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1、http://www.paper.edu.cnOntheruinproblemsofMarkov-modulatedriskmodel¤XinZhangSchoolofMathematicalSciences,NankaiUniversityTianjin300071,ChinaApril23,2006AbstractInthispaper,weconsiderthecompoundpoissonriskmodelin°uencedbyanexternalMarko-vianenvironmentprocess
2、,i.e.Markov-modulatedcompoundPoissonmodel.TheexplicitLaplacetransformsofGerber-Shiufunctionsareobtained,whiletheexplicitGerber-Shiufunctionsarederivedinthetwo-statecase.Keywords:Markovadditiveprocess;Laplacetransform;Gerber-Shiufunction;Markov-modulatedcompo
3、undPoissonmodel.AMS2005SubjectClassi¯cation:Primary60J80.1IntroductionIntheliteraturemuchattentionhasbeendevotedtotheMarkov-modulatedriskandqueuemod-els,see,e.g.,byReinhard(1984),Asmussen(1989),AdanandKulkarni(2003),Miyazawa(2004),YiandShuanming(2005),Albrec
4、herandBoxma(2005).Reinhard(1984)consideredaclassofsemi-Markovriskmodelsinwhichtheclaimfrequenciesandclaimamountsarein°uencedbyanexternalMarkovianenvironmentprocess.Asystemofintegro-di®erentialequationsfornon-ruinprobabilities,whentheclaimsizedistributionsare
5、exponential,wasderived.YiLuandShuanmingLi(2005)solvedtheaboveintegro-di®erentialequationsinthetwo-statemodelandgavetheexplicitformulasfornon-ruinprobabilities.Asmussen(1989)treatedclassicalriskproblemswhenthearrivalsoftheclaimsaredescribedbyaCoxprocesswheret
6、heintensityprocessisa¯nite-stateMarkovprocess.TheCram¶er-Lundberg¤SupportedbytheNaturalScienceFoundationofChina(GrantNo.10571092).1http://www.paper.edu.cnapproximationanddi®usionapproximationswithcorrectiontermswereconsidered.InthebookofAsmussen(2000),thecom
7、poundpoissonriskmodelin°uencedbyanexternalMarkovianenvi-ronmentprocess,i.e.Markov-modulatedcompoundpoissonmodel,wasconsideredsu±ciently.TheruinprobabilityandtheCram¶er-LundbergboundswerebeautifullysolvedinthechapterVIofthebook.AlbrecherandBoxma(2005)analyzes
8、everalpopularriskmodelsthroughsemi-markovmodel,suchascompoundPoissonmodel,SparreAndersenmodelswithphase-typeinterclaimtimes.Miyazawa(2004)considerareal-valuedMarkovadditiveprocesswhichlinearmove
