2002（SCI）OPTIMAL CONSUMPTION AND PORTFOLIO WITH BOTH FIXED AND PROPORTIONAL TRANSACTION COSTS

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1、SIAMJ.CONTROLOPTIM.c2002SocietyforIndustrialandAppliedMathematicsVol.40,No.6,pp.1765–1790OPTIMALCONSUMPTIONANDPORTFOLIOWITHBOTHFIXEDANDPROPORTIONALTRANSACTIONCOSTS∗BERNTØKSENDAL†ANDAGNESSULEM`‡Abstract.Weconsideramarketmodelwithonerisk-freeandoneriskyasset,i

2、nwhichthedynamicsoftheriskyassetaregovernedbyageometricBrownianmotion.Inthismarketweconsideraninvestorwhoconsumesfromthebankaccountandwhohastheopportunityatanytimetotransferfundsbetweenthetwoassets.Wesupposethatthesetransfersinvolveafixedtransactioncostk>0,ind

3、ependentofthesizeofthetransaction,plusacostproportionaltothesizeofthetransaction.Theobjectiveistomaximizethecumulativeexpectedutilityofconsumptionoveraplanninghorizon.Weformulatethisproblemasacombinedstochasticcontrol/impulsecontrolproblem,whichinturnleadstoa

4、(nonlinear)quasi-variationalHamilton–Jacobi–Bellmaninequality(QVHJBI).WeprovethatthevaluefunctionistheuniqueviscositysolutionofthisQVHJBI.Finally,numericalresultsarepresented.Keywords.portfolioselection,transactioncost,impulsecontrol,quasi-variationalinequali

5、ties,viscositysolutionsAMSsubjectclassifications.Primary,93E20,91B28;Secondary,60H30,49L25,35R45PII.S03630129003760131.Introduction.Let(Ω,F,P)beaprobabilityspacewithagivenfiltration{Ft}t≥0.WedenotebyX(t)theamountofmoneytheinvestorhasinthebankattimetandbyY(t)the

6、amountofmoneyinvestedintheriskyassetattimet.WeassumethatintheabsenceofconsumptionandtransactionstheprocessX(t)growsdeterministicallyatexponentialrater,whileY(t)isageometricBrownianmotion;i.e.,(1.1)dX(t)=rX(t)dt,X(0)=x,(1.2)dY(t)=αY(t)dt+σY(t)dW(t),Y(0)=y,wher

7、eW(t)isone-dimensionalFt-Brownianmotionandα>r>0andσ=0areconstants.Supposethatatanytimettheinvestorisfreetochooseaconsumptionratec(t)≥0.Thisconsumptionisautomaticallydrawnfromthebankaccountholdingwithnoextracosts.Moreover,atanytimetheinvestorcandecidetotransf

8、ermoneyfromthebankaccounttothestockandconversely.Assumethatapurchaseofsizeofstocksincursatransactioncostconsistingofasumofafixedcostk>0(independentofthesizeofthetransaction)plusacostλpro