of a smooth density of the laws of these functionals with respect to the Lebesgue measure. In the particular case of Lévy processes these equations can be simplified and, in a number of cases, solved explicitly.
Keywords: publishing / Differential / Independent Increments / Hspace / Exponential Functionals / 0.1667em

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