{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:47:49Z","timestamp":1740491269787,"version":"3.38.0"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2012,12]]},"abstract":"<jats:p>In the setting of the classical Cram\u00e9r\u2013Lundberg risk insurance model, Albrecher and Hipp (2007) introduced the idea of tax payments. More precisely, if <jats:italic>X<\/jats:italic> = {<jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub>: <jats:italic>t<\/jats:italic>\u2265 0} represents the Cram\u00e9r\u2013Lundberg process and, for all <jats:italic>t<\/jats:italic>\u2265 0, <jats:italic>S<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub>=sup_{<jats:italic>s<\/jats:italic>\u2264 <jats:italic>t<\/jats:italic>}<jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>s<\/jats:italic>\n               <\/jats:sub>, then Albrecher and Hipp studied <jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub> - \u03b3 <jats:italic>S<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub>,<jats:italic>t<\/jats:italic>\u2265 0, where \u03b3\u2208(0,1) is the rate at which tax is paid. This model has been generalised to the setting that <jats:italic>X<\/jats:italic> is a spectrally negative L\u00e9vy 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 <jats:italic>S<\/jats:italic> = {<jats:italic>S<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub>: <jats:italic>t<\/jats:italic>\u2265 0} to vary as a function of the current value of <jats:italic>S<\/jats:italic>. Specifically, they considered the so-called <jats:italic>perturbed<\/jats:italic> spectrally negative L\u00e9vy process, <jats:italic>U<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub>:=<jats:italic>X<\/jats:italic>\n               <jats:sub>\n                  <jats:italic>t<\/jats:italic>\n               <\/jats:sub> -\u222b<jats:sub>(0,<jats:italic>t<\/jats:italic>]<\/jats:sub>\u03b3(<jats:italic>S<\/jats:italic>_u)d<jats:italic>S<\/jats:italic> \n               <jats:sub>\n                  <jats:italic>u<\/jats:italic>\n               <\/jats:sub>,<jats:italic>t<\/jats:italic>\u2265 0, under the assumptions that \u03b3:[0,\u221e)\u2192 [0,1) and \u222b<jats:sub>0<\/jats:sub>\n               <jats:sup>\u221e<\/jats:sup> (1-\u03b3(s))d <jats:italic>s<\/jats:italic> =\u221e. 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 \u03b3:[0,\u221e)\u2192\u221d. Moreover, we show that, with appropriately chosen \u03b3, the perturbed process can pass continuously (i.e. creep) into (-\u221e, 0) in two different ways.<\/jats:p>","DOI":"10.1017\/s0021900200012845","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T10:49:29Z","timestamp":1459248569000},"page":"1005-1014","source":"Crossref","is-referenced-by-count":0,"title":["Spectrally Negative L\u00e9vy Processes Perturbed by Functionals of their Running Supremum"],"prefix":"10.1017","volume":"49","author":[{"given":"Andreas E.","family":"Kyprianou","sequence":"first","affiliation":[]},{"given":"Curdin","family":"Ott","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2016,3,1]]},"reference":[{"doi-asserted-by":"publisher","key":"S0021900200012845_ref9","DOI":"10.1007\/s004400050113"},{"volume-title":"Introductory Lectures on Fluctuations of L\u00e9vy Processes with Applications","year":"2006","key":"S0021900200012845_ref8"},{"doi-asserted-by":"publisher","key":"S0021900200012845_ref7","DOI":"10.1239\/jap\/1261670694"},{"volume-title":"L\u00e9vy Matters II","year":"2012","first-page":"97","key":"S0021900200012845_ref6"},{"doi-asserted-by":"publisher","key":"S0021900200012845_ref1","DOI":"10.1007\/s11857-007-0004-4"},{"volume-title":"L\u00e9vy Processes","year":"1996","key":"S0021900200012845_ref4"},{"doi-asserted-by":"publisher","key":"S0021900200012845_ref3","DOI":"10.1214\/aoap\/1075828052"},{"doi-asserted-by":"publisher","key":"S0021900200012845_ref2","DOI":"10.1239\/jap\/1214950353"},{"volume-title":"Stochastic Integration with Jumps","year":"2002","key":"S0021900200012845_ref5"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900200012845","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,4,22]],"date-time":"2017-04-22T00:24:59Z","timestamp":1492820699000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900200012845\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,12]]},"references-count":9,"journal-issue":{"issue":"04","published-print":{"date-parts":[[2012,12]]}},"alternative-id":["S0021900200012845"],"URL":"https:\/\/doi.org\/10.1017\/s0021900200012845","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"type":"print","value":"0021-9002"},{"type":"electronic","value":"1475-6072"}],"subject":[],"published":{"date-parts":[[2012,12]]}}}