## 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.

### MSC:

 60G51 Processes with independent increments; Lévy processes 60K05 Renewal theory 60K15 Markov renewal processes, semi-Markov processes 91B30 Risk theory, insurance (MSC2010)

### Citations:

Zbl 1119.62103; Zbl 1144.60032; Zbl 1210.60098
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### References:

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