Journal of Theoretical Probability

, Volume 22, Issue 2, pp 418–440 | Cite as

On an Explicit Skorokhod Embedding for Spectrally Negative Lévy Processes

Article

Abstract

We present an explicit solution to the Skorokhod embedding problem for spectrally negative Lévy processes. Given a process X and a target measure μ satisfying an explicit admissibility condition we define functions φ± such that the stopping time T=inf {t>0:Xt∈{−φ(Lt),φ+(Lt)}} induces XTμ, where (Lt) is the local time in zero of X. We also treat versions of T which take into account the sign of the excursion straddling time t. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]). In particular, we compute explicitly the quantities introduced in Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]) in our setup.

Our method relies on some new explicit calculations relating scale functions and the Itô excursion measure of X. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.

Keywords

Skorokhod embedding Levy process Excursion theory Scale function Minimal stopping time 

Mathematics Subject Classification (2000)

60G44 60G40 60G51 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avram, F., Kyprianou, A.E., Pistorius, M.R.: Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14(1), 215–238 (2004) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bertoin, J.: An extension of Pitman’s theorem for spectrally positive Lévy processes. Ann. Probab. 20(3), 1464–1483 (1992) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bertoin, J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996) MATHGoogle Scholar
  4. 4.
    Bertoin, J.: Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7(1), 156–169 (1997) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bertoin, J., Le Jan, Y.: Representation of measures by balayage from a regular recurrent point. Ann. Probab. 20(1), 538–548 (1992) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bingham, N.H.: Fluctuation theory in continuous time. Adv. Appl. Probab. 7(4), 705–766 (1975) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cox, A., Hobson, D., Obłój, J.: Pathwise inequalities of the local time: applications to Skorokhod embeddings and optimal stopping. Ann. Appl. Probab. (to appear). ArXiV:math.PR/0702173 (2007) Google Scholar
  8. 8.
    Cox, A., Obłój, J.: Classes of Skorokhod embeddings for simple symmetric random walk. ArXiv:math.PR/0609330 (2006) Google Scholar
  9. 9.
    Itô, K.: Poisson point processes attached to Markov processes. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, Calif., 1970/1971. Probability Theory, vol. III, pp. 225–239. Univ. California Press, Berkeley (1970) Google Scholar
  10. 10.
    Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin (2006) Google Scholar
  11. 11.
    Lambert, A.: Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. Henri Poincaré Probab. Stat. 36(2), 251–274 (2000) MATHCrossRefGoogle Scholar
  12. 12.
    Millar, P.W.: Exit properties of stochastic processes with stationary independent increments. Trans. Am. Math. Soc. 178, 459–479 (1973) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Monroe, I.: On embedding right continuous martingales in Brownian motion. Ann. Math. Stat. 43, 1293–1311 (1972) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Obłój, J.: The Skorokhod embedding problem and its offspring. Probab. Surv. 1, 321–392 (2004) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Obłój, J.: An explicit Skorokhod embedding for functionals of excursions of Markov processes. Stoch. Process. Their Appl. 117, 409–431 (2007) MATHCrossRefGoogle Scholar
  16. 16.
    Pistorius, M.R.: A potential-theoretical review of some exit problems of spectrally negative Lévy processes. In: Séminaire de Probabilités XXXVIII. Lecture Notes in Math., vol. 1857, pp. 30–41. Springer, Berlin (2005) Google Scholar
  17. 17.
    Pistorius, M.R.: An excursion theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. In: Séminaire de Probabilités XL. Lecture Notes in Math., vol. 1899, pp. 287–308. Springer, Berlin (2007) CrossRefGoogle Scholar
  18. 18.
    Rost, H.: The stopping distributions of a Markov process. Invent. Math. 14, 1–16 (1971) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Skorokhod, A.V.: Studies in the Theory of Random Processes. Addison-Wesley, Reading (1965). Translated from the Russian by Scripta Technica, Inc. MATHGoogle Scholar
  20. 20.
    Vallois, P.: Le problème de Skorokhod sur R: une approche avec le temps local. In: Séminaire de Probabilités, XVII. Lecture Notes in Math., vol. 986, pp. 227–239. Springer, Berlin (1983) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Dept. of MathematicsImperial College LondonLondonUK
  2. 2.Dept. of MathematicsKing’s College LondonLondonUK

Personalised recommendations