(Non)Differentiability and Asymptotics for Potential Densities of Subordinators

Leif Döring (Technische Universität Berlin)
Mladen Savov (Oxford University)

Abstract


For subordinators with positive drift we extend recent results on the structure of the potential measures and the renewal densities. Applying Fourier analysis a new representation of the potential densities is derived from which we deduce asymptotic results and show how the atoms of the Lévy measure translate into points of (non)differentiability of the potential densities.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 470-503

Publication Date: March 17, 2011

DOI: 10.1214/EJP.v16-860

References

  1. F. Avram, A.E. Kyprianou and M.R. Pistorius. Exit problems for spectrally negative Leevy processes and applications to (Canadized) Russian options. Annals of Applied Probability, 2005. Math. Review 2005c:60053
  2. J. Bertoin. Levy Processes. Cambridge Tracts in Mathematics 121 (1998) Math. Review 98e:60117
  3. S.K. Chiu and C. Yin. Passage times for a spectrally negative Levy process with applications to risk theory. to appear in Bernouilli. Math. Review number not available.
  4. T. Chan, E.A. Kyprianou and M. Savov. Smoothness of Scale Functions for Spectrally negative Levy Processes to appear in Probability Theory and Related Fields. Math. Review number not available.
  5. R. Doney and M. Savov. Right-inverses of Levy Processes. Annals of Probability 4 (2010), 1390-1400. Math. Review number not available.
  6. K. van Harn and F.W. Steutel. Stationarity of delayed subordinators. Stoch. Models. 17 (2001), 369-374 Math. Review 2002g:60050
  7. H. Kesten. Hitting probabilities of single points for processes with stationary independent increments Mem. of the American Mathematical Society 93 (1969). Math. Review 42 #6940
  8. J.F.C. Kingman. Regenerative Phenomena. Wiley Series in Probability and Mathematical Statistics (1972). Math. Review 50 #3353
  9. J.F.C. Kingman. The stochastic theory of regenerative events. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 180-224. Math. Review 32 #1758"
  10. F. Spitzer. Principles of Random Walk. Graduate Texts in Mathematics, Springer 34 (1976) Math. Review 52 #9383"
  11. T. W. Koerner. Fourier analysis. Cambridge University Press, Cambridge (1989). Math. Review 90j:42001
  12. R. Song and Z. Vondracek. Potential theory of special subordinators and subordinate killed stable processes. J. Theor. Prob. 19 (2006), 817-847. Math. Review 2008g:60237"
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.