×

Phase-type Fitting of scale functions for spectrally negative Lévy processes. (English) Zbl 1291.60094

Summary: We study the scale function of the spectrally negative phase-type Lévy process. Its scale function admits an analytical expression and so do a number of its fluctuation identities. Motivated by the fact that the class of phase-type distributions is dense in the class of all positive-valued distributions, we propose a new approach to approximating the scale function and the associated fluctuation identities for a general spectrally negative Lévy process. Numerical examples are provided to illustrate the effectiveness of the approximation method.

MSC:

60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)
65C50 Other computational problems in probability (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Bertoin, J., (Lévy Processes, Cambridge Tracts in Mathematics, vol. 121, (1996), Cambridge University Press Cambridge)
[2] Doney, R. A., (Fluctuation Theory for Lévy Processes, Lecture Notes in Mathematics, vol. 1897, (2007), Springer Berlin), Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6-23, 2005, Edited and with a foreword by Jean Picard
[3] Kyprianou, A. E., (Introductory Lectures on Fluctuations of Lévy Processes with Applications, Universitext, (2006), Springer-Verlag Berlin) · Zbl 1104.60001
[4] Kuznetsov, A.; Kyprianou, A.; Rivero, V., The theory of scale functions for spectrally negative Lévy processes, (Springer Lecture Notes in Mathematics, vol. 2061, (2013)), 97-186 · Zbl 1261.60047
[5] Surya, B. A., Evaluating scale functions of spectrally negative Lévy processes, J. Appl. Probab., 45, 1, 135-149, (2008) · Zbl 1140.60027
[6] Asmussen, S., (Applied Probability and Queues, Applications of Mathematics (New York), vol. 51, (2003), Springer-Verlag New York), Stochastic Modelling and Applied Probability
[7] Asmussen, S., Fitting phase-type distributions via the EM algorithm, Scand. J. Statist., 23, 419-441, (1996) · Zbl 0898.62104
[8] Feldmann, A.; Whitt, W., Fitting mixtures of exponentials to long-tail distributions to analyze network performance models, Perform. Eval., 31, 245-279, (1998)
[9] Chan, T.; Kyprianou, A. E.; Savov, M., Smoothness of scale functions for spectrally negative Lévy processes, Probab. Theory Related Fields, (2009) · Zbl 1259.60050
[10] Loeffen, R. L., On optimality of the barrier strategy in de finetti’s dividend problem for spectrally negative Lévy processes, Ann. Appl. Probab., 18, 5, 1669-1680, (2008) · Zbl 1152.60344
[11] Kyprianou, A. E.; Palmowski, Z., Distributional study of de finetti’s dividend problem for a general Lévy insurance risk process, J. Appl. Probab., 44, 2, 428-443, (2007) · Zbl 1137.60047
[12] 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) · Zbl 1042.60023
[13] Asmussen, S.; Avram, F.; Pistorius, M. R., Russian and American put options under exponential phase-type Lévy models, Stochastic Process. Appl., 109, 1, 79-111, (2004) · Zbl 1075.60037
[14] Albrecher, H.; Avram, F.; Kortschak, D., On the efficient evaluation of ruin probabilities for completely monotone claim distributions, J. Comput. Appl. Math., 233, 10, 2724-2736, (2010) · Zbl 1201.91088
[15] Kammler, D. W., Chebyshev approximation of completely monotonic functions by sums of exponentials, SIAM J. Numer. Anal., 13, 5, 761-774, (1976) · Zbl 0333.41016
[16] Cumani, A., On the canonical representation of homogeneous Markov processes modelling failure-time distributions, Microelectron. Reliab., 22, 3, 583-602, (1982)
[17] Dehon, M.; Latouche, G., A geometric interpretation of the relations between the exponential and generalized Erlang distributions, Adv. Appl. Probab., 14, 4, 885-897, (1982) · Zbl 0492.60018
[18] Jacod, J.; Shiryaev, A. N., (Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 288, (2003), Springer-Verlag Berlin)
[19] Pistorius, M. R., On maxima and ladder processes for a dense class of Lévy process, J. Appl. Probab., 43, 1, 208-220, (2006) · Zbl 1102.60044
[20] Feller, W., An introduction to probability theory and its applications. vol. II, (1971), John Wiley & Sons Inc. New York · Zbl 0219.60003
[21] Pistorius, M. R., On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum, J. Theoret. Probab., 17, 1, 183-220, (2004) · Zbl 1049.60042
[22] Egami, M.; Yamazaki, K., On the continuous and smooth fit principle for optimal stopping problems in spectrally negative Lévy models, Adv. Appl. Probab., (2014), forthcoming · Zbl 1398.60062
[23] K. Yamazaki, Contraction options and optimal multiple-stopping in spectrally negative Lévy models, 2012. arXiv:1209.1790.
[24] Kyprianou, A. E.; Pistorius, M. R., Perpetual options and canadization through fluctuation theory, Ann. Appl. Probab., 13, 3, 1077-1098, (2003) · Zbl 1039.60044
[25] Dempster, A. P.; Laird, N. M.; Rubin, D. B., Maximum likelihood from incomplete data via the EM algorithm, J. R. Stat. Soc. Ser. B Stat. Methodol., 39, 1, 1-38, (1977) · Zbl 0364.62022
[26] Wu, C.-F. J., On the convergence properties of the EM algorithm, Ann. Statist., 11, 1, 95-103, (1983) · Zbl 0517.62035
[27] Avram, F.; Palmowski, Z.; Pistorius, M. R., On the optimal dividend problem for a spectrally negative Lévy process, Ann. Appl. Probab., 17, 1, 156-180, (2007) · Zbl 1136.60032
[28] Horváth, A.; Telek, M., Approximating heavy tailed behaviour with phase type distributions, (Advances in Algorithmic Methods for Stochastic Models, (2000))
[29] Lang, A.; Arthur, J. L., Parameter approximation for phase-type distributions, (Matrix-analytic Methods in Stochastic Models, Lecture Notes in Pure and Applied Mathematics, (1996), CRC Press)
[30] Asmussen, S.; Madan, D. B.; Pistorius, M. R., Pricing equity default swaps under an approximation to the CGMY Lévy model, J. Comput. Finance, 11, 2, 79-93, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.