  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.
