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On scale functions for Lévy processes with negative phase-type jumps

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Abstract

We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

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References

  1. Asmussen, S.: Stationary distributions for fluid flow models with or without Brownian noise. Commun. Stat. Stochastic Models 11(1), 21–49 (1995). https://doi.org/10.1080/15326349508807330

    Article  Google Scholar 

  2. Asmussen, S.: Applied Probability and Queues. Applications of Mathematics. Stochastic Modelling and Applied Probability, vol. 51, 2nd edn. Springer, New York (2003)

    Google Scholar 

  3. Asmussen, S.: Lévy processes, phase-type distributions, and martingales. Stoch. Models 30(4), 443–468 (2014). https://doi.org/10.1080/15326349.2014.958424

    Article  Google Scholar 

  4. Asmussen, S., Albrecher, H.: Ruin Probabilities. Advanced Series on Statistical Science and Applied Probability, vol. 14, 2nd edn. World Scientific Publishing Co. Pte. Ltd., Hackensack (2010). https://doi.org/10.1142/9789814282536

    Book  Google Scholar 

  5. Asmussen, S., Avram, F., Pistorius, M.R.: Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109(1), 79–111 (2004). https://doi.org/10.1016/j.spa.2003.07.005

    Article  Google Scholar 

  6. Asmussen, S., Laub, P.J., Yang, H.: Phase-type models in life insurance: fitting and valuation of equity-linked benefits. Risks 7(1), 17 (2019)

    Article  Google Scholar 

  7. Avram, F., Grahovac, D., Vardar-Acar, C.: The \(W\), \(Z\) scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems. ESAIM Probab. Stat. 24, 454–525 (2020). https://doi.org/10.1051/ps/2019022

    Article  Google Scholar 

  8. Bertoin, J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  9. Bini, D.A.: Numerical computation of polynomial zeros by means of Aberth’s method. Numer. Algorithms 13(3–4), 179–200 (1996). https://doi.org/10.1007/BF02207694

    Article  Google Scholar 

  10. Bladt, M., Nielsen, B.F.: Matrix-Exponential Distributions in Applied Probability. Probability Theory and Stochastic Modelling, vol. 81. Springer, New York (2017). https://doi.org/10.1007/978-1-4939-7049-0

    Book  Google Scholar 

  11. Bladt, M., Nielsen, B.F., Samorodnitsky, G.: Calculation of ruin probabilities for a dense class of heavy tailed distributions. Scand. Actuar. J. 7, 573–591 (2015). https://doi.org/10.1080/03461238.2013.865257

    Article  Google Scholar 

  12. D’Auria, B., Ivanovs, J., Kella, O., Mandjes, M.: First passage of a Markov additive process and generalized Jordan chains. J. Appl. Probab. 47(4), 1048–1057 (2010). https://doi.org/10.1239/jap/1294170518

    Article  Google Scholar 

  13. Dȩbicki, K., Mandjes, M.: Queues and Lévy Fluctuation Theory. Springer, Berlin (2015)

    Book  Google Scholar 

  14. Egami, M., Yamazaki, K.: Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 1–22 (2014). https://doi.org/10.1016/j.cam.2013.12.044

    Article  Google Scholar 

  15. Hubalek, F., Kyprianou, E.: Old and new examples of scale functions for spectrally negative Lévy processes. In: Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability, vol. 63, pp. 119–145. Birkhäuser/Springer, Basel (2011). https://doi.org/10.1007/978-3-0348-0021-1_8

  16. Ivanovs, J.: Markov-modulated Brownian motion with two reflecting barriers. J. Appl. Probab. 47(4), 1034–1047 (2010). https://doi.org/10.1239/jap/1294170517

    Article  Google Scholar 

  17. Ivanovs, J.: One-sided Markov additive processes and related exit problems, PhD thesis. BOXPress (2011)

  18. Ivanovs, J., Palmowski, Z.: Occupation densities in solving exit problems for Markov additive processes and their reflections. Stoch. Process. Appl. 122(9), 3342–3360 (2012). https://doi.org/10.1016/j.spa.2012.05.016

    Article  Google Scholar 

  19. Kuznetsov, A., Kyprianou, A.E., Pardo, J.C.: Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22(3), 1101–1135 (2012). https://doi.org/10.1214/11-AAP787

    Article  Google Scholar 

  20. Kuznetsov, A., Kyprianou, A.E., Rivero, V.: The theory of scale functions for spectrally negative Lévy processes. In: Lévy Matters II. Lecture Notes in Mathematics, vol. 2061, pp. 97–186. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31407-0_2

  21. Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006)

    Google Scholar 

  22. Landriault, D., Willmot, G.E.: On series expansions for scale functions and other ruin-related quantities. Scand. Actuar. J. 4, 292–306 (2020). https://doi.org/10.1080/03461238.2019.1663444

    Article  Google Scholar 

  23. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, American Statistical Association, Alexandria (1999). https://doi.org/10.1137/1.9780898719734

  24. Mordecki, E.: The distribution of the maximum of a Lévy processes with positive jumps of phase-type. In: Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko (Kyiv, 2002), vol. 8, pp. 309–316 (2002)

  25. Pistorius, M.R.: A potential-theoretical review of some exit problems of spectrally negative Lévy processes. In: Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics, vol. 1857, pp. 30–41. Springer, Berlin (2005). https://doi.org/10.1007/978-3-540-31449-3_4

  26. Potra, F.: On q-order and r-order of convergence. J. Optim. Theory Appl. 63(3), 415–431 (1989)

    Article  Google Scholar 

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Acknowledgements

The financial support of Sapere Aude Starting Grant 8049-00021B “Distributional Robustness in Assessment of Extreme Risk” from Independent Research Fund Denmark is gratefully acknowledged.

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  1. Department of Mathematics, Aarhus University, Aarhus, Denmark

    Jevgenijs Ivanovs

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  1. Jevgenijs Ivanovs
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Correspondence to Jevgenijs Ivanovs.

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Ivanovs, J. On scale functions for Lévy processes with negative phase-type jumps. Queueing Syst 98, 3–19 (2021). https://doi.org/10.1007/s11134-021-09696-w

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