Advertisement

Parisian Excursion Below a Fixed Level from the Last Record Maximum of Lévy Insurance Risk Process

  • Budhi A. SuryaEmail author
Chapter
Part of the MATRIX Book Series book series (MXBS, volume 2)

Abstract

This paper presents some new results on Parisian ruin under Lévy insurance risk process, where ruin occurs when the process has gone below a fixed level from the last record maximum, also known as the high-water mark or drawdown, for a fixed consecutive periods of time. The law of ruin-time and the position at ruin is given in terms of their joint Laplace transforms. Identities are presented semi-explicitly in terms of the scale function and the law of the Lévy process. They are established using recent developments on fluctuation theory of drawdown of spectrally negative Lévy process. In contrast to the Parisian ruin of Lévy process below a fixed level, ruin under drawdown occurs in finite time with probability one.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author would like to thank a number of anonymous referees and associate editors for their useful suggestions and comments that improved the presentation of this paper. This paper was completed during the time the author visited the Hugo Steinhaus Center of Mathematics at Wrocław University of Science and Technology in Poland. The author acknowledges the support and hospitality provided by the Center. He thanks to Professor Zbigniew Palmowski for the invitation, and for some suggestions over the work discussed during the MATRIX Mathematics of Risk Workshop in Melbourne organized by Professors Konstantin Borovkov, Alexander Novikov and Kais Hamza to whom the author also like to thanks for the invitation. This research is financially supported by Victoria University PBRF Research Grants # 212885 and # 214168 for which the author is grateful.

References

  1. 1.
    Agarwal, V., Daniel, N., Naik, N.: Role of managerial incentives and discretion in hedge fund performance. J. Financ. 64, 2221–2256 (2009)CrossRefGoogle Scholar
  2. 2.
    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, 215–238 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  4. 4.
    Broadie, M., Chernov, M., Sundaresan, S.: Optimal debt and equity values in the presence of Chapter 7 and Chapter 11. J. Financ. LXII, 1341–1377 (2007)Google Scholar
  5. 5.
    Chan, T., Kyprianou, A.E., Savov, M.: Smoothness of scale functions for spectrally negative Lévy processes. Probab. Theory Rel. 150, 691–708 (2011)CrossRefGoogle Scholar
  6. 6.
    Chesney, M., Jeanblanc-Picqué, M., Yor, M.: Brownian excursions and Parisian barrier options. Adv. Appl. Probab. 29, 165–184 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Czarna, I., Palmowski, Z.: Ruin probability with Parisian delay for a spectrally negative Lévy process. J. Appl. Probab. 48, 984–1002 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dassios, A., Wu, S.: Perturbed Brownian motion and its application to Parisian option pricing. Finance Stoch. 14, 473–494 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Francois, P., Morellec, E.: Capital structure and asset prices: some effects of bankruptcy procedures. J. Bus. 77, 387–411 (2004)CrossRefGoogle Scholar
  10. 10.
    Goetzmann, W.N., Ingersoll Jr., J.E., Ross, S.A.: High-water marks and hedge fund management contracts. J. Financ. 58, 1685–1717 (2003)CrossRefGoogle Scholar
  11. 11.
    Kusnetzov, A., Kyprianou, A.E., Rivero, V.: The Theory of Scale Functions for Spectrally Negative Lévy Processes, Lévy Matters II. Springer Lecture Notes in Mathematics. Springer, Berlin (2013)Google Scholar
  12. 12.
    Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006)zbMATHGoogle Scholar
  13. 13.
    Kyprianou, A.E., Surya, B.A.: Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. Finance Stoch. 11, 131–152 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lambert, A.: Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. Henri Poincaré 2, 251–274 (2000)CrossRefGoogle Scholar
  15. 15.
    Landriault, D., Renaud, J-F., Zhou, X.: An insurance risk model with Parisian implementation delays. Methodol. Comput. Appl. Probab. 16, 583–607 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Loeffen, R., Czarna, I., Palmowski, Z.: Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19, 599–609 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Loeffen, R., Palmowski, Z., Surya, B.A.: Discounted penalty function at Parisian ruin for Lévy insurance risk process. Insur. Math. Econ. 83, 190–197 (2017)CrossRefGoogle Scholar
  18. 18.
    Mijatović, A., Pistorius, M.R.: On the drawdown of completely assymetric Lévy process. Stoc. Proc. Appl. 122, 3812–3836 (2012)CrossRefGoogle Scholar
  19. 19.
    Palmowski, Z., Tumilewicz, J.: Pricing insurance drawdown-type contracts with underlying Lévy assets. Insur. Math. Econ. 79, 1–14 (2018)CrossRefGoogle Scholar
  20. 20.
    Surya, B.A.: Evaluating scale function of spectrally negative Lévy processes. J. Appl. Probab. 45, 135–149 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhang, H., Leung, T., Hadjiliadis, O.: Stochastic modeling and fair valuation of drawdown insurance. Insur. Math. Econ. 53, 840–850 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsVictoria University of WellingtonWellingtonNew Zealand

Personalised recommendations