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On continuity properties of the law of integrals of Lévy processes. (English) Zbl 1180.60042

Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLI. Some papers are selected contributions of the seminars in Nancy 2005 and Luminy 2006. Berlin: Springer (ISBN 978-3-540-77912-4/pbk). Lecture Notes in Mathematics 1934, 137-159 (2008).
The authors consider a bivariate Lévy process \((\xi,\eta)\) such that the integral \(\int_0^\infty e^{-\xi_{t-}}d\eta_t\) converges almost surely. They characterise, in terms of their Lévy measures, those \((\xi,\eta)\) for which the law of this integral has atoms. They consider then almost surely convergent integrals of type \(I:=\int^\infty_0 g(\xi_t)dy_t\), where \(g\) is a deterministic function, and \(y\) is an almost surely strictly increasing stochastic process, independent of the Lévy process \(\xi\). In the particular case of \(y_t\equiv t\), they give sufficient conditions ensuring that \(I\) has no atoms, and under further conditions, that \(I\) is absolutely continuous. These results are extended to some more general processes \(y\).
For the entire collection see [Zbl 1140.60002].

MSC:

60G51 Processes with independent increments; Lévy processes
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