# On continuity properties of the law of integrals of Lévy processes - Zurich Open Repository and Archive

Bertoin, Jean; Lindner, A; Maller, R (2008). On continuity properties of the law of integrals of Lévy processes. Lecture Notes in Mathematics, 1934:137-159.

## Abstract

Let $(\xi,\eta)$ be a bivariate L\'evy 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 L\'evy 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 $\xi$.

## Abstract

Let $(\xi,\eta)$ be a bivariate L\'evy 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 L\'evy 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 $\xi$.

## Citations

22 citations in Web of Science®
18 citations in Scopus®

## Altmetrics

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2008 29 May 2013 15:19 05 Apr 2016 16:47 Springer 0075-8434 https://doi.org/10.1007/978-3-540-77913-1_6 http://www.ams.org/mathscinet-getitem?mr=2483729http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1180.60042

Preview
Content: Submitted Version
Language: English
Filetype: PDF
Size: 287kB
View at publisher

## TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.