The Annals of Probability

The law of the supremum of a stable Lévy process with no negative jumps

Violetta Bernyk, Robert C. Dalang, and Goran Peskir

Full-text: Open access

Abstract

Let X=(Xt)t≥0 be a stable Lévy process of index α∈(1, 2) with no negative jumps and let St=sup0≤stXs denote its running supremum for t>0. We show that the density function ft of St can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann–Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for ft. Recalling the familiar relation between St and the first entry time τx of X into [x, ∞), this further translates into an explicit series representation for the density function of τx.

Article information

Source
Ann. Probab. Volume 36, Number 5 (2008), 1777-1789.

Dates
First available in Project Euclid: 11 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1221138766

Digital Object Identifier
doi:10.1214/07-AOP376

Mathematical Reviews number (MathSciNet)
MR2440923

Zentralblatt MATH identifier
1185.60051

Subjects
Primary: 60G52: Stable processes 45D05: Volterra integral equations [See also 34A12]
Secondary: 60J75: Jump processes 45E99: None of the above, but in this section 26A33: Fractional derivatives and integrals

Keywords
Stable Lévy process with no negative jumps spectrally positive running supremum process first hitting time first entry time weakly singular Volterra integral equation polar kernel Riemann–Liouville fractional differential equation Abel equation Wiener–Hopf factorization

Citation

Bernyk, Violetta; Dalang, Robert C.; Peskir, Goran. The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36 (2008), no. 5, 1777--1789. doi:10.1214/07-AOP376. https://projecteuclid.org/euclid.aop/1221138766


Export citation

References

  • [1] Abramowitz, M. and Stegun, I. A. (1992). Handbook of Mathematical Functions. Dover, New York.
  • [2] Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stochastic Process. Appl. 109 79–111.
  • [3] Bernyk, V., Dalang, R. C. and Peskir, G. (2007). Predicting the ultimate supremum of a stable Lévy process. Research Report No. 28, Probab. Statist. Group Manchester.
  • [4] Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • [5] Bingham, N. H. (1973). Maxima of sums of random variables and suprema of stable processes. Z. Wahrsch. Verw. Gebiete 26 273–296.
  • [6] Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York.
  • [7] Borovkov, K. and Burq, Z. (2001). Kendall’s identity for the first crossing time revisited. Electron. Comm. Probab. 6 91–94.
  • [8] Doney, R. A. (2008). A note on the supremum of a stable process. Stochastics 80 151–155.
  • [9] Erdélyi, A. (1954). Tables of Integral Transforms 1. McGraw-Hill, New York.
  • [10] Gradshteyn, I. S. and Ryzhik, I. M. (1994). Table of Integrals, Series, and Products. Academic Press, New York.
  • [11] Heyde, C. C. (1969). On the maximum of sums of random variables and the supremum functional for stable processes. J. Appl. Probab. 6 419–429.
  • [12] Hochstadt, H. (1973). Integral Equations. Wiley, New York.
  • [13] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • [14] Mordecki, E. (2002). The distribution of the maximum of the Lévy process with positive jumps of phase-type. Theory Stoch. Process. 8 309–316.
  • [15] Podlubny, I. (1999). Fractional Differential Equations. Academic Press, New York.
  • [16] Polyanin, A. D. and Manzhirov, A. V. (1998). Handbook of Integral Equations. Chapman and Hall, Boca Raton, FL.
  • [17] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.