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arXiv · 1402.6559

On the range of exponential functionals of Lévy processes

Anita Behme·Alexander Lindner·Makoto Maejima
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Abstract

We characterize the support of the law of the exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$ of two one-dimensional independent Lévy processes $ξ$ and $η$. Further, we study the range of the mapping $Φ_ξ$ for a fixed Lévy process $ξ$, which maps the law of $η_1$ to the law of the corresponding exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$. It is shown that the range of this mapping is closed under weak convergence and in the special case of positive distributions several characterizations of laws in the range are given.

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Anita Behme, Alexander Lindner, Makoto Maejima. 2014-10-13. On the range of exponential functionals of Lévy processes. https://arxiv.org/abs/1402.6559

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