关于带扰动广义Cox保险风险模型的破产概率

关于带扰动广义Cox保险风险模型的破产概率 关于带扰动广义Cox保险风险模型的破产 概率 第19卷第6期 2007年12月 重庆邮电大学(自然科学版) JournalofChongqingUniversityofPostsandTelecommunications(NaturalScience) Vo1.19No.6 Dec.2007 Onruinprobabilityforageneralized Coxinsuranceriskmodelwithperturbation ZHANDe—Sheng.'!,TANGJia—Shan. (1.CollegeofProfessi(real_rechnology,AnhuiUniversityofTechnology,Maanshan243011,P.R.China; 2.CollegeofMathematicsandComputerScience,NanjingNormalUniversity,Nanjing210097,P.R.China; 3.CollegeofMathematicsandPhysics.NanSingUniversityofPostsandTelecommunications.Nanjing210003,P.R. Abstract:Anewinsuranceriskmodelwasinvestigated,inwhichtheinsuranlsarrivalandtheclaimsarrivalaredriv— enbytwoindependentgeneralizedCoxprocesses.ThemodelisalsoperturbedbyaBrownianmotion.Usingthe martingalemethod.anupperboundfortheruinprobabilityofthemodelisgiven. Keywords:ruinprobability;upperbound:insuranceriskmodel;martingalemethod CLCnumber:0211.6DocumentCode:AArticleID:1673825X(2007)06—0782—03 关于带扰动广义Cox保险风险模型的破产概率 占德胜,唐加山. (1.安徽工业大学职,lk技术学院.安徽马鞍山243011;2.南京师范大学数学与计算机 科学学院,江苏南京210097 3南京邮电大学数理学院,江苏南京210003) 摘要:研究了一种新的带布朗运动干扰的保险风险模型.在模型中,投保人以及索 赔都成批到达,到达的点过程 是2个独立的Cox过程.利用鞅方法.给出了该模型破产概率的一个上界. 关键词:破产概率;上界;保险风险模型;鞅方法 1IntrOductiOn Insurancerisktheoryisoneoftheimportantre— searchareainbothoftheactuarialscienceandthe appliedprobability.Ruinprobahilityforriskrood— elsisoneofthehottopicsintherisktheoryduring thepastdecades.FortheclassicalCramer—Iund— bergmodel,theriskprocessorthesurplusprocess iswrittenby U()一"+一?Y(1) where"?0representstheinitialcapitalofthein— surer,c>0thepremiUIllincomerate,N()the claimnumberprocesswhichismodeledbyahomo— geneousPoissonprocessandYtheclaimsizesatis— fyingthatY,_i一1,2,…arei.i.d.nonnegative randomvariables. Undercertainconditions.theruinprobabilities andmanyotherresultsoftherelatedproblemsfor thiskindofmodelareobtainedr1.2].However, fromthepracticalpointofview,thisclassicalrisk modelisunrealistic.Therefore,variantextensions oftheclassicalmodelhavebeenpresentedinthe literatureandthenewonesaremorerealistically motivated[1].Specifically.ifthepremiumprocess *Receiveddate:2007—03—0l ischangedfromf,thesecondtermintheright M(t) handsideofequation(1),to?Y,,whereM(f)is ahomogeneousPoissonprocessandYisthepre— miumchargedfromthei-thpolicyholder,i一1,2, … ,thenthenewriskmodeIiScalleddoublePois— soninsuranceriskmodel(seee.g.[3]).Ifthein— surantarrivingprocessM(t)andtheclaimnumber processN(t)arechangedtomoregeneralproces— ses,Coxprocesses,thenthenewmodeliscalled doubleCoxinsuranceriskmodel(forexamplesee J4,5).Besidestheextensionofthepoint process,riskmodelswithinterferencearealsoin— vestigatedinrecentyears(seee.g.[6]). Inallthemodelsmentionedabove,ifthereis oneunfortunatethinghappened,thenthereisonly oneinsurantwhowillclaimforalOSScoveredby thepolicy.However,inpracticalsituation,forin— stancetheiDSUranceforvehicles,ifonetrafficacci— denthappens.thentherearealwaysatIeasttwo insurantswhowillapplyforthecompensationfor thelosses[7,8].Motivatedbythispracticalap- plication,inthisnote,weinvestigateanewmod— el.inwhichthearrivingprocessoftheinsurants andtheclaimnumberprocessaremodeledbytwo 第6期ZHANDeSt1eng?eta1:()nrLfinprobabilityforageneralizedCoxinsuranceriskmodelwitt1 I)erfurbation?783. independentgeneralizedCoxprocessesandthe modelisalsoperturbedbyaBrownianmotion.By usingthemartingalemethod,wegiveanupper boundoftheruinprobabilityofthemode1. 2Mainresultsanditsproofs Inthissection,wegivethemainresultsandits proofsofthispaper. 2.1Modeldescription Ietl(f),f?0}and!(),f?0}betwoinde pendentnonnegativestochasticprocesses,and l(f),f?0and!(),f?0}betwoindependent standardPoissonprocesses,whichareindependent from,(f),?0},=1,2.Then(f)一 ,?(A(t))isaCoxprocesswiththeintensity processgivenby(f),whereA()=i()dsis theaccumulatedintensityfunction,一1,2.Iet {/,2(Z),Z?1}(i一1,2)betwoindependentstochas— ticseriesof..d.randomvariableswiththecom— mondistributionsgivenbyP(/,2l(?)一J)一P,and P(2(?)一)一q,,J一1,2,…,Then ?… N,(f)一/,2i(z) ,一】 isageneralizedhomogeneousCoxprocesses,一1,2. Iet一{,,?1}andy一{y,?1}betwose riesofi.i.d.randomvariables,thentheinsurance riskmodelinvestigatedinthispaperisdefinedas follows (,)N2(, u(f)一"+?,一?y,+(f)(2) Where>0isaconstantandB(f),f?0}isa standardBrownianmotion.Supposethatallsto— chasticprocessesandrandomseriesareindepend— entfromeachother.Forthesurvivaloftheinsur— ?1Nn ancecompany,wesupposethatE(?,,?y,) ,一』J—J >0. Theintuitivemeaningofourriskmodelisasfol lows:"representstheinitialcapitaloftheinsurer, N()thenumberofpointatwhichthereisonein— surantorabathofinsurantsarriveduringthetime intervalr0,f].Thenumberofinsurantsatthel-th pointisdenotedby/,2l(Z).isthepremium chargedfromtheJthinsurant.N2(f)standsfor thenumberofpointatwhichthereisoneorabath ofclaimingoccurringduringthetimeperiod[0,f]. Thenumberofclaimingatthelthpointisdenotedby /,22(Z)andy_istheclaimsizeoftheJthclaim.The perturbationitemaB(f)representstheuncertaingain and/orlossoftheinsurancecompany. 2.2Ruinprobability Definetheruintimeoftheriskmodelas: T:一infU(f)<0U(0)一"} f?() Then(")一P(T<IU(0)一")istheruinprob abilityoftheinsuranceriskmode1.Forthec.n venienceinwhatfollows,weintroducethefollow ingnotation. 以I 一以1(),f?0}, VA2 . 歹一以2(f),?0}, {Nl(),?f}, 一 {N!(),?f},一B(),?f}, if一nVV2V Ifdefine D:一X(卜l1_l+…+XZ—l,2,… Q/:一卜1)..1+…+X?,Z一1,2,…(3) where/,2(0)一0(i=1,2),thenitiseasytoprove that{Df,Z?1}and{Q,Z?1}aretwoindependent seriesesofi.i.d.randomvariables,withthecorn mortdistributionsgivenbyF『j()一?PF(') 一1 andFQ(z)一?qFi'-Cr),for?0,whereF(,) isthevalueatofthedistributionfunctionofthe lfoldconvolutionofthedistributionoftherandom variableZ.Furthermore,similartoIemma2.2in [9],wehavethefollowinglemma. LemmalFortherandomvariablesdefinedin formula(3),theinsuranceriskmodeldefinedin formula(2)isequivalenttothefollowing '{,lt,{l? "+?D一?Q+(f)(4)1,1 Beforestatingthemainresult,wegiveakey lemma.Forthesake,forr?0,wedenoteand supposetheyexist, h】(r)一le…Ff)(d_1,)一1, J0 h2(r)= Lemma2Ifwe M : Ie…FQ(d)一1J0 define exp(一()) 1' exp(hl(r)以l()+!(t)A2(f)+Tt) thenM,?0}isamartingale. ProofTheproofisastandardoneandissimi lartothatintheliterature.Wegiveasketchhere. N, Firstlv,wepr()ve1hatE(e/l.,.I)一eA1(t)hI', ,2? thenprovethatE(e,jI!)一eAz(tHQ'.Fina1 ly,usingtheconditionalexpectationandproperties oftheBrwonianmotion,provethemartingale property,i.e.E(M)一Mfor0??f.This completestheproof. Hereisthemajnresl】1t ? 784?重庆邮电大学(自然科学皈)第19卷 Theorem1Fortheinsuranceriskmodelde{lrle informula(4).theruinprobabilityininfinitytime hasanupperboundgivenby(")?(11(7)?e一, where c(r)一E(exp(矗(,)n()^:(r)^()斗—)). ProofBasedontheresultobtainedinIemma 2.theproofofthetheoremissimilartothatof Theorem2.2inl10.Thesketchoftheproofisas thefollowing. Foranyt<,itcanbeshowi]thatexp(7") 一 M.?E.(MT^,I丁?t)P(丁?t.).Itimplies from(丁)<OthatE,(M7了?t)?infexp(一 (l(r)AI(t)一h2(r)A!(t)一去f7一)).Therefore, P(丁?t0)?exp(r")SuI)exp(hI(7)A](f), ?lf 1h ,2(r)A,2(f)+?toy:r).Takingexpectationonboth gettheresult.The Itiseasytoseetllatforaconcrete,一.(,(r)de— finedinTheorem3isaconstantan(1thedecayrate oftheupperboundise一".Apparently.1arger, resultsinmorepreciousasymptoticbehavior.If define R—sup{r,('(7')一一, similarlytothediscussionafterTheorcm2.2in L1O,wehaveasimilarcoro1lary. CorollarylForany0??R.tileruinproba— bility(")forthegeneralizedCoxinsurancerisk modelhasthefollowingupperbound. (")?C(R一?)e. Remark1Theconcreteformoftheresultsob— rainedinthisnoteissireilartothoseintIlelitera— ture(seeTheorem2.2an(1Corollary2.3inr10), however,thecontentiverydifferent.Specifical— lY,asanexample,(,一)define(1inf()rmula(5)is closelyrelatedtoboththesizeoftl1e1)atchclai— mingandthelOSIclaime(1bytheifiSHrants.wl1ilea similarfunctionh(r)iI1r10]isonlydepex1denton theclajms;zeoftl1ensl1rFli1ts. 3Concludingremarks Inthisnote.weinvesligateanewinSllrancerisk model,inwhicbtheinsurantsarriv,1]andt1]e claimsarrivalaredriYenbvtwoindependentCox processes.Ateachofinsurallts}lrrivalpoint,there isabatchofinstlran1.swhopaythepremiuman(1at eachofelajHiSpoint.thereis}lbgItchofiIlSt1l-a1I1s whoclai12]theJOSt..per',urbatjollofBrwoRiaH motionisalsointrodtlced.Usingtilemartingale method,wegiveanupperboundoftheruinproba— bilityfortheriskmode1. References: :1] :2] :3] 10 L,HENGShixue.I,he5urveyforresearcheso{ruin theoryJ(inChinese).AdvancesinMathematics (China).2002,31(5):403—422. EEBRECHTSP.KIUPPEIBERGC,MIK0SCHT. 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YUWen一ang.Theprobabilityofruinmultitype-in— suranceriskmode}withcompoundgeneralizedhomo— geneousPoissonprocess[J~(inChinese).Comm.On . Xpp1.Math.AndComput.2003,17(3):63—69. XIANGYang.RuinprobabilitiesinaCoxriskmodel perturbedbybrownianmorionJ](inChinese).Jour— nalofHuaihuaUniversilv.2005,24(2):3—5. 作者简介 占德胜(I968一).男,安徽省怀宁人,讲师,南 京师范大学硕士研究生,主要研究方向为 偏微分方程:E—mail:zhandesheng559@ 163.eoHl. 唐加山(1968一).男,安徽天长人,教授,博 士.硕士生导师,主要研究方向为应用概 率,随机过程,排队论以及信号与信息处理 等,发论文3o余篇.E—mail:jiashant@ yahoo.ca. w g一 dC