Spectrally negative Lévy processes perturbed by functionals of their running supremum

Abstract

In the setting of the classical Cramér--Lundberg risk insurance model, Albrecher and Hipp (2007) introduced the idea of tax payments. More precisely, if X = {X_t: t#x2264; 0} represents the Cramér--Lundberg process and, for all t#x2264; 0, S_t=sup_{s≥ t}X_s, then Albrecher and Hipp studied X_t - γ S_t,t#x2264; 0, where γ∈(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 Albrecher, Renaud and Zhou (2008). Finally, Kyprianou and Zhou (2009) extended this model further by allowing the rate at which tax is paid with respect to the process S = {S_t: t#x2264; 0} to vary as a function of the current value of S. Specifically, they considered the so-called perturbed spectrally negative Lévy process, U_t:=X_t -∫_(0,t]γ(S_u)dS _u,t#x2264; 0, under the assumptions that γ:[0,∞)→ [0,1) and ∫₀^∞ (1-γ(s))d s =∞. 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 γ:[0,∞)→∝. Moreover, we show that, with appropriately chosen γ, the perturbed process can pass continuously (i.e. creep) into (-∞, 0) in two different ways.