Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T06:32:17.011Z Has data issue: false hasContentIssue false
  • English
  • Français

One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications

Part of: Limit theorems Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Shai Covo
Shai Covo*
Affiliation:
Bar Ilan University
*
∗ Postal address: Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, Israel. Email address: green355@netvision.net.il
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function Fs (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂nFs (x; t) ∂ xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

Keywords

SubordinatorLévy processcompound Poisson processtruncated Lévy measureordered jumpsgamma processsmall jumpsweak convergenceDickman distribution

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 367 - 392
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

This paper is part of the author's PhD thesis, prepared at Bar Ilan University under the supervision of Professor E. Merzbach. This work was supported by the Doctoral Fellowship of Excellence, Bar Ilan University.

References

Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich.Google Scholar
Asmussen, S. and Rosiński, J. (2001). Approximations of small Jumps of Lévy processes with a view towards simulation. J. Appl. Prob. 38, 482–493.Google Scholar
Basawa, I. V. and Brockwell, P. J. (1982). Nonparametric estimation for nondecreasing Lévy processes. J. R. Statist. Soc. B 44, 262–269.Google Scholar
Covo, S. (2008). On approximations of small Jumps of subordinators with particular emphasis on a Dickman-type limit. Submitted.Google Scholar
Eliazar, I. and Klafter, J. (2003). On the extreme flights of one-sided Lévy processes. Physica A 330, 8–17.Google Scholar
Emmer, S. and Klüppelberg, C. (2004). Optimal portfolios when stock prices follow an exponential Lévy process. Finance Stoch. 8, 17–44.Google Scholar
Griffiths, R. C. (1988). On the distribution of points in a Poisson Dirichlet process. J. Appl. Prob. 25, 336–345.Google Scholar
Hensley, D. (1986). The convolution powers of the Dickman function. J. London Math. Soc. 33, 395–406.Google Scholar
Holst, L. (2001). The Poisson–Dirichlet distribution and its relatives revisited. Preprint, Royal Institute of Technology, Stockholm. Available at http://www.math.kth.se/matstat/.Google Scholar
Howlader, H. A. and Balasooriya, U. (2003). Bayesian estimation of the distribution function of the Poisson model. Biometrical J. 45, 901–911.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.Google Scholar
Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The variance gamma process and option pricing. Europ. Finance Rev. 2, 79–105.Google Scholar
Penrose, M. D. and Wade, A. R. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. Appl. Prob. 36, 691–714.Google Scholar
Perman, M. (1993). Order statistics for Jumps of normalised subordinators. Stoch. Process. Appl. 46, 267–281.Google Scholar
Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855–900.Google Scholar
Rosiński, J. (2001). Series representations of Lévy Processes from the perspective of point processes. In Lévy Processes, eds Barndorff-Nielsen, O. E. et al. Birkhäuser, Boston, MA, pp. 401–415.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68). Cambridge University Press.Google Scholar
Takács, L. (1955). On stochastic processes connected with certain physical recording apparatuses. Acta Math. Acad. Sci. Hungar. 6, 363–380.Google Scholar
Vervaat, W. (1972). Success Epochs in Bernoulli Trials (with Applications in Number Theory). Mathematisch Centrum, Amsterdam.Google Scholar
Wheeler, F. S. (1990). Two differential-difference equations arising in number theory. Trans. Amer. Math. Soc. 318, 491–523.Google Scholar