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**Spectrally negative Lévy processes perturbed by functionals of their running supremum.**
*(English)*
Zbl 1260.60094

Summary: In the setting of the classical Cramer-Lundberg risk insurance model, H. Albrecher and C. Hipp [Bl. DGVFM 28, No. 1, 13-28 (2007; Zbl 1119.62103)] introduced the idea of tax payments. More precisely, if \(X = \{X_t : t\geq 0\}\) represents the Cramer-Lundberg process and, for all \(t\geq 0\), \(S_t = \sup_{s\leq t}X_s\), then they studied \(X_t - \gamma S_t\), \(t\geq 0\), where \(\gamma\in(0,1)\) is the rate at which tax is paid. This model has been generalised to the setting that \(X\) is a spectrally negative Lévy process by H. Albrecher et al [J. Appl. Probab. 45, No. 2, 363-375 (2008; Zbl 1144.60032)]. Finally, A. E. Kyprianou and X. Zhou [J. Appl. Probab. 46, No. 4, 1146–1156 (2009; Zbl 1210.60098)] extended this model further by allowing the rate at which tax is paid with respect to the process \(S = \{S_t : t\geq 0\}\) to vary as a function of the current value of \(S\). Specifically, they consider the so-called perturbed spectrally negative Levy process
\[
U_t=X_t-\int_{(0,t]}\gamma(S_u)\,{\mathrm d} S_u,\qquad t\geq 0,
\]
under the assumptions \(\gamma :[0,\infty)\rightarrow [0,1)\) and \(\int_0^\infty (1-\gamma(s)){\mathrm d}s =\infty\).

In this article, we show that a number of the identities in [loc. cit.] are still valid for a much more general class of rate functions \(\gamma:[0,\infty)\rightarrow \mathbb{R}\). Moreover, we show that, with appropriately chosen \(\gamma\), the perturbed process can pass continuously (i.e., creep) into \((-\infty, 0)\) in two different ways.

In this article, we show that a number of the identities in [loc. cit.] are still valid for a much more general class of rate functions \(\gamma:[0,\infty)\rightarrow \mathbb{R}\). Moreover, we show that, with appropriately chosen \(\gamma\), the perturbed process can pass continuously (i.e., creep) into \((-\infty, 0)\) in two different ways.

### MSC:

60G51 | Processes with independent increments; Lévy processes |

60K05 | Renewal theory |

60K15 | Markov renewal processes, semi-Markov processes |

91B30 | Risk theory, insurance (MSC2010) |

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\textit{A. E. Kyprianou} and \textit{C. Ott}, J. Appl. Probab. 49, No. 4, 1005--1014 (2012; Zbl 1260.60094)

### References:

[1] | Albrecher, H. and Hipp, C. (2007). Lundberg’s risk process with tax. Blätter DGVFM 28, 13-28. · Zbl 1119.62103 |

[2] | Albrecher, H., Renaud, J.-F. and Zhou, X. (2008). A Lévy insurance risk process with tax. J. Appl. Prob. 45, 363-375. · Zbl 1144.60032 |

[3] | Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215-238. · Zbl 1042.60023 |

[4] | Bertoin, J. (1996). Lévy Processes . Cambrdige University Press. · Zbl 0861.60003 |

[5] | Bichteler, K. (2002). Stochastic Integration with Jumps . Cambridge University Press. · Zbl 1002.60001 |

[6] | Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061 ), Springer, Berlin, pp. 97-186. · Zbl 1261.60047 |

[7] | Kyprianou, A. E. and Zhou, X. (2009). General tax structures and the Lévy insurance risk model. J. Appl. Prob. 46, 1146-1156. · Zbl 1210.60098 |

[8] | Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001 |

[9] | Perman, M. and Werner, W. (1997). Perturbed Brownian motions. Prob. Theory Relat. Fields 108 , 357-383. · Zbl 0884.60082 |

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