Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-23T02:52:58.812Z Has data issue: false hasContentIssue false
  • English
  • Français

Evaluating first-passage probabilities for spectrally one-sided Lévy processes

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

L. C. G. Rogers
L. C. G. Rogers*
Affiliation:
University of Bath
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Fast stable methods for inverting multidimensional Laplace transforms have been developed in recent years by Abate, Whitt and others. We use these methods here to compute numerically the first-passage-time distribution for a spectrally one-sided Lévy process; the basic algorithm is not easy to apply, and we have to develop a variant of it. The numerical performance is as good as the original algorithm.

Keywords

Lévy processspectrally one-sidedLaplace transformfirst-passage distribution

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G51: Processes with independent increments; L'evy processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1173 - 1180
Copyright
Copyright © by the Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abate, J., and Whitt, W. (1992a). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10, 5–88.CrossRefGoogle Scholar
Abate, J., and Whitt, W. (1992b). Solving probability transform functional equations for numerical inversion. Operat. Res. Lett. 12, 275–281.CrossRefGoogle Scholar
Abate, J., and Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7, 36–43.CrossRefGoogle Scholar
Abate, J., Choudhury, G. L., and Whitt, W. (1999). An introduction to numerical transform inversion and its application to probability models. In Computational Probability, ed. Grassman, W. Kluwer, Boston, pp. 257–323.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.CrossRefGoogle Scholar
Borodin, A. N., and Salminen, P. (1996). Handbook of Brownian Motion — Facts and Formulae. Birkhäuser, Basel.CrossRefGoogle Scholar
Choudhury, G. L., Lucantoni, D. M., and Whitt, W. (1994). Multidimensional transform inversion with applications to the transient M/G/1 queue. Ann. Appl. Prob. 4, 719–740.CrossRefGoogle Scholar
Dubner, H., and Abate, J. (1968). Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. J. Assoc. Comput. Mach. 15, 115–123.CrossRefGoogle Scholar
Grimmett, G. R., and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd edn. Oxford University Press.Google Scholar
Hilberink, B., and Rogers, L. C. G. (2000). Optimal capital structure and endogenous default. Preprint, University of Bath.Google Scholar
Hosono, T. (1981). Numerical inversion of Laplace transform and some applications to wave optics. Radio Sci. 16, 1015–1019.CrossRefGoogle Scholar
Rogers, L. C. G., and Williams, D. (2000). Diffusions, Markov Processes and Martingales, Vol. 1. Cambridge University Press.Google Scholar
Simon, R. M., Stroot, M. T., and Weiss, G. H. (1972). Numerical inversion of Laplace transforms with application to percentage labeled experiments. Comput. Biomed. Res. 6, 596–607.CrossRefGoogle Scholar