On Continuity Properties of the Law of Integrals of Lévy Processes

Jean Bertoin (1) , Alexander Lindner (2) , Ross Maller (3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 M4-TUM - Centre for Mathematical Sciences
3 CMA - Centre for Mathematical Analysis, and School of Finance

Let $(\xi,\eta)$ be a bivariate Lévy process such that the integral $\int_0^\infty e^{-\xi_{t-}} \, d\eta_t$ converges almost surely. We characterise, in terms of their \LL measures, those Lévy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form $I:=\int_0^\infty g(\xi_t) \, dt$, where $g$ is a deterministic function. We give sufficient conditions ensuring that $I$ has no atoms, and under further conditions derive that $I$ has a Lebesgue density. The results are also extended to certain integrals of the form $\int_0^\infty g(\xi_t) \, dY_t$, where $Y$ is an almost surely strictly increasing stochastic process, independent of $\xi$.

Mots clés: Integral of a Lévy process; continuous distributions; density

Domaines: Probabilités [math.PR]

Jean Bertoin, Alexander Lindner, Ross Maller. On Continuity Properties of the Law of Integrals of Lévy Processes. 2006. ⟨hal-00023383⟩