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Modern Stochastics: Theory and Applications


On distributions of exponential functionals of the processes with independent increments
Volume 7, Issue 3 (2020), pp. 291–313
Pub. online: 8 September 2020      Type: Research Article      Open accessOpen Access
Received
3 February 2020
Revised
31 July 2020
Accepted
31 July 2020
Published
8 September 2020

Abstract

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.

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© 2020 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Process with independent increments exponential functional Kolmogorov-type equation smoothness of the density

MSC2010
60G51 91G80

Funding
This research was partially supported by Defimath project of the Research Federation of “Mathématiques des Pays de la Loire” and by PANORisk project “Pays de la Loire” region.

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