Spectrally negative Levy processes perturbed by functionals of their running supremum

TL;DR

This paper extends the analysis of spectrally negative Lévy processes perturbed by functionals of their running supremum, showing broader applicability of existing identities and exploring conditions for continuous crossing into negative territory.

Contribution

It generalizes previous models by allowing more flexible rate functions for perturbation and investigates the process's ability to creep into negative values.

Findings

Many identities from prior models remain valid under general rate functions.

The process can pass into negative values continuously with suitable choices of the rate function.

The paper broadens the theoretical understanding of perturbed spectrally negative Lévy processes.

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.