On distributions of exponential functionals of the processes with independent increments

2018/04/19 by Vostrikova, L. · 1 citation
#60G51 #91G80 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1804.07069

Abstract

The aim of this paper is to study the laws of the exponential functionals of the processes X with independent increments, namely I_t= ∫ ₀^texp(-X_s)ds, t≥ 0, and also I_∞= ∫ ₀^∞exp(-X_s)ds. Under suitable conditions we derive the integro-differential equations for the density of I_t and I_∞. We give sufficient conditions for the existence of smooth density of the laws of these functionals. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.

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