Spectrally negative Levy processes perturbed by functionals of their running supremum

Abstract

In the setting of the classical Cramer-Lundberg risk insurance model, Albrecher and Hipp (2007) 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 Albrecher and Hipp (2007) study $X_t - \gamma S_t$, $t\geq 0$, 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 L\'evy process by Albrecher et al. \cite{albr_ren_zhou}. Finally Kyprianou and Zhou (2009) extend 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, \[ U_t=X_t-\int_{(0,t]}\gamma(S_u)\,{\rm d} S_u,\qquad t\geq 0, \] under the assumptions $\gamma :[0,\infty)\rightarrow [0,1)$ and $\int_0^\infty (1-\gamma(s)){\rm d}s =\infty$. In this article we show that a number of the identities in Kyprianou and Zhou (2009) 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 (ie. creep) into $(-\infty, 0)$ in two different ways.