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.


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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