Some properties of exponential integrals of Lévy processes and examples

2006/06/04 by Kondo, Hitoshi, Maejima, Makoto, Sato, Ken-iti
#60E07 #60G51 #60H05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.math/0606084

Abstract

The improper stochastic integral Z=∫₀^∞-exp(-X_s-)dY_s is studied, where \(X_t, Y_t), t \geqslant 0 \ is a Lévy process on \mathbb R ^1+d with \X_t \ and \Y_t \ being \mathbb R-valued and \mathbb R ^d-valued, respectively. The condition for existence and finiteness of Z is given and then the law \mathcal L(Z) of Z is considered. Some sufficient conditions for \mathcal L(Z) to be selfdecomposable and some sufficient conditions for \mathcal L(Z) to be non-selfdecomposable but semi-selfdecomposable are given. Attention is paid to the case where d=1, \X_t\ is a Poisson process, and \X_t\ and \Y_t\ are independent. An example of Z of type G with selfdecomposable mixing distribution is given.

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