固定交易费用情形下的最优分红与再保险张鑫南开大学数学科学学院，天津300071摘要：在这篇文章中，我们考虑金融公司在具有交易费用情况下的最优分红问题。除了分红控制变量，金融公司还可以采取比例再保险以及投资于由无风险资产和风险资产构成的金融市场以降低自身的风险。由于固定交易费用的存在，这个问题变为一个混合经典-脉冲随机控制问题。我们明确的解决了这个问题并构造了最优的值函数和最优策略。关键词：应用概率论；脉冲控制；分红；交易费用中图分类号：O211.9TheOptimizationofDividendsandReinsurancewithFixedTransactionCostsZHANGXinSchoolofMathematicalSciences,NankaiUniversity,Tianjin300071Abstract:Inthispaper,weconsiderthedividendoptimizationproblemforafinancialcorporationwithtransactioncosts.Besidesthedividendcontrol,thefinancialcorporationtakesproportionalreinsurancetoreduceriskandinvestsitsreserveinafinancialmarketconsistingofrisk-freeasset(bond)andriskyasset(stock).Becauseofthepresenceoffixedtransactioncosts,theproblembecomesamixedclassical-impulsestochasticcontrolproblem.Wesolvethisproblemexplicitlyandconstructthevaluefunctiontogetherwiththeoptimalpolicy.基金项目：theSpecializedResearchFundfortheDoctoralProgramofHigherEducation(SRFDPGrantNo.20100031120002),NationalNaturalScienceFoundationofChina(NSFCgrantNo.11001139,11171164)作者简介：ZhangXin（1985-），male，associateprofessor，majorresearchdirection：ActuarialScience,MathematicalFinance.Correspondenceauthor：ZhangXin（1985-），male，associateprofessor，majorresearchdirection：ActuarialScience,MathematicalFinance.-1-Keywords:AppliedProbability;Impulsecontrol;Dividends;Transactioncosts.0IntroductionApplyingcontinuoustimestochasticprocesstomodelfinancialproblemsandinparticularto(re)-insuranceproblemshasattractedsignificantinterestoverrecentdecades.Thereexistsanextensiveliteratureonapplicationofstochasticcalculusandstochasticcontroltechniquetovariousinsuranceproblemssuchasmaximizingtheexpectedpresentvalueofthedividendspaidoutuptothetimeofbankruptcy.AnexcellentsurveyofstochasticmodelsfortheoptimaldividendpolicycanbefoundinTaksar[1].Forothercontributionstothisproblem,seeAsmussenandTaksar[2],PaulsenandGjessing[3],HøjgaardandTaksar[4,5,6],Asmussenetal.[7],Choullietal.[8,9].Inallofthesepapers,thereserveprocessofaninsurancecompanyisdrivenbyaBrownianmotionwithconstantdriftanddiffusioncoefficients.Moreovertheyassumecostlesstradingandtheoptimalstrategyconsistingoftradingorinterventioncanactateverytimeinstant.Takingtransactioncostsintoaccount,howtosolvetheoptimaldividendproblemliesinthetheoryofimpulsecontrol(seeBensoussanandLions[10]).Therearetwocommontransac-tioncosts:proportionalcosts(seeforinstanceCadenillas[11,Section4])andfixedcosts(seeCadenillas[11,Section6]).ThetheoryofimpulsecontrolwasintroducedbyBensoussanandLions[12]andextendedbyRichard[13].OtherimportanttheoreticalcontributionsincludeHarrisonetal.[14],BensoussanandLions[10],Dixit[15],Dumas[16]andKorn[17,18].Jeanblanc-PicqueandShiryaev[19]firstappliedimpulsecontroltheorytoobtaintheoptimaldividendpolicywhenthereisafixedcosteachtimeforthedividendpaidout.CadenillasandZapatero[20]applyclassicalandimpulsecontroltodealwithexchangerateproblems.TheclassicalandimpulsestochasticcontrolfortheoptimizationofthedividendandriskpolicyisstudiedbyCadenillasetal.[21].-2-∑∞Inthispaper,weextendtheresultsofCadenillasetal.[21]byinvolvingtheinvestmentchoice,thatis,besidestheproportionalreinsuranceanddividendspay...