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Spectrally Negative Lévy Processes Perturbed by Functionals of their Running Supremum

Part of: Special processes Mathematical economics

Published online by Cambridge University Press:  30 January 2018

Andreas E. Kyprianou  and
Curdin Ott
Andreas E. Kyprianou*
Affiliation:
University of Bath
Curdin Ott*
Affiliation:
University of Bath
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.
∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.
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Abstract

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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 = {Xt: t≥ 0} represents the Cramér–Lundberg process and, for all t≥ 0, St=sup_{s≤ t}Xs, then Albrecher and Hipp studied Xt - γ St,t≥ 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 = {St: t≥ 0} to vary as a function of the current value of S. Specifically, they considered the so-called perturbed spectrally negative Lévy process, Ut:=Xt -∫(0,t]γ(S_u)dSu,t≥ 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.

Keywords

Spectrally negative Lévy processexcursion theorycreepingruin

MSC classification

Primary: 60K05: Renewal theory 60K15: Markov renewal processes, semi-Markov processes 91B30: Risk theory, insurance
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 1005 - 1014
Copyright
© Applied Probability Trust 

References

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