Abstract
Let (ξ, η) 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 Lévy 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 := ∫ ∞0 g(ξ 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 ∫ ∞0 g(ξ t ) dY t , where Y is an almost surely strictly increasing stochastic process, independent of ξ.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bertoin, J. (1996) Lévy Processes. Cambridge University Press, Cambridge.
Carmona, P., Petit, F. and Yor, M. (1997) On the distribution and asymptotic results for exponential functionals of Lévy processes. In: Exponential Functionals and Principal Values Related to Brownian Motion (M. Yor, Ed.), pp. 73–130. Biblioteca de la Revista Matemática Iberoamericana.
Carmona, P., Petit, F. and Yor, M. (2001) Exponential functionals of Lévy processes. In: Lévy Processes, Theory and Applications (O.E. Barndorff-Nielsen, T. Mikosch and S. Resnick, Eds.), pp. 41–55. Birkhäuser, Boston.
Davydov, Yu. A. (1987) Absolute continuity of the images of measures. J. Sov. Math. 36, 468–473.
Davydov, Yu.A., Lifshits, M.A. and Smorodina, N.V. (1998) Local Properties of Distributions of Stochastic Functionals. Translations Math. Monographs 173, AMS, Providence.
Dufresne, D. (1990) The distribution of a perpetuity, with application to risk theory and pension funding. Scand. Actuar. J. 9, 39–79.
Erickson, K.B. and Maller, R.A. (2004) Generalised Ornstein-Uhlenbeck processes and the convergence of Lévy integrals. In: Séminaire de Probabilités XXXVIII (M. Émery, M. Ledoux, M. Yor, Eds.), pp. 70–94. Lect. Notes Math. 1857, Springer, Berlin.
Erickson, K.B. and Maller, R.A. (2007) Finiteness of integrals of functions of Lévy processes. Proc. London Math. Soc. 94, 386–420.
Gjessing, H.K. and Paulsen, J. (1997) Present value distributions with applications to ruin theory and stochastic equations. Stoch. Proc. Appl. 71, 123–144.
Grincevičius, A. (1980) Products of random affine transformations. Lithuanian Math. J. 20 no. 4, 279–282. Translated from Litovsk. Mat. Sb. 20 no. 4, 49–53, 209 (Russian).
de Haan, L. and Karandikar, R.L. (1989) Embedding a stochastic difference equation into a continuous-time process. Stoch. Proc. Appl. 32, 225–235.
Haas, B. (2003). Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106, 245–277.
Kesten, H. (1969) Hitting probabilities of single points for processes with stationary independent increments. Memoirs Amer. Math. Soc. 93.
Klüppelberg, C., Lindner, A. and Maller, R. (2004) A continuous time GARCH process driven by Lévy process: stationarity and second order behaviour. J. Appl. Probab. 41, 601–622.
Kondo, H., Maejima, M. and Sato, K. (2006) Some properties of exponential integrals of Lévy processes and examples. Electr. Comm. Probab. 11, 291–303.
Lifshits, M.A. (1984) An application of the stratification method to the study of functionals of processes with independent increments. Theory Probab. Appl. 29, 753–764.
Lindner, A. and Maller, R. (2005) Lévy integrals and the stationarity of generalised Orustein-Uhlenbeck processes. Stoch. Proc. Appl. 115, 1701–1722.
Nourdin, I. and Simon, T. (2006) On the absolute continuity of Lévy processes with drift. Ann. Probab. 34, 1035–1051.
Paulsen, J. (1993) Risk theory in a stochastic economic environment. Stoch. Proc. Appl. 46, 327–361.
Protter, P.E. (2004) Stochastic Integration and Differential Equations. 2nd edition. Springer, Berlin.
Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.
Shtatland, E.S. (1965) On local properties of processes with independent increments. Theor. Prob. Appl. 10, 317–322.
Yor, M. (2001) Exponential Functionals of Brownian Motion and Related Processes. Springer, New York.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bertoin, J., Lindner, A., Maller, R. (2008). On Continuity Properties of the Law of Integrals of Lévy Processes. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol 1934. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77913-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-77913-1_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77912-4
Online ISBN: 978-3-540-77913-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)