A Wiener-Hopf factorization related potential measure for spectrally negative Lévy process

Abstract

For spectrally negative Lévy process (SNLP), we find an expression, in terms of scale functions, for a potential measure involving the maximum and the last time of reaching the maximum up to a draw-down time. As applications, we obtain a potential measure for the reflected SNLP and recover a joint Laplace transform for the Wiener-Hopf factorization for SNLP.

Keywords

Spectrally negative Lévy process (SNLP) / potential measure / draw- down time / excursion theory / scale function / Wiener-Hopf factorization

References

[1] Avram F, Vu N, Zhou X. On taxed spectrally negative Lévy processes with draw-down stopping. Insurance Math Econom, 2017, 76: 69–74

[2] Bertoin J. Lévy Processes. Cambridge Tracts in Math, 121. Cambridge: Cambridge Univ Press, 1996

[3] Biffis E, Kyprianou A E. A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Math Econom, 2010, 46: 85–91

[4] Hadjiliadis O, Vecer J. Drawdowns preceding rallies in a Brownian motion process. Quant Finance, 2006, 6: 403–409

[5] Kuznetsov A, Kyprianou A E, Rivero V. The theory of scale functions for spectrally negative Lévy processes. In: Cohen S, Kuznetsov A, Kyprianou A E, Rivero V, eds. Lévy Matters II. Lecture Notes in Math, Vol 2061. Berlin: Springer, 2012, 97–186

[6] Kyprianou A E. Gerber-Shiu Risk Theory. EAA Ser. Cham: Springer, 2013

[7] Kyprianou A E. Fluctuations of Lévy Processes with Applications. 2nd ed. Berlin: Springer, 2014

[8] Landriault D, Li B, Li S. Drawdown analysis for the renewal insurance risk process. Scand Actuar J, 2017, 3: 267–285

[9] Lehoczky J. Formulas for stopped diffusion processes with stopping times based on the maximum. Ann Appl Probab, 1977, 5: 601–607

[10] Li B, Vu N, Zhou X. Exit problems for general draw-down times of spectrally negative Lévy processes. J Appl Probab, 2019, 56: 441–457

[11] Meilijson I. The time to a given drawdown in Brownian motion. In: Azéma J, Émery M, Ladoux M, Yor M, eds. Séminaire de Probabilités XXXVII. Lecture Notes in Math, Vol 1832. Berlin: Springer, 2003, 94–108

[12] Pistorius M R. An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. In: Donati-Martin C, Émery M, Rouault A, Stricker C, eds. Séminaire de Probabilitiés XL. Lecture Notes in Math, Vol 1899. Berlin: Springer, 2007, 287–307

[13] Suprun V. Problem of ruin and resolvent of terminating processes with independent increments. Ukrainian Math J, 1976, 28: 39–45

[14] Wang W, Chen P, Li S. Generalized expected discounted penalty function at general drawdown for Lévy risk processes. Insurance Math Econom, 2020, 91: 12–25

[15] Wang W, Zhou X. Draw-down Parisian ruin for spectrally negative Lévy process. Adv in Appl Probab (to appear)

[16] Wang W, Zhou X. A drawdown reflected spectrally negative Lévy process. J Theoret Probab (to appear)

RIGHTS & PERMISSIONS

Higher Education Press