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Optimal Investment and Consumption Portfolio Choice Problem for Assets Modeled by Levy Processes

Permanent Link: http://ufdc.ufl.edu/UFE0042720/00001

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Title: Optimal Investment and Consumption Portfolio Choice Problem for Assets Modeled by Levy Processes
Physical Description: 1 online resource (111 p.)
Language: english
Creator: SANKARPERSAD,RYAN GOVINDRA
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

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Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: We consider an extension of Merton's optimal portfolio choice and consumption problem for a portfolio in which the underlying risky asset is an exponential Levy process. The investor is able to move money between a risk free asset and a risky asset and consume from the risk free asset. Given the dynamics of the total wealth of the portfolio we consider the problem of finding portfolio weights and a consumption process which optimizes the investors expected utility of consumption over the investment period. The problem is solved in both the finite and infinite horizon cases for a family of HARA utility functions using the techniques of stochastic control theory. The general closed form solutions are found for for the case of a power utility function and then for a more generalized utility. We consider a variety of Levy processes and make a comparison of the optimal portfolio weights. Our results are consistent with the expectation that the greater the inherent uncertainty of a process leads to a smaller fraction of wealth invested in the risky asset. In particular an investor is more careful when the risky asset is a discontinuous Levy process when compared to the continuous case such as those found in a geometric Brownian motion model.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by RYAN GOVINDRA SANKARPERSAD.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Yan, Liqing.

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Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042720:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042720/00001

Material Information

Title: Optimal Investment and Consumption Portfolio Choice Problem for Assets Modeled by Levy Processes
Physical Description: 1 online resource (111 p.)
Language: english
Creator: SANKARPERSAD,RYAN GOVINDRA
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We consider an extension of Merton's optimal portfolio choice and consumption problem for a portfolio in which the underlying risky asset is an exponential Levy process. The investor is able to move money between a risk free asset and a risky asset and consume from the risk free asset. Given the dynamics of the total wealth of the portfolio we consider the problem of finding portfolio weights and a consumption process which optimizes the investors expected utility of consumption over the investment period. The problem is solved in both the finite and infinite horizon cases for a family of HARA utility functions using the techniques of stochastic control theory. The general closed form solutions are found for for the case of a power utility function and then for a more generalized utility. We consider a variety of Levy processes and make a comparison of the optimal portfolio weights. Our results are consistent with the expectation that the greater the inherent uncertainty of a process leads to a smaller fraction of wealth invested in the risky asset. In particular an investor is more careful when the risky asset is a discontinuous Levy process when compared to the continuous case such as those found in a geometric Brownian motion model.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by RYAN GOVINDRA SANKARPERSAD.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Yan, Liqing.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042720:00001


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OPTIMALINVESTMENTANDCONSUMPTIONPORTFOLIOCHOICEPROBLEMFORASSETSMODELEDBYLEVYPROCESSESByRYANG.SANKARPERSADADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011RyanG.Sankarpersad 2

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ACKNOWLEDGMENTS IwouldliketothankmydissertationadvisorDrLiqangYanforhishelpthroughouttheresearchprocess,yourguidancewascrucialtomygrowthasamathematician.AlsoDrYan'scoursesinstochasticcalculusandmathematicalnancehelpeddevelopmyinterestintheeldofnance.Iwouldalsoliketoindividuallythankallmembersofmydissertationcommittee.DrMichaelJury'syearlongclassinMeasureTheorywasfundamentaltomyunderstandingofmoreadvancedconceptsrequiredforthisdissertation.DrMuraliRao'scourseinProbabilitytheoryhelpedsolidifymyunderstandingofmeasuretheoryandwasagreatintroductiontothetheoryofrandomvariables.DrStanUryasev'scoursesonxedincomederivatives,portfoliotheoryandriskmanagementtechniquesintroducedmuchofthenancialtheorythatIhaveusedinthisdissertationandthroughoutthejobapplicationprocess.DrJosephGlover'spoignantquestionsduringmyOralExamandinsightintotheeldofLevyProcesseswerehelpfulinguidingmyresearchoverthelastfewyearsofresearch.IwouldalsoliketothankDrJayRitterfromtheDepartmentofFinanceforteachingmetheclassicaltheoryofcorporatenanceduringhisyearlongcourse.Lastbutnotleast,Iwouldliketothankmyfamily(Mom,Dad,RiannaandReshma)foralltheloveandsupportthroughoutthedissertationprocess,youwereallinstrumentaltomysuccess. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1GENERALFRAMEWORKANDPROBLEMSETUP ............... 9 1.1Introduction ................................... 9 1.2StochasticCalculusPreliminaries ....................... 11 1.3ItoDiffusionandit'sGenerator ........................ 14 2STATEMENTOFMAINPROBLEM ......................... 22 2.1OptimalControlTheory ............................ 22 2.2DynamicProgrammingMethodology ..................... 23 2.3HamiltonJacobiBellman ........................... 27 2.4VericationTheorem .............................. 31 3MERTONPROBLEM ................................ 35 3.1ClassicMertonSolution ............................ 35 3.2InniteHorizonPowerUtility .......................... 38 3.3InniteHorizonLogUtility ........................... 45 3.4FiniteHorizonPowerUtility .......................... 50 3.5FiniteHorizonLogUtility ............................ 58 4LEVYPROCESS ................................... 61 4.1WhyuseaLevyProcesswithJumps? .................... 61 4.2PreliminariesofLevyProcess ......................... 62 4.3Ito-LevyDiffusionandit'sGenerator ..................... 67 4.4VericationTheoremforLevyProcesswithJumps ............. 73 5OPTIMALCONTROLPROBLEMINTHEJUMPCASE ............. 77 5.1ClassicMertonProblemwithJumps ..................... 77 5.2JumpCasewithLogUtility .......................... 84 5.3FiniteHorizonPowerUtilitywithJumps ................... 88 5.4GeneralizationofBequestFunctionintheJumpCase ........... 93 4

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6NUMERICALRESULTSANDCONCLUSION ................... 97 6.1JumpdiffusionandLevytriplets ........................ 97 6.2OptimalforPowerUtility .......................... 99 6.2.1\(1,1)Process ............................. 100 6.2.2CompoundPoissonProcesswithExponentialDensity ....... 101 6.3ExplicitValueFunctions ............................ 104 6.4Conclusion ................................... 106 REFERENCES ....................................... 109 BIOGRAPHICALSKETCH ................................ 111 5

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LISTOFTABLES Table page 2-1Tablewithcomparisonoftwoworldsusedtosolveoptimalcontrolproblem .. 26 3-1MaximumdifferencesbetweennitehorizonandinnitehorizonfordifferentvaluesofT ...................................... 54 6-1Tableforasafunctionof,where=.16,r=.06,=.5 .......... 104 6-2Tableforasafunctionof,where=.16,r=.06,=.5 .......... 105 6

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LISTOFFIGURES Figure page 3-1Valuefunctionforpowerutilitywithnojumpswithparameters,=.16,r=.06,=1,=.5,=.3 ............................... 41 3-2Wealthprocessforpowerutilitywithnojumpswithparameters,w=1,=.16,r=.06,=1,=.5,=.3 .......................... 43 3-3Valuefunctionforlogutilitywithnojumpswithparameters,=.16,r=.06,=1,=.5,=.3 ............................... 47 3-4WealthprocessforLogutilitywithnojumpswithparameters,w=1,=.16,r=.06,=1,=.3 .............................. 49 3-5PlotsofthevaluefunctionforvaluesofT=1,2,100comparedtotheplotofthevaluefunctionforinnitehorizon ........................ 54 5-1PlotsofthevaluefunctionforvaluesofT=1,2.5,5,7.5,100comparedtotheplotofthevaluefunctionintheinnitehorizoncase. ............... 92 5-2Plotsofthevaluefunctionforvaluesof=1,.5,.25,.125,1e-09comparedtotheplotofthevaluefunctionfor=0 ....................... 96 7

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMALINVESTMENTANDCONSUMPTIONPORTFOLIOCHOICEPROBLEMFORASSETSMODELEDBYLEVYPROCESSESByRyanG.SankarpersadMay2011Chair:LiqangYanMajor:Mathematics WeconsideranextensionofMerton'soptimalportfoliochoiceandconsumptionproblemforaportfolioinwhichtheunderlyingriskyassetisanexponentialLevyprocess.Theinvestorisabletomovemoneybetweenariskfreeassetandariskyassetandconsumefromtheriskfreeasset.Giventhedynamicsofthetotalwealthoftheportfolioweconsidertheproblemofndingportfolioweightsandaconsumptionprocesswhichoptimizestheinvestorsexpectedutilityofconsumptionovertheinvestmentperiod.Theproblemissolvedinboththeniteandinnitehorizoncasesforafamilyofhyperbolicabsoluteriskaversionutilityfunctionsusingthetechniquesofstochasticcontroltheory.Thegeneralclosedformsolutionsarefoundforforthecaseofapowerutilityfunctionandthenforamoregeneralizedutility.WeconsideravarietyofLevyprocessesandmakeacomparisonoftheoptimalportfolioweights.Wendthatourresultsareconsistentwithexpectationsthatthegreatertheinherentuncertaintyofagivenprocessleadstoasmallerfractionofwealthinvestedintheriskyasset.InparticularaninvestorismorecarefulwhentheriskyassetisadiscontinuousLevyprocesswhencomparedtothecontinuouscasesuchasthosefoundinageometricBrownianmotionmodel. 8

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CHAPTER1GENERALFRAMEWORKANDPROBLEMSETUP 1.1Introduction Givenaportfolioofassetstheproblemofndingoptimalweightsforeachoftheassetshasbeenextensivelystudiedandhasmanyrealworldapplications.TherstmajorbreakthroughinthisareawasmadebyHenryMarkowitzinhis1952JFpaper[ 17 ]inwhichhewasabletondoptimalportfolioweightsforeachoftheassetsintheportfolioinadiscretetimesetting.Markowitzanalysiswasaoneperiodmodelthataimedtominimizetheriskunderaconstraintontheexpectedreturnoftheportfolio.Hewasabletondasocalledefcientfrontierwhichwastheintersectionofthesetofalladmissibleportfolioswithaportfoliowithminimumriskandmaximumexpectedreturn.Onemajorcomputationalrestrictionofthismodelisthatoneneedsalargedatasettocomputethenecessarycovariancesbetweeneachoftheassetsintheportfolio.AsuccessfulattempttoextendMarkowitz'smodeltoacontinuoustimesettingwasmadebyRobertMertoninhis1972[ 18 ]workforwhichhereceivedtheNobelprize.InthisworkMertonshowedthatifassetsintheportfolioweremodeledusinggeometricBrownianmotion,theportfolioofnassetscouldbereducedtoaportfolioofonlytwoassetsusinghissocalledmutualfundtheorem.Themutualfundportfolioconsistingoftwoassetsismadeupofariskfreeassetandariskyasset.TheriskfreeassetistypicallyisaU.S.treasurybillwhiletheriskyassetisalinearcombinationofthen)]TJ /F5 11.955 Tf 12.91 .01 Td[(1remainingassetsintheportfolio.ThisideamirrorsthediscretetimeideaofMarkowitz'smarketportfoliowhichismadeupofacombinationofassets.OncethissimplicationhasbeenmadeMertonconsiderstheproblemofndingoptimalconsumptionandportfolioweightsforthefamilyofhyperbolicabsoluteriskaversionutilityfunctions.OncethesetofoptimalcontrolsarefoundMertonthenndstheportfolioweightsforeachofthenassetsintheportfoliosothatacomparisoncanbemadewiththeoptimalMarkowitzmean-varianceportfolio.Thisprocedurewillnotbethe 9

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amajorfocusofourpaperandwerefertheinterestedreaderto[ 18 ]foradiscussioninthisdirection.TherehavebeenmanystudieswhichhaveconsideredrelaxingtheconditionsofMerton'soriginalpaper.DavisandNorman[ 10 ]1990paperrelaxestheconditionofnotransactioncostsandpresentmoregeneralizedresultstoincludeconvextransactioncostfunctions.FlemingandPang[ 11 ]considertheMertonproblemwithaVasieckinterestratemodelandpresentresultsusingasubsolution/supersolutionmethodology.Paperssuchas[ 16 ]and[ 27 ]introduceilliquidassetsintothemarketandpresentresultstotheoriginalMertonproblem.EachofthesepaperspresentstheclassicMertonproblemandit'ssolutioninsomecontext,atrendthatwewillcontinueinthispaper.HoweverwewillmoveinanewdirectionbyconsideringtheMertonproblemwithaexponentialLevyprocesssothatthepriceprocessisallowedtohavejumps(inparticularisallowedtobediscontinuous). AlthoughwepresentsomeresultsinthegeometricBrowniancase(classicMertoncase),themainresultsofourpaperareforpriceprocesseswithdiscontinuitiessuchasthoseconsideredinMerton's1976paper[ 19 ].InthispaperMertonconsiderspricesprocesseswhichareallowedtohavePoissonjumpsandderivesaclosedformsolutionforthepriceofvanillaoptions.Thispaperwasamajorbreakthroughintheuseofdiscontinuouspriceprocesses,andalthoughwillnotbeconcerneddirectlywithoptionpricingformulaswewillusetheideaspresentedinthispaper.ThemainresultsofourpapercomesfrommodelingtheriskyassetinourportfoliobyaLevyprocesswithjumpsandndingcontrolswhichoptimizeexpectedutilityofconsumption.ThisworkcloselyfollowsresearchperformedbyBerntOksendalwhicharesummarizedinhisbookappliedstochasticcontrolofjumpdiffusions[ 21 ].OksendalsolvestheMertonprobleminthecasewhentheunderlyingpriceprocessisanexponentialLevyprocess,asolutionwhichwepresentforcomparisonandcompleteness.OncewehavepresentedthesolutionincasescoveredunderOksendal'spresentationweproceedtogeneralizetheutilityfunctionsandderiveclosedformsolutionstothecorrespondingproblem.A 10

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derivationoftheclosedformsolutionisprovidedalongwithnumericalresultstoverifytheaccuracyofthesolutionsprovided. Theformulationandsolutionofourproblemrequiresthestudyofstochasticcalculusandothertheorieswhichrequireknowledgeofstochasticcalculus.Acompletesurveyofstochasticcalculusisnotprovidedanditisassumedthereaderisfamiliarwiththegeneraltheoryfoundinsuchbooksas[ 20 ]and[ 22 ].Inparticularweassumethereaderhashadsomeintroductoryexposuretostochasticdifferentialequationsandtheassociatedprobabilitytheoryrequiredtosolvetheseequations.Webeginwithapresentationofthestochasticcalculusnecessaryforouranalysis,startingwithsomeintroductorydenitionsandtheorems. 1.2StochasticCalculusPreliminaries Thissectionlaysthefoundationformuchofthepaper,manyoftheseresultsarefoundinintroductorystochasticcalculusbookssuchas[ 20 ]and[ 22 ].Weassumethereaderisfamiliarwithaprobabilityspace(,F,P)whereisthesetofallpossibleoutcomes,Fisthe-algebra,andPisaprobabilitymeasure.Astochasticprocess(Xt)t0isasequenceofrandomvariablesdenedoftheprobabilityspace(,F,P)takingvaluesinR.InparticularforxedtimetthemapXt(!)isarandomvariableforeach!2,whileforaxedrealizationoftheevent!2themapXt(!)isarealvaluedfunctionforeacht0.Asisstandardwewillsuppressthedependenceon!throughoutthepaperandwritetherandomvariableasXt.Tostudyastochasticprocess(Xt)t0denedon(,F,P)weneedawayofencompassingtheinformationtheprocessXthasaccumulateduptimetsothatdecisionscanbemadeabouttheprocess.Informationaboutrandomvariablesisclassiedusingtheconceptofa-algebra,henceforastochasticprocesswewillneedtoconsiderasequenceofincreasing-algebrastypicallyreferredtoasaltration. 11

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Denition1.1. Givenaprobabilityspace(,F,P),altrationisafamilyF=(Ft)t0of-algebrassuchthatFsFt80s
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(ii) fort00andh0(stochasticcontinuity)limt!hP(jBt)]TJ /F6 11.955 Tf 11.95 0 Td[(Bhj>)=0 FormuchofourworkwewillbeconsideringtheltrationgeneratedbyBrownianmotion,inparticularwewillbeworkingwiththeltrationFB=(Bs;s2[0,t]).WhenconsideringourstochasticdifferentialequationsasadiffusionprocesstheBrownianmotiontermcontainsallsourcesofrandomness.Hencetheltration(Bs;s2[0,t])willcontainallinformationnecessarytomakedecisionsabouttherandomcomponentofthediffusionprocess.Onceweareabletoencapsulatetheinformationofaprocessintheformofa-algebraweareabletodeveloptheideaoftheconditionalexpectationgivena-algebra.Thisconceptsallowsustondasubsetofstochasticprocesseswhoseexpectedfuturevaluebasedongiveninformationisequaltoitscurrentvaluecalledamartingale.Themathematicalformalitiesofwhicharecapturedinthefollowingdenition. Denition1.5. Astochasticprocess(Xt)t0iscalledamartingalewithrespecttotheltration(Ft)t0ifforallt0 E[jXtj]<1i.e.Xtisintegrable XtisFt-measurableforallt foranys
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usetheseextensivelywhileprovingresultsforgeneralLevyprocesseslateron.WewouldliketostudythesetofallItoprocesseswhichcanbewrittenintermsofthesetwomartingalessothatwemayreplicatethemodernportfoliotheory.Todothiswemustwritedownthestochasticdiffusionequationswhichmodelsstockpricesofourportfolio,hencewemustgiveaprecisemathematicaldenitionofadiffusionequation. 1.3ItoDiffusionandit'sGenerator Tostudyaportfolioofassetsinacontinuoustimesettingweneedtodescribethedynamicsofthegivenassets,thiscanbedonebyusingthetheoryofstochasticdifferentialequations.Eachassetwillbemodeledusingadiffusionprocesswhichcanbethoughtofasaparticlewhosetrajectoryisinuencedbyanexternalsourceofrandomness.Thisrandomnessmaycomefrombeattributedtooneormanyexternalsources,fortheremainderofthepaperwewillassumethattherewillbeonlyonesourceofrandomness.Thismeansthatourtheoremswillbestatedintheonedimensionalcase,sinceaonedimensionalBrownianmotionwillbethedrivingforceofrandomnessinourdiffusionequations.Themultidimensionalversionsofthetheoremsweuseinthissectioncanbefoundinanintroductorystochasticcalculusbookssuchas[ 20 ]or[ 22 ].ThesourceofrandomnessismodeledbyaddingaBrownianmotiontermtoastandarddiffusionprocess,moreprecisely Denition1.6. AnItodiffusionisastochasticprocess(Xt)[s,T]satisfyingastochasticdifferentialequationoftheformdXt=(t,Xt)dt+(t,Xt)dBtt2[s,T];Xs=x2R (1) whereBtisanone-dimensionalBrownianmotionand:[s,T]R!R,:[s,T]R!RsatisfythefollowingglobalLipschitzandatmostlineargrowthconditionsforallx,y2R,t2[s,T]j(t,x))]TJ /F3 11.955 Tf 11.96 0 Td[((t,y)j+j(t,x))]TJ /F3 11.955 Tf 11.96 0 Td[((t,y)jCjx)]TJ /F6 11.955 Tf 11.95 0 Td[(yj 14

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j(t,x)j+j(t,x)jD(1+jxj) MoreoverifthesolutionXt(!)isadaptedtotheltration(Bs;st)thenEs,xZTsjXtj2dt<1 TheuniquesolutionofEquation( 1 )willbedenotedbyXs,xtforallts,ifs=0wewillusethenotationXxt.Wenotethatthedriftcoefcientandthediffusioncoefcientdependonthetimeparametert,withatransformationwecanreducethiscasetothetimedependentcase.Thereductionwillbeperformedlateroninthepaper,hencemanyofthedenitionsandtheoremswillbestatedwithoutexplicittimedependence.ThesolutionXs,xtofEquation( 1 )isreferredtoastimehomogeneousthefollowingreason,Xs,xs+h=x+Zs+hs(r,Xs,xr)dr+Zs+hs(r,Xs,xr)dBr (1) Makingthechangeofvariablesu=r)]TJ /F6 11.955 Tf 11.96 0 Td[(swemaywriteEquation( 1 )asXs,xs+h=x+Zh0(u+s,Xs,xu+s)du+Zh0(u+s,Xs,xu+s)dBu+s=x+Zh0(u+s,Xs,xu+s)du+Zh0(u+s,Xs,xu+s)d Bu whered Bu=Bu+s)]TJ /F6 11.955 Tf 12.18 0 Td[(BsisaBrownianmotionwiththesameP-distributionasBu.Since(Bu)u0and( Bu)u0havethesameP-distributionastochasticdifferentialequationoftheformdXt=(t,Xt)dt+(t,Xt)dBt;X0=x whosesolutionX0,xhcanbewrittenasX0,xh=x+Zh0(u,X0,xu)du+Zh0(u,X0,xu)dBu (1) wehavethattheequationsgoverningbothoftheseItoprocessesareessentiallythesamefromaprobabilitystandpoint.Inparticular,comparingEquations( 1 )and( 1 ) 15

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weseethatthelatterisjusttheformerequationwiths=0sothatX0,xhhasthesameP-distributionasXs,xs+hi.e.(Xt)t0istimehomogeneous.Wewillhenceforthrefertobothstochasticdifferentialequationsinterchangeably.AnimportantpropertyofItodiffusionswhichwillbeofgreatuseisthesocalledMarkovpropertywhichallowsustousethepresentbehaviorofadiffusiontomakedecisionsaboutthefuturewithoutconsideringpastbehavior.InordertostatetheStrongMarkovpropertyweneedthefollowingdenitionofarandomtime Denition1.7. Let(Ft)t0bealtration,afunction:![0,1)iscalledastoppingtimewithrespectto(Ft)t0iff!:(!)tg2Ft Astoppingtimeisarandomvariableforwhichthesetofallpaths(events)!2withf(!)tgcanbedecidedgiventheltrationFt.Thedenitionallowsustoconsideronlytherandomtimesforwhichwemaydecidedwhetherornotthetimehasbeenreachedgiventheinformationuptotimeti.e.givenFt.OncewehavedenedthisrandomtimewemaystatetheStrongMarkovproperty Theorem1.1. (StrongMarkovpropertyforItodiffusions)Let(Xt)t0beanItodiffusion,astoppingtimeandf:Rn!RbeaBorelmeasurablefunction,thenfor!2,h0Ex[f(X+h)jF](!)=EX(!)[f(Xh)] Throughoutmuchofthetheoryofmathematicalnanceweassumethatthepriceprocessofagivenassetisthesolutionofastochasticdifferentialequation,henceastochasticprocess.Wewouldliketoconsiderfunctionsonthesestochasticprocess,sothenaturalquestionistoaskwhetherafunctiononastochasticprocessisitselfastochasticprocess.Inordertoanswerthisquestionweneedtobeabletowritedownthedifferentialequationassociatedwiththisnewprocess.ThismeansthattheclassicaltheoryofNewtoniancalculusmustbeextendedtoincludestochasticterms.Thisisdone 16

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byIto'sformulawhichisageneralizationofthechainrulefromNewtoniancalculus,wewillextendtoamoregeneralversionlaterinthepaper. Theorem1.2. (1-dimensionalItoformula)GivenanItoprocessXtoftheformdXt=(t,Xt)dt+(t,Xt)dBt whereBt2RisaBrownianmotion.Ifg(t,x)2C1,2([0,1),R)thenYt=g(t,Xt) isanItoprocesswithdYt=@g @t(t,Xt)dt+@g @x(t,Xt)dXt+1 2@g2 @x2(t,Xt)dXtdXt Proof. AproofofIto'slemmacanbefoundonpage46of[ 20 ] AmajorcomponentofthetheoryofstochasticoptimalcontrolistheinnitesimalgeneratorofanItodiffusion.Thiscanbethoughtofasandextensionofthederivativefromthedeterministiccalculus,wherethelimitdenitionhasasimilarformwithextensionstoincludetostochasticnatureofanItodiffusion. Denition1.8. Let(Xt)tsbeanItodiffusioninRoftheformdXt=(t,Xt)dt+(t,Xt)dBtXs=x thentheinnitesimalgeneratorAofXtisdenedbyA(s,x)=limt!sEs,x[(t,Xt)])]TJ /F3 11.955 Tf 11.96 0 Td[((s,x) t)]TJ /F6 11.955 Tf 11.95 0 Td[(s forallx2R,s2[0,1)and2C1,20([0,1),R).ThesubsetofL2([0,1)R)forwhichthelimitA(s,x)existsforalls,xwillbecalledDA. Itturnsoutthatthisderivativeinthestochasticsenseisrelatedtotheclassicalderivativeinaninherentway,whichcanbefoundbyapplyingIto'sLemma.ThefollowingtheoremshowsthattheinnitesimalgeneratorofanItodiffusionturnsouttobea 17

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secondorderdifferentialoperatorinthecaseofdiffusionsdrivenbyBrownianmotion.ItwillalsobeshownlaterthatinthecaseofdiffusionsdrivenbyamoregeneralLevyprocesstheinnitesimalgeneratorwillendupbeinganintegro-differentialoperator. Theorem1.3. LetXtbeanItodiffusionoftheformdXt=(t,Xt)dt+(t,Xt)dBtXs=x andconsiderthedifferentialoperatorAonC1,20([0,1),R)givenbyA=@ @t+@ @x+1 22@2 @x2 andlet2C1,20([0,1),R)besuchthatforallts,s2[0,1)andx2REs,xZtsjA(r,Xr)drj<1,andEs,x"Zts@ @x(r,Xr)(r,Xr)2dr#<1 then2DAandA(s,x)=A(s,x). Proof. Since2C1,20([0,1),R)wemayapplyIto'sformulatocomputed(t,Xt)=@ @t(t,Xt)+@ @x(t,Xt)dXt+1 2@2 @x2(t,Xt)dXtdXt=@ @t(t,Xt)+@ @x(t,Xt)[(t,Xt)dt+(t,Xt)dBt]+1 22(t,Xt)@2 @x2(t,Xt)dt=A(t,Xt)dt+@ @x(t,Xt)(t,Xt)dBt Integrationofbothsidesoftheequationwithrespecttothepropermeasureswendthat(t,Xt))]TJ /F3 11.955 Tf 11.95 0 Td[((s,Xs)=ZtsA(r,Xr)dr+Zts@ @x(r,Xr)(r,Xr)dBr usingthefactthatXs=xandtakingtheexpectationEs,xofbothsidesoftheequationweareabletocomputethenumeratorofthelimitEs,x[(t,Xt)])]TJ /F3 11.955 Tf 11.96 0 Td[((s,x)=Es,xZtsA(r,Xr)dr+Es,xZts@ @x(r,Xr)(r,Xr)dBr 18

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Giventhatthefunction@ @x(r,Xr)(r,Xr)isBF-measurable,Fr-adapted,andEs,xhRts)]TJ /F16 7.97 Tf 6.67 -4.43 Td[(@ @x(r,Xr)(r,Xr)2dri<1astandardresultfromstochasticcalculusgivesthatEs,xZts@ @x(r,Xr)(r,Xr)dBr=0 hencewehavethatEs,x[(t,Xt)])]TJ /F3 11.955 Tf 11.95 0 Td[((s,x)=Es,xZtsA(r,Xr)dr dividingbothsidesbyt)]TJ /F6 11.955 Tf 11.95 0 Td[(sandtakinglimitsgivesA(s,x)=:limt!sEs,x[(t,Xt)])]TJ /F5 11.955 Tf 11.96 0 Td[((s,x) t)]TJ /F6 11.955 Tf 11.95 0 Td[(s=limt!sEs,xhRtsA(r,Xr)dri t)]TJ /F6 11.955 Tf 11.95 0 Td[(s=limt!sEs,x1 t)]TJ /F6 11.955 Tf 11.96 0 Td[(sZtsA(r,Xr)dr=Es,xlimt!s1 t)]TJ /F6 11.955 Tf 11.96 0 Td[(sZtsA(r,Xr)dr=Es,xd dtZtsA(r,Xr)dr=Es,x[A(t,Xt)]=:E[A(t,Xt)jXs=x]=A(s,x) wherethefourthequalityfollowsfromdominatedconvergencetheoremwhichwemayapplyusingtheassumptionthatEs,xhRtsjA(r,Xr)drji<1sothatthetermintheexpectationisessentiallybounded. GiventheformoftheinnitesimalgeneratorofanItodiffusionwemayusethefollowingtheoremtocomputeexpectationsofrstexittimesofthediffusionfromagivenregion.Wewillnotexplicitlycomputeanyexpectationsofexittimes,howeverwewillneedDynkin'sformulatoproveresultsaboutexittimesofcontrolledItodiffusions. 19

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Theorem1.4. (OnedimensionalDynkin'sFormula)LetXtbeasolutionofanItodiffusionoftheformdXt=(t,Xt)dt+(t,Xt)dBtXs=x andbeastoppingtimewithEs,x[]<1.If2C1,20([0,1)R),thenEs,x[(,X)]=(s,x)+ExZsA(r,Xr)dr Proof. Since2C1,20([0,1)R)wemayapplyIto'sformulaasintheproofoftheoremtogetEs,x[(t,Xt)])]TJ /F3 11.955 Tf 11.96 0 Td[((s,x)=Es,xZtsA(r,Xr)dr+Es,xZts@ @x(r,Xr)(r,Xr)dBr Lettingt=wehavethatEs,x[(,X)]=(s,x)+Es,xZsA(r,Xr)dr+Es,xZs@ @x(r,Xr)(r,Xr)dBr TonishtheproofweneedtonowshowthatEs,xRs(r,Xr)@ @x(r.Xr)dBr=0.Tosimplifynotationletg(r,Xr)=(r,Xr)@ @x(r,Xr),thenwehavethatgisaboundedBorelFr-measurablefunctioni.e.jgjMforsomeM>0.ConsiderthefamilyoffunctionsnR^ksg(r,Xr)dBrok,IclaimthisfamilyisuniformlyintegrablewithrespecttothemeasurePx.ToshowthisfamilyisuniformlyintegrableitisenoughtoapplytheoremC.3from[ 20 ],henceweneedtoshowthefollowingintegralisnite,Es,x"Z^ksg(r,Xr)dBr2#=Es,xZ^ksg2(r,Xr)drM2Es,x[^k]M2E2[]<1 wheretherstequalityfollowsfromtheIto'sIsometry.Thenitenessofthisintegralfromtheuseofaquadratictestfunctiongivesusthatthefamilyisuniformlyintegrablesothatlimk!1Es,xZ^ksg(r,Xr)dBr=Es,xlimk!1Z^ksg(r,Xr)dBr 20

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hencecombingtheresultsabovewehavethatEs,xZsg(r,Xr)dBr=Es,xlimk!1Z^ksg(r,Xr)dBr=limk!1Es,xZ^ksg(r,Xr)dBr=limk!1Es,xZksfr
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CHAPTER2STATEMENTOFMAINPROBLEM 2.1OptimalControlTheory Wehavenowsetupafoundationforourproblem,thissectionwillstatethemainproblemandbegintotalkabouthowwemaygoaboutndingasolutionusingoptimalcontroltheory.Tobeginthissectionwegivesomedetailsoncontrolsandhowtheyareimplementedinourproblem. Denition2.1. GivenaBorelmeasurablesetURacontrolprocessisaBF-measurablestochasticprocess(ut)tstakingvaluesinUi.e.u:[s,1)!U Denition2.2. Givenacontrolprocess(ut)tsacontrolledprocessXtisasolutionofthestochasticdifferentialequationdXt=(t,Xt,ut)dt+(t,Xt,ut)dBtXs=x>0 (2) where:RRU!R,:RRU!RandBtisanone-dimensionalBrownianmotionandforallx,y2R,ts,u,v2Uwehavej(t,x,u))]TJ /F3 11.955 Tf 11.96 0 Td[((t,y,v)j+j(t,x,u))]TJ /F3 11.955 Tf 11.96 0 Td[((t,y,v)jC1jx)]TJ /F6 11.955 Tf 11.95 0 Td[(yj+C2ju)]TJ /F6 11.955 Tf 11.95 0 Td[(vjj(t,x,u)j+j(t,x,u)jD(1+jxj+juj) Theparameterut2UinEquation( 2 )isusedtocontroltheprocessXt,givenaBorelsetURwevarytheparameterut2UtocontrolthebehavioroftheprocessXt.Thedecisiononhowtovarythecontrolutisbasedontheinformationuptotimet,sothatutisFt)]TJ /F1 11.955 Tf 9.3 0 Td[(measurable.Sincethecontrolprocessdependsontheunderlying!wehaveamendedourdenitionofanItodiffusiontoincludeappropriateconditionsfortheexistenceanduniquenessofacontrolleddiffusion.InordertodothiswehavesetitupsothattheglobalLipschitzandlineargrowthconditionsholdforallcontrolsut2U.ToapplythetheoryofdiffusionsweneedEquation( 2 )tobeanItodiffusion,thisispossibleifwerestrictthecontrolsweusetoasubsetofcontrolscalledMarkovcontrols 22

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Denition2.3. LetXtbeasolutiontoEquation( 2 ),aMarkovcontrolisacontrolwhichdoesnotdependontheinitialstateofthesystem(s,x)butonlydependsonthecurrentstateofthesystematanytimeti.e.thereexitsafunction u:Rn+1!URksuchthatu(t,w)= u(t,Xt(w)) TheMarkovcontrolisimportantinourstudybecausetherestrictiontothissubsetofallcontrolsallowsustoturntheequationgoverningthewealthprocessintoanItodiffusion.Thefactthatthewealthprocessisadiffusionmakesuseofimportanttheoremsfromthetheoryofdiffusionspossible,withoutthiswewouldnotbeabletoproceedusingourmethodology. 2.2DynamicProgrammingMethodology Considerthefollowingfollowingcontrolledprocesson[0,G]dXt=(t,Xt,ut)dt+(t,Xt,ut)dBtX0=x0 (2) withgivenperformancefunction Ju(x0)=Ex0ZG0f(r,Xr,ur)dr+g(G,XG)fG<1g (2) whereG=infft>0:(t,Xt)=2Ggisastoppingtime,andthesetG[0,G]Riscalledthesolvencyset.Thecontinuousfunctionf:[0,G]RR!Riscalledtheutilityfunctionandthecontinuousfunctiong:RR!Riscalledthebequestfunction.Wewouldliketondanoptimalcontrolufromasetofadmissiblecontrolssothatthisperformancefunctionismaximized.Tondouroptimalcontrolwemustusethetechniqueofdynamicprogramming,asthenamesuggestweneedtoconsiderafamilyofoptimalcontrolproblemsfromwhichwewillselectanoptimalcontrol.Inordertodothisweneedtoconsiderdifferentinitialtimesandstatesalongagiventrajectoryofthecontrolleddiffusion.IfweweretoconsideradiffusionXtstartingatx0asinEquation( 2 )thenanadmissiblecontroluis(Ft)t0measurablewhichmeansthatthecontrollerhasallinformationaboutthesystemuptotimetsothatXtisalmostsurelydeterministic 23

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underaprobabilitymeasureP(jFt).Thismeansforallstatesofthesystemwealreadyknowtheinitialtimepointandtheinitialvalueatthistimepoint.Tobeabletousethemethodofdynamicprogrammingwemustvarytheinitialtimeandstatesofthesystemandchoosethebestpossiblecontrolfromasetofadmissiblecontrols.TodothisweconsidercontrolleddiffusionsoftheformdXt=(t,Xt,ut)dt+(t,Xt,ut)dBtXs=x (2) wheret2[s,bG]andtheperformancefunctionhastheformJu(s,x)=Es,xZbGsf(r,Xr,ur)dr+g(bG,XbG)fbG<1g (2) wherebG=infft>s:(t,Xt)=2Gg.Beforemovingontothestatementoftheoptimalcontrolproblemweneedtodiscussexactlywhatmathematicalpropertiesanadmissiblecontrolshouldhave.Wegivethegeneralmathematicalformulationofanadmissiblecontrolovertheset[s,bG]here,wewilluseaslightlydifferentsetlateronwhenwewritethetimehomogeneousversionofthecontrolledprocess. Denition2.4. Wesaythatthecontrolprocessuisadmissibleandwriteu2A[s,bG]if uismeasurable,(Ft)ts-adaptedwithEs,xhRbGsjurj2dri<1 Equation( 2 )hasauniquestrongsolution(Xt)tsforallXs=xandEs,xsupstbGjXtj2<1 Ju(s,x)iswelldenedi.e.Es,xZbGsf(r,Xr,ur)dr+g(bG,XG)fbG<1g<1 Thestochasticcontrolproblemmaynowbestated,givencontinuousutilityfunctionfandbequestfunctiongwewouldliketondoptimalcontroluandvaluefunction(s,x) 24

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suchthat(s,x)=supu2A[s,bG]Ju(s,x)=Ju(s,x) (2)(bG,x)=g(bG,x) (2) subjecttotheconstraintofEquation( 2 ).TobegintheanalysisoftheoptimalcontrolprobleminEquation( 2 ),weintroducesomenotationthatwillmakeourproblematimehomogeneouscontrolproblem.WerstbeginbyrewritingEquation( 2 )usingthesubstitutionYt=264s+tXs+t375t0 whichgivesusatimehomogeneouscontrolledprocessYtsatisfying dYt= (Yt,us+t)dt+ (Yt,us+t)dBs+tY0=(s,x)=:y (2) WemayalsorecasttheperformancefunctioninEquation( 2 )usingournewnotation,lettingr=s+twehave ZbGsf(r,Xr,ur)dr=ZbG)]TJ /F9 7.97 Tf 6.58 0 Td[(s0f(s+t,Xs+t,us+t)dt=ZG0f(Yt,us+t)dt whereG=bG)]TJ /F6 11.955 Tf 11.96 0 Td[(s,hencetheperformancefunctionbecomes Jus+t(y)=EyZG0f(Yt,us+t)dt+g(YG)fG<1g (2) SincedBs+thasthesamedistributionasdBtwemaymakethisreplacementinthecontrolledprocessequation.WenotethatthecontrolfromEquation( 2 )isatimeshiftedversionofthecontrolfromEquation( 2 ),afterthesubstitutionforYtweseethatwearenowworkingwiththecontrolprocessus+tineachequation.Thesolutiontotheoptimalcontrolproblemisfoundbyworkingintwodifferentworlds,thisideaisatthecenterofourentireanalysishenceweshallspendsometimediscussinghowtogobackandforthbetweeneachworld.Wesummarizeeachoftheequationsandthe 25

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vitalinformationineachworldinTable 2)]TJ /F5 11.955 Tf 11.95 0 Td[(1 ,thistableisveryusefulwhencomparingequationsbetweenthedifferentworlds.Themainideaofworkingbetweenthetwo Table2-1. Tablewithcomparisonoftwoworldsusedtosolveoptimalcontrolproblem Twoworlds OriginalXtspaceTransformedYt=(s+t,Xs+t)spacedXt=(t,Xt,ut)dt+(t,Xt,ut)dBtdYt= (Yt,us+t)dt+ (Yt,us+t)dBtXs=xY0=(s,x)=:yJ(s,x)=Es,x"ZbGsf(t,Xt,ut)dt+g(bG,XbG)#J(y)=EyZG0f(Yt,us+t)dt+g(YG)ut=u(t,Xt)us+t=u(s+t,Xs+t)=u(Yt)ut2A[s,bG]us2A[0,G] worldsisthatwewillbesolvingtheoptimalcontrolproblemintheYtspacewherethecontrolisus+tandthenwewilltransformbacktotheXtspacewherethecontrolisut.TheconnectionbetweenthetwoworldsthatwewillneedisthefactthatintheYtworldwehavethatus=u(s,x)=u(s,Xs)sinceXs=xintheXtworld.NowrenamingthevariablewehavethattheoptimalcontrolintheXtworld(denotedsimplyut)isgivenbyut=u(t,Xt)whereuistheoptimalcontrolwendintheYtworld.Hence,theoptimalcontrolproblemissolvedbytransformingintotheYtworldwhereanoptimalcontrolu(s,w)willbefound,thenusingthatoptimalcontrolweareabletotransformbacktotheXtworldtosolvefortheoptimalcontrolut(t,Xt)whichisasolutionofthecontrolproblemintheXtworld.Howeverfornowwemakenodistinctionbetweenthedifferentversionsof,anduandstatethecontrolprobleminthenewnotation.Givenutilityfunctionfandbequestfunctiongwewouldliketondoptimalcontroluandvaluefunction(y)suchthat (y)=supu2A[0,G]Ju(y)=Ju(y) (2) where J(y)=EyZG0fu(Yt,us+t)dt+g(YG)fG<1g (2) 26

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subjecttotheconstraint dYt=(Yt,us+t)dt+(Yt,us+t)dBtY0=(s,x)=:y (2) InournewnotationaMarkovcontrol(wemakenodistinctionbetweenuand u)isonesuchthatus+t=u(s+t,Xs+t)=u(Yt)hencewemayrewriteEquation( 2 )asdYt=(Yt,u(Yt))dt+(Yt,u(Yt))dBtY0=(s,x)=:y (2) WenowhavetherighthandsideofEquation( 2 )onlydependsonthestateofthesystemYtattimetanddoesnotdependontimeexplicitly,i.e.thesolutionYtofEquation( 2 )isatimehomogeneousItodiffusion.Giventhatthestateofoursystemhasthisformwemaynowapplythetheoryofdiffusionstosolveouroptimalcontrolproblem.Beforestatingthemaintheoremsnecessarytosolvetheoptimalcontrolproblemwementionthatthenotationbetweenthetwoworldswillbeusedinterchangeablyfortheremainderofthepaper,anyconfusionwiththenotationcanberesolvedbyconsultingTable 2)]TJ /F5 11.955 Tf 11.95 0 Td[(1 .ThemaintheoremsofthepaperwillbestatedintheYtworldsincemuchoftheworkwillbedoneinthissettingandtheresultswillthenbetransformedbacktotheXtworldtoprovideasolutiontotheoriginalproblem. 2.3HamiltonJacobiBellman TheHamiltonJacobiBellmantheoremisattheheartofmuchoftheanalysisthatwillbeperformedthroughoutthepaper,itusewillbeofanindirectnaturebutitisofgreatimportancehenceweprovidethecompletestatementandproof. Theorem2.1. LetJu(y)beasinEquation( 2 ),whereu=u(Y)isaMarkovcontroland(y)=supuJu(y) 27

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Supposingthefunction2C2G\C( G)isboundedforallnitestoppingtimesGa.s.Pyandally2G.IfanoptimalMarkovcontroluexistsand@GisregularforYutthensupu2A[0,G][fu(y)+Au(y)]=0forally2G (2) and(y)=g(y)forally2@G (2) Proof. Since@GisregularforYutwehaveG=0foranyy2@G,sothat(y)=g(YG)fG<1g=g(y) henceEquation( 2 )holdsforally2@G.ToproveEquation( 2 )xy=(s,x)2GandtakeutobeanyMarkovcontrol.IfGisastoppingtimethenwemaycompute Ey[Ju(Y)]=EyEYZG0fu(Yr)dr+g(Y)=EyEyZGfu(Yr)dr+g(Y)jF=EyEyZG0fu(Yr)dr+g(YG))]TJ /F15 11.955 Tf 11.95 16.28 Td[(Z0fu(Yr)drjF=EyZG0fu(Yr)dr+g(YG))]TJ /F15 11.955 Tf 11.96 16.28 Td[(Z0fu(Yr)dr=EyZG0fu(Yr)dr+g(YG))]TJ /F6 11.955 Tf 11.96 0 Td[(EyZ0fu(Yr)dr=Ju(y))]TJ /F6 11.955 Tf 11.95 0 Td[(EyZ0fu(Yr)dr Thisshowsthat Ju(y)=EyZ0fu(Yr)dr+Ey[Ju(Y)] (2) whichisanequalityusedtoproveBellman'sprincipleofoptimalitywhichwedonotdirectlyprovehere,butwewillusetheaboveequalitytonishtheproof.Firstwemustdenethepropercontinuationregion,letUGoftheformU=f(r,x)2G:r
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wheres<>:vif(r,x)2Uu(r,x)x2GnU wherev2A[0,G]isanarbitrarycontrol.Withthiscontinuationregionwehavethat (Y)=Ju(Y)=Ju(Y) (2) Usingthefactthat(y)isthesupremumandEquation( 2 )wehave (y)Ju(y)=EyZ0fu(Yr)dr+Ey[Ju(Y)]=EyZ0fu(Yr)dr+Ey[(Y)] (2) Sinceweassumed(y)2C2(G)andisastoppingtimewemayapplyDynkin'sformulatoget Ey[(Y)]=(y)+EyZ0Au(Yr)dr (2) IfwenowplugEquation( 2 )intoEquation( 2 )weget (y)EyZ0fu(Yr)dr+(y)+EyZ0Au(Yr)dr Combiningexpectationandsubtracting(y)frombothsideswehave EyZ0fu(Yr)dr+Au(Yr)dr0 Nowwelett1!sandusingthefactthatf()andAu()arecontinuousatywemayperformtheintegrationtogetthat (fu(y)+Au(y))Ey[]0 29

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dividingoutEy[]wehavethatfu(y)+Au(y)0forallstoppingtimesu2A[0,G].Thesupremumisobtainedattheoptimalcontroluwhere (y)=Ju(y)=EyZG0fu(Yr)(Yr)dr+g(YG) (2) isasolutionofthecombinedDirichlet-PoissonproblemsothatAu(y)(y)=)]TJ /F6 11.955 Tf 9.3 0 Td[(fu(y)(y)forally2G ThistheoremgivesanecessaryconditionthatstatesifanoptimalcontroluexiststhenthefunctionF(u)=fu(y)+Au(y)attainsitmaximumvalue0atu=u.Howeveritdoesnotaddressthequestionofsufciencyoftheoptimalcontrolui.e.ifforeverypointy2Gwendau(y)suchthatF(u(y))=0,willu(y)beanoptimalcontrol?Thisquestioncanbeansweredbythefollowingconverse Theorem2.2. Let2C2(G)\C( G)satisfythefollowingconditions fu(y)+Au(y)0y2Gandforallu2A[0,G] limt!G(Yt)=g(YG)fG<1ga.s.Py f(Y)gGisuniformlyPyintegrableforallMarkovcontrolsandally2G. then(y)Ju(y)forallMarkovcontrolsu2Aandy2G.Moreoverifforally2GwendaMarkovcontrolu=u0(y)suchthat fu0(y)(y)+Au0(y)(y)=0 (2) thenut=u0(Yt)isoptimaland(y)=(y)=Ju(y). Proof. LetR<1anddenethestoppingtimeTR=minfR,G,infft>0;jYtjRgg 30

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sothatwehavelimR!1TR=G.WemayapplyDynkin'sformulaandusethehypothesisthatAu(y))]TJ /F6 11.955 Tf 21.92 0 Td[(fu(y)forally2Gtogetthat Ey[(YTR)]=(y)+EyZTR0Au(Yr)dr(y))]TJ /F6 11.955 Tf 11.96 0 Td[(EyZTR0fu(Yr)dr Rearrangingofthisequationandcombiningtermswithexpectationsgivesusthat (y)EyZTR0fu(Yr)dr+(YTR) (2) ApplyingFatou'sLemmatothisequationandusingtheremaininghypothesiswehave (y)liminfR!1EyZTR0fu(Yr)dr+(YTR)EyliminfR!1ZTR0fu(Yr)dr+(YTR)=EyZT0fu(Yr)dr+limR!1(YTR)=EyZT0fu(Yr)dr+limt!G(Yt)=EyZT0fu(Yr)dr+g(YG)fG<1g=Ju(y) Tocompletetheproof,ifweareabletondacontrolu0(y)suchthatfu0(y)(y)+Au0(y)(y)=0thenwemayapplythesameargumentasabovetout=u0(Yt)sothattheinequalitybecomesequalityi.e.(y)=Ju(y)anduisanoptimalcontrol. 2.4VericationTheorem TheHamiltonJacobiBellmanequationhasmanytechnicalconditionsthataregenerallydifculttocheckexplicitly,inparticularwedonotknowthesolutionapriori.EachversionoftheHamiltonJacobiBellmanequationrequiresthatweknowthatthesolutionsatisfy2C2(G)\C( G).ThismakessolvingtheoptimalcontrolproblemusingtheHamiltonJacobiBellmantheoremsdirectlyratherdifcult,inordertosolve 31

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thisproblemweintroducetheideaofvericationtheorems.UsingtheHamiltonJacobiBellmantheoremswouldrequireustoknowthesolutionandcheckalltherequiredtechnicalconditions,oncethisisdonethenweareabletodetermineifacontroluisanoptimalcontrolandshowthatisactuallyasolutiontotheHamiltonJacobiBellmanequation.Thevericationtheoremreversetheorderofthisprocess.WerstbeginwithasolutiontotheHamiltonJacobiBellmanequationfromwhichweareabletoalsogetacandidatefortheoptimalcontrolu.Oncewehaveshownthatsatisestheconditionsofthevericationtheoremwecanthenshowthatu=sothattheoptimalcontrolproblemissolved. Theorem2.3. Letu2A[s,G]and(s,x)2Gandsupposethefollowingconditionsaresatisedforalls2[0,G]andx2R 2C1,2([0,G)R)iscontinuouson[0,G]Randsatisesthequadraticgrowthconditionj(s,x)jC(1+jxj2) satisestheHamiltonJacobiBellmanequationsupu2A[s,G][fu(s,x)+Au(s,x)]=0s2[0,G)(G,x)=g(G,x) fuiscontinuouswithjfu(s,x)jCf(1+jxj2+jjujj2)forsomeconstantCf>0. ju(s,x)j2C(1+jxj2+jjujj2)forsomeconstantC>0. then(s,x)(s,x)forall(s,x)2G.Moreoverifuo(s,x)isthemaxofu7!fu(s,x)+Au(s,x)andu=u0(s,Xs)isadmissiblethen(s,x)=(s,x)forall(s,x)2Ganduisandoptimalstrategyi.e.(s,x)=Ju(s,x). Proof. Fixs2[0,G]andx2R,togetthattheprocessXisboundedwedenethestoppingtimen=G^infft>s:jXt)]TJ /F6 11.955 Tf 11.95 0 Td[(Xsjng Letu2A[s,G]beanadmissiblecontrolandXs=xthenbyIto'slemmawehavethat (n,Xn)=(s,x)+ZnsAur(r,Xr)dr+Znsx(r,Xr)ur(r,Xr)dBr (2) 32

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Sincex(s,Xs)iscontinuousontheset[s,n]wehavethatthereexistsaconstantCxsuchthatjx(s,x)j2Cx.UsingthefactthaturisadmissibleandthatXrisboundedon[s,n]wendthatEs,xZnsjx(r,Xr)ur(r,Xr)j2drEs,xZnsjx(r,Xr)j2jur(r,Xr)j2drEs,xZGsCxC(1+jXrj2+jjurjj2)dr<1 henceRnsx(r,Xr)ur(r,Xr)dBrisamartingalewhichgivesthatEs,xZnsx(r,Xr)ur(r,Xr)dBr=0 TakingexpectationsonbothsidesofEquation( 2 )andusingthelastcalculationwegetthat Es,x[(n,Xn)]=Es,x(s,x)+ZnsAur(r,Xr)dr addingEs,xRnsfur(r,Xr)drtobothsidesofthisequationandcombiningtermsintoasingleexpectationwendthat Es,xZnsfur(r,Xr)dr+(n,Xn)=Es,x(s,x)+Zns(fur(r,Xr)+Aur(r,Xr))drEs,x[(s,x)]=(s,x) (2) wherewehaveusedthatfactthatforanycontrolur2Awehavethatfur(r,Xr)+Aur(r,Xr)0.IfwecannowshowthatthelefthandsidegoestoJu(s,x)asn!1theproofwillbecomplete.Inordertodothisweneedtoletn!1sothatn!G,ourmainconcernherewillbebringingthelimitinsideoftheexpectationoperator.Toaccomplishthiswewouldliketouseddominatedconvergencetheorem,henceweneedtocheckthatthehypothesisaremeti.e.Znsfur(r,Xr)dr+(n,Xn)Znsjfur(r,Xr)jdr+j(n,Xn)jCfZGs(1+jXrj2+jjurjj2)dr+C(1+jXGj2)2L1 33

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anapplicationofdominatedconvergencegivesandthefactthat(G,XG)=g(G,XG)Es,xZnsfur(r,Xr)dr+(n,Xn))166(!Es,xZGsfur(r,Xr)dr+g(G,XG)=Ju(s,x) Lettingn!1onbothsidesofEquation( 2 )andusingthisresultgivesusthatJu(s,x)(s,x)forallu2A sincetherighthandsidedoesnotdependonuwetakethesupremumofbothsidesoverthesetofalladmissiblecontrolstoget(s,x)=supu2AJu(s,x)(s,x) IfwewerenowabletondamaximizerutheonlypartoftheproofabovethatwouldbedifferentisthefactthatinsteadofaninequalityintheargumentwewouldhaveequalityinEquation( 2 )sothatwemaycontinuetheargumenttogetthat(s,x)=Ju(s,x)=(s,x)completingbothpartsoftheproof. HencethevericationtheoremgivesusawayofndingasolutiontotheoptimalcontrolproblemwithouthavingtocheckthecomplicatedhypothesesoftheHamiltonJacobiBellmanequations.Thevericationmethodologyofsolvingtheoptimalcontrolproblemusingthevericationtheoremisasfollows WritedowntheHamiltonJacobiBellmanequationandtheappropriateboundarycondition. TaketherstorderderivativeswithrespecttothecontrolvariablesintheHamiltonJacobiBellmanequation,andsolvefortheoptimalcontrolcandidateu0. PlugtheoptimalcontrolsbackintotheHamiltonJacobiBellmanequationtogetanonlinearpartialdifferentialequationwhichhastobesolvedforacandidateofthevaluefunctionsubjecttotheboundarycondition Showthatsatisestheconditionsofthevericationtheoremandthatus=u0(s,Xs)isanadmissiblecontrolforalls2[0,G],then=anduisanoptimalcontrolstrategy. 34

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CHAPTER3MERTONPROBLEM 3.1ClassicMertonSolution ThissectionformulatesandsolvesaversionoftheclassicoptimalcontrolproblemthatMertonsolvedinhis1971paper[ 18 ].InhispaperMertonassumedageneralfamilyofutilityfunctionscalledtheHyperbolicabsoluteriskaversionutilities.IwillpresentasubsetofthisfamilywhichavoidsthecompletegeneralityasMertonoriginallypresentedbutwhichstillcapturestheessenceofhisresults.Theoriginalproblemconsidersaportfolioofnassets,howeverwiththeassumptionoflognormallydistributedassetswemayuseasocalledmutualfundtheoremwhichallowsustoinsteadconsideraportfolioof2assets.Theassetsunderconsiderationareariskfreeasset(suchasaU.S.treasurybond)andariskyasset(suchasashareofstock)whichmaybewrittenasalinearcombinationofthenassets.Tobegintheanalysislet(,F,F[0,1),P)bealteredprobabilityspace.TheriskfreeRtassetevolvesaccordingtothedifferentialequation dRt=rRtdt;R0=1 (3) wherer0isaconstantwhichrepresentstheriskfreerateofinterest,whiletheriskyassetStevolvesaccordingtoaGeometricBrownianmotion dSt=Stdt+StdBt;S0=s0 (3) where0and0representtherateofthereturnandvolatilityoftheassetStrespectively. Denition3.1. (Tradingstrategy)Atradingstrategyisatwodimensionalstochasticprocessft=(0t,1t)gt2[0,1),suchthattisBF-measurableandFt-adapted. Financiallyweinterpret0tasthenumberofsharesintheriskfreeassetRt,while1tisthenumberofsharesintheriskyassetStattimet.Hencethetotalwealthattimetof 35

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theportfolioofassetsmaybewrittenas Wt=0tRt+1tSt (3) LetctbeanFtadaptedprocessthattheinvestorisabletochooseattimet,whichrepresentstherateatwhichmoneycanbemovedfromtheriskfreeassettotheriskyassetwithintheportfolio. Denition3.2. SelfnancingAtradingstrategy(0t,1t)iscalledselfnancingifthecorrespondingwealthprocess(Wt)t0iscontinuousandadaptedsuchthat Wt=w0+Zt00udRu+Zt01udSu)]TJ /F15 11.955 Tf 11.96 16.28 Td[(Zt0cudu (3) Theassumptionofaselfnancingportfolioofassetsbasicallystatesthatnosourcesofexternalcapitalcanbeaddedtotheportfolio,anycapitalgainsmustbereinvestedintotheportfolio.Wealsohaveunderlyingassumptionsthattherearenotransactioncoststoredistributecapitalbetweenthetwoassets.Lettbethefractionoftotalwealthoftheportfolioinvestedintheriskyasset,thenwemaywritet=1tSt Wt whilethefractionofwealthintheriskfreeassetis0tRt Wt.AssumingthatWt>0foralltwemaydivideEquation( 3 )byWttoseethat1)]TJ /F3 11.955 Tf 9.47 0 Td[(t=0tRt Wt.WiththeseassumptionandthenotationweabovewemaywritethechangeisthewealthprocessusingEquation( 3 )as dWt=0tdRt+1tdSt)]TJ /F6 11.955 Tf 11.95 0 Td[(ctdt=r0tRtdt+1t[Stdt+StdBt])]TJ /F6 11.955 Tf 11.96 0 Td[(ctdt=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(t)rWtdt+tWtdt+tWtdBt)]TJ /F6 11.955 Tf 11.96 0 Td[(ctdt=()]TJ /F6 11.955 Tf 11.95 0 Td[(r)tWtdt+[rWt)]TJ /F6 11.955 Tf 11.96 0 Td[(ct]dt+tWtdBt (3) 36

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Equation( 3 )alongwithinitialconditionW0=00R0+10S0=:wwillserveasthecontrolleddiffusionfortheoptimizationproblem.Thecontrolprocessinthisproblemisgivenbythevectorut=(t,ct),wheretisthefractionofwealthintheriskyassetandctistheconsumptionprocess.Thefreedomofchoicefortcanbethoughtofasaninvestorchoosingthefractionoftotalwealthoftheportfoliohe/shewouldlikeinvestedintheriskyasset.Ateachpointintimetheinvestormuchchoosethisfractionalongwiththeconsumptionprocessctastomaximizesomeperformancefunction.ThereallocationofportfolioweightstattimetisthecontinuoustimeversionoftheproblemsolvedbyHenryMarkowitz[ 17 ],whichwasdoneinadiscretetimesettingandisfamouslyreferredtoasModernPortfolioTheory.Wewillassumeatraderhascompleteinformationinthemarketuptotimetsothatthecontrolprocessut=(t,ct)areadaptedwithrespecttothestandardltration.Wemaynowdenethevaluefunctionfortheoptimizationproblemintheclassicalcase,welookatseveralcases.Inthischapterwewillconsidercostfunctionalsonaninnitehorizonwithutilityfunctionsoftheformf(t,Wt,ut)=e)]TJ /F16 7.97 Tf 6.59 0 Td[(tU(ct)wherethereisnoexplicitdependenceonWt.However,sincectisastochasticprocesstherewillbeabuiltindependenceonWtwhichwillshowupwhenwendthecontrolprocessexplicitly.Giventhisformoftheutilityfunctionthecostfunctionalbecomes J,c(s,w)=Es,wZ1se)]TJ /F16 7.97 Tf 6.59 0 Td[(tU(ct)dt=e)]TJ /F16 7.97 Tf 6.58 0 Td[(sEyZ10e)]TJ /F16 7.97 Tf 6.59 0 Td[(tU(cs+t)dt (3) wherethe(utility)functionUisincreasing,differentiableandconcaveinct.Theprocessctistheconsumptionratethattheinvestormustchoosewhenrebalancingtheportfoliosoastomaximizethecostfunctional.Sotheoptimalcontrolproblemweareconsideringinthissectionisoneinwhichtheinvestortriestomaximizetheexpectedutilityofconsumptionoverthegiventradingperiodbychoosingtheappropriatecontrols. 37

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Inmathematicaltermswewouldliketond(s,w)suchthat (s,w)=sup,cJ,c(s,w)=sup,cEs,wZ1se)]TJ /F16 7.97 Tf 6.59 0 Td[(tU(ct)dt (3) Toseehowthevaluefunction(s,w)behavesineachofthetwoworldsweperformthefollowingcalculationwherethecontrolprocessisut=(t,ct)(s,w)=sup,cEs,wZ1sf(t,Wt,ut)dt=sup,cEs,wZ1se)]TJ /F16 7.97 Tf 6.59 0 Td[(tU(ct)dt=sup,cEs,wZ10e)]TJ /F16 7.97 Tf 6.59 0 Td[((s+r)U(cs+r)dr=sup,cEs,wZ10f(s+r,Ws+r,us+rdr=sup,cEyZ10f(Yr,us+r)dr=sup,cEyZ10f(Yr,u(Yr))dr=(y) TheequalitybeforethelastcomesfromthefactthatwewillbeusingMarkovcontrolsthroughoutouranalysis,hencewehavethatus+r=u(s+r,Ws+r)=u(Yr).ThisargumentprovidesthegluetobindtogethertheoptimalcontrolproblembetweentheXtandYtspaces.InparticularweareabletondtheoptimalcontrolsusingtheHamiltonJacobiBellmanequationintheYtspace,thentransformbackintotheXtspacetondtheoptimalcontrolprocess.Italsoshowsthatthevaluefunctionsineachofthetwospaceswillbethesameas(s,w)=(y),hencewewillusetheseinterchangeablyfromthispointon.TosolvetheoptimalcontrolproblemwewillperformmostofouranalysisintheYtspace,oncetheproblemissolvedinthisspacewewillonlytransformbacktotheXtspacethendtheoptimalcontrolprocessut.InparticulartheItodiffusionandHamiltonJacobiBellmanequationwillliveintheYtspace,oncetheoptimalcontrolsarefoundwewillusethevericationtheoremtowritethecontrolsintheXtspace. 3.2InniteHorizonPowerUtility TherstoptimalcontrolproblemweconsiderisequivalenttothecontrolproblemconsideredbyMertonin[ 18 ],inparticularweconsidertheutilityfunctionf(t,Wt,ut)=e)]TJ /F16 7.97 Tf 6.58 0 Td[(tct .Mertonsolvesthisprobleminamoregeneralsetting,butwewillconsideronlyaspecialcaseofthesolutionsothatwemaymakedirectcomparisonswiththeremainder 38

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ofthepaper.ThecostfunctionalsinboththeXtandYtspacearegivenby J,c(s,w)=Es,wZ1se)]TJ /F16 7.97 Tf 6.59 0 Td[(tct dt=e)]TJ /F16 7.97 Tf 6.59 0 Td[(sEyZ10e)]TJ /F16 7.97 Tf 6.59 0 Td[(tcs+t dt (3) GiventhecostfunctionalintheYtspacewewouldliketondoptimalcontrolsc,andvaluefunction(s,w)suchthat(s,w)=sup,cJ,c(s,w)=J,c Inordertondtheoptimalcontrolsandthevaluefunction,weneedtoapplythevericationtheorem.Todothiswewillassumethatisboundedandoptimalcontrolscandexists,thenthevaluefunctionmustsatisfythetheHamiltonJacobiBellmanequationfortheItodiffusioninEquation( 3 )forthewealthprocessi.e.satises sup,ce)]TJ /F16 7.97 Tf 6.59 0 Td[(sc +s+()]TJ /F6 11.955 Tf 11.96 0 Td[(r)ww+(rw)]TJ /F6 11.955 Tf 11.95 0 Td[(c)w+1 22w22ww=0 (3) Therstorderoptimalconditionsarefoundbytakingrstderivativeswithrespecttocandandsolvingforthecriticalpointsrespectivelysothatthepossibleoptimalcontrolsare c=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(esw1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1=)]TJ /F5 11.955 Tf 9.29 0 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(r)w 2www (3) WiththeseoptimalcontrolstheHamiltonJacobiBellmanequationbecomese)]TJ /F16 7.97 Tf 6.58 0 Td[(s )]TJ /F6 11.955 Tf 5.48 -9.68 Td[(esw )]TJ /F12 5.978 Tf 5.75 0 Td[(1+s)]TJ /F5 11.955 Tf 13.15 8.09 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)22w 2ww+rww)]TJ /F15 11.955 Tf 11.95 9.68 Td[()]TJ /F6 11.955 Tf 5.48 -9.68 Td[(esw1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1w+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)22w 22ww=0=)(es)1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1w+s)]TJ /F5 11.955 Tf 13.15 8.09 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)22w 22ww+rww)]TJ /F15 11.955 Tf 11.96 9.68 Td[()]TJ /F6 11.955 Tf 5.48 -9.68 Td[(es1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1w=0=)1)]TJ /F3 11.955 Tf 11.95 0 Td[( (es)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1w+s)]TJ /F5 11.955 Tf 13.15 8.08 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)22w 22ww+rww=0 Hencewehavethatouroptimalcontrolproblemisreducedtosolvingthispartialdifferentialequation.Bytheargumentaboveweknowthatthevaluefunctionhastheform(s,w)=e)]TJ /F16 7.97 Tf 6.59 0 Td[(s(0,w),itwasshownbyMertonandotherssincethenthat 39

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(0,w)=Kp0wwhereKp0isaconstantsothat(s,w)=Kp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw.ItturnsoutthatwemayndthevalueoftheconstantK0explicitly.Tothisendwendthenecessaryderivativesandplugintotheequationwendthatourequationbecomes1)]TJ /F3 11.955 Tf 11.96 0 Td[( (es)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1(Kp0) )]TJ /F12 5.978 Tf 5.75 0 Td[(1w)]TJ /F6 11.955 Tf 11.96 0 Td[(Kp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw)]TJ /F5 11.955 Tf 13.15 8.08 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2Kp0e)]TJ /F16 7.97 Tf 6.58 0 Td[(s 22()]TJ /F5 11.955 Tf 11.96 0 Td[(1)w+rKp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw=0=)Kp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(s1)]TJ /F3 11.955 Tf 11.95 0 Td[( (Kp0)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1() )]TJ /F12 5.978 Tf 5.76 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F5 11.955 Tf 16.45 8.09 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22()]TJ /F5 11.955 Tf 11.95 0 Td[(1)+rw=0=)Kp0=0(Kp0)1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1=1 1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(r+()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2 22()]TJ /F5 11.955 Tf 11.95 0 Td[(1) TakingthenontrivialsolutionwendthatKp0=1 1 1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(r)]TJ /F5 11.955 Tf 16.46 8.09 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22(1)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F10 7.97 Tf 6.58 0 Td[(1 Hencethesolutiontotheoptimalcontrolproblemisgivenby (s,w)=1 1 1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(r)]TJ /F5 11.955 Tf 16.45 8.09 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2 22(1)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw (3) Thissolutionisonlyvalidwhen >r+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22(1)]TJ /F3 11.955 Tf 11.95 0 Td[() (3) sothattheterminbracketsispositiveandwehavearealsolution.ChoosingparameterssothatthisconditionissatisedwepresentagraphofthevaluefunctioninFigure 3)]TJ /F5 11.955 Tf 11.95 0 Td[(1 ,wewillusethisgraphtocomparetotheothercasesandtoverifytheconsistencyofourresults.Inthiscaseweareabletondtheoptimalcontrolsexplicitlysincethevaluefunctionisissmoothenough.Theexplicitsolutionsarefoundbyplugging(s,w)backintoEquation( 3 )toget c=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(esw1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1=Kp0w (3) 40

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Figure3-1. Valuefunctionforpowerutilitywithnojumpswithparameters,=.16,r=.06,=1,=.5,=.3 fortheoptimalconsumptionrateand =)]TJ /F5 11.955 Tf 9.3 0 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(r)w 2www=)]TJ /F6 11.955 Tf 11.95 0 Td[(r 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[() (3) Hencewehavethattheconsumptionratect=Kp0Wtisalinearfunctionofthewealthprocess.SinceKp00,intuitivelythismeansthatanincreaseinwealthprocessshouldleadtoanincreaseinconsumptionbytheinvestor.Thefractionofwealthintheriskyassett=)]TJ /F9 7.97 Tf 6.59 0 Td[(r 2(1)]TJ /F16 7.97 Tf 6.59 0 Td[()turnsouttobeaconstantintime,whichmeansthattheinvestorshouldcontinuallychangehis/herportfolioweightthroughouttheinvestmentperiodsothattheweightisequaltothisconstanttooptimizethegivenutilityfunction.Thisisdifculttodoinarealworldsettingasitisdifculttocontinuouslyexecutetradestokeeptheportfolioweightconstant,butisusefulinthesensethattheinvestorknowsexactlyhowtochoosehis/herconsumptionastimeevolvesthroughtheinvestmentperiod.Inactualitytheinvestorwouldexecutetradesatanitesetoftimes,tryingtokeeptheweightoftheriskyassetasclosetoconstanttaspossibletogetthebestresult.TheexistenceoftheoptimalcontrolsandthesolutionoftheoptimalcontrolproblemalldependsonthestochasticdifferentialEquation( 3 )haveasolution.Withexplicit 41

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solutionsfortheconsumptionandfractionoftheriskyassetgivenby ct=Kp0Wtt=)]TJ /F6 11.955 Tf 11.95 0 Td[(r 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[() (3) wemayplugtheseintoEquation( 3 )toseewhyasolutionispossibleandwemayalsondtheexplicitsolutionifnecessary, dWt=()]TJ /F6 11.955 Tf 11.95 0 Td[(r)tWtdt+[rWt)]TJ /F6 11.955 Tf 11.96 0 Td[(ct]dt+tWtdBt=()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Wtdt+rWtdt)]TJ /F6 11.955 Tf 11.95 0 Td[(KP0Wtdt+)]TJ /F6 11.955 Tf 11.95 0 Td[(r (1)]TJ /F3 11.955 Tf 11.96 0 Td[()WtdBt=()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[()+r)]TJ /F6 11.955 Tf 11.96 0 Td[(KP0Wtdt+)]TJ /F6 11.955 Tf 11.96 0 Td[(r (1)]TJ /F3 11.955 Tf 11.96 0 Td[()WtdBt HencethewealthprocessisaGeometricBrownianmotionoftheform dWt Wt=()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[()+r)]TJ /F6 11.955 Tf 11.95 0 Td[(KP0dt+)]TJ /F6 11.955 Tf 11.95 0 Td[(r (1)]TJ /F3 11.955 Tf 11.95 0 Td[()dBtWs=w (3) whosesolutionexistsandisuniquegivenWs=w.Equation( 3 )isaGeometricBrownianmotionwhosesolutionisgivenby Wt=we()]TJ /F13 5.978 Tf 5.75 0 Td[(r)2 2(1)]TJ /F17 5.978 Tf 5.76 0 Td[()+r)]TJ /F9 7.97 Tf 6.58 0 Td[(KP0)]TJ /F12 5.978 Tf 7.78 3.26 Td[(1 2()]TJ /F13 5.978 Tf 5.76 0 Td[(r)2 2(1)]TJ /F17 5.978 Tf 5.75 0 Td[()2(t)]TJ /F9 7.97 Tf 6.59 0 Td[(s)+)]TJ /F13 5.978 Tf 5.76 0 Td[(r (1)]TJ /F17 5.978 Tf 5.75 0 Td[()p t)]TJ /F9 7.97 Tf 6.59 0 Td[(sN(0,1) (3) whereN(0,1)isastandardnormaldistribution.GiventhisformandvaluesfortheparameterswehavesimulatedthiswealthprocessandtheresultisgiveninFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(2 .Theinterpretationoftheconstantvalueofisthatfortheinvestoroptimizethecostfunctiontheymustmaintainthisconstantfractionofwealthintheriskyassetovertheinvestmentperiod.Foramoreprecisemathematicalinterpretationwerecallthatthefractionofwealthintheriskyassetisgivenbyt=1tSt 0tRt+1tSt=)]TJ /F6 11.955 Tf 11.96 0 Td[(r 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[()=:0 42

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Figure3-2. Wealthprocessforpowerutilitywithnojumpswithparameters,w=1,=.16,r=.06,=1,=.5,=.3 fortheinvestortokeepthisfractionoftotalwealthequaltotheconstanthe/shemustsatisfythefollowingcondition1tSt=0 1)]TJ /F3 11.955 Tf 11.96 0 Td[(00tRt Thismeanstheinvestormustkeephis/herwealthinvestedintheriskyequaltoaconstantlinearmultipleofthewealthinvestedintheriskfreeasset.Hencetheinvestormustconstantlyrebalancethetradingstrategy(0t,1t)iftheyhopetomaximizetheexpectedutilityofconsumptionovertheinvestmentperiod.Thecontinualrebalancingofthepositionsoftheassetintheportfoliorequiresaconsiderationoftransactioncostsasthiswillsurelybeafactorinhowoftentheinvestoriswillingtoperformtherebalancing.Thisisamatterthathasbeenstudiedinpaperssuchas[ 10 ]and[ 21 ]andonewhichwedonotaddresshere,onenalstepremainstocompletelyclassifythesolutionandthatistheissueofcheckingtheconditionsofthevericationtheorem.Toshowthatthesolution(s,w)isoptimalweneedtoshowthattheconditionsofthevericationtheoremaresatisedsothatthemethodologyislegitimateforthesolution.Werstneedtoshowthatthecontrolprocessut=(ct,t)isanelementofA[s,1)tomakesurethat 43

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wehaveanadmissiblecontrol.Thepreciseformofthecontrolprocessisut=KP0Wt,)]TJ /F6 11.955 Tf 11.96 0 Td[(r 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[() whichismeasurableandFt-adaptedbyinspection.Wenextcheckthatthecontrolprocesssatisesthefollowingintegrabilitycondition Es,wZ1sjjutjj2dt=Es,w"Z1s((Kp0Wt)2+)]TJ /F6 11.955 Tf 11.96 0 Td[(r 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[()2)dt#=(Kp0)2Es,wZ1sW2tdt+)]TJ /F6 11.955 Tf 11.95 0 Td[(r 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[()2Ps,w[s,1)<1 ThersttermisnitesinceWtisauniquestrongsolution,andthesecondtermsisnitesinceweareworkingonaprobabilityspace.HencethecontrolprocessinthiscaseisappropriatelybehavedasfarintegrabilityintheL2sensegoes.Thenalconditionforadmissibilityistocheckthenitenessofthecostfunctionalwhichonlydependsonthefactthattheoptimalconsumptionprocesssatisesct=Kp0Wt,hencethisgivesusthat J,c(s,w)=Es,wZ1se)]TJ /F16 7.97 Tf 6.58 0 Td[(t(Kp0Wt) dt(Kp0) Es,wZ1sWtdt(Kp0) Es,wZ1s(1+W2t)dt(Kp0) Ps,w[s,1)+Es,wZ1sW2tdt<1 Hencethecontrolprocessut2A[s,1),tonishthevericationofthetechnicalconditionswemustverifythatthehypothesesofthevericationtheoremaresatised.Firstwenotethat(s,w)2C1,20,andsatisestheHamiltonJacobiBellmanequationbyconstructionoftheconstantKp0.Thegrowthconditionalsoholdssince j(s,w)j=jesKp0wjKp0jwjKp0(1+jwj2) 44

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Usingasimilarcalculationforfc,(s,w)wemayshowthatjfc,(s,w)j=e)]TJ /F16 7.97 Tf 6.59 0 Td[(sc )]TJ /F10 7.97 Tf 6.59 0 Td[(1jcj)]TJ /F10 7.97 Tf 6.58 0 Td[(1(1+jcj2) andtheconditionforc,(s,w)satisesjc,(s,w)j2=jwj2=22jwj222(1+jwj2) ThecalculationsallowustousethevericationTheorem 2.3 forthiscaseoftheoptimalcontrolproblem.Henceanapplicationofthistheoremshowsthatthecandidate(s,w)isactuallytheoptimalsolutiontothecontrolproblemi.e.(s,w)=(s,w).Fortheremainderofthepaperthecheckofthetechnicalconditionswillbelessrigorousthanthissection,asmanyoftheargumentsaresimilartotheaboveroutine.Thepowerutilityfunctionconsideredinthissectionwillbethemainfocusofthepaper,howeverthereisaspecialcaseofthefamilyofhyperbolicabsoluteriskaversionutilityfunctionsthatprovideclosedformsolutionsintheinnitehorizoncase.Thisspecialcaseisthatoflogarithmicutilityofwhichwewillconsiderinthefollowingsection. 3.3InniteHorizonLogUtility Thesecondcaseweconsiderintheinnitehorizoncaseisthatofalogutilityfunctioni.e.U(c)=logc.Thisisanotherexampleofautilityfunctionfromthehyperbolicabsoluteriskaversionfamilywhichexhibitsdecreasingabsoluteriskaversion,thecostfunctionalhastheform J,c(s,w)=Es,wZ1se)]TJ /F16 7.97 Tf 6.59 0 Td[(tlog(ct)dt=e)]TJ /F16 7.97 Tf 6.58 0 Td[(sEyZ10e)]TJ /F16 7.97 Tf 6.59 0 Td[(tlog(cs+t)dt (3) InthiscasetheHamiltonJacobiBellmanequationonlychangesintherstterm sup,ce)]TJ /F16 7.97 Tf 6.58 0 Td[(slogc+s+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)ww+(rw)]TJ /F6 11.955 Tf 11.96 0 Td[(c)w+1 22w22ww=0 Sinceweareworkingoveraninnitehorizonthevaluefunctionhastheform(s,w)=e)]TJ /F16 7.97 Tf 6.58 0 Td[(sb(w),sowemayrewritetheHamiltonJacobiBellmanequationwithoutexplicit 45

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dependenceonthetimevariable sup,clogc)]TJ /F3 11.955 Tf 11.95 0 Td[(b+()]TJ /F6 11.955 Tf 11.96 0 Td[(r)wbw+(rw)]TJ /F6 11.955 Tf 11.96 0 Td[(c)bw+1 22w22bww=0 (3) hencetherstorderconditionforchangesonlyslightlywhiletheconditionforcisdifferentandisgivenby c=1 bw=)]TJ /F5 11.955 Tf 9.3 0 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(r)bw 2wbww (3) Tryingafunctionoftheformb(w)=(Kl0logw+Kl1)andpluggingtheseintotheHamiltonJacobiBellmanequationwendthatlog1 bw)]TJ /F3 11.955 Tf 11.95 0 Td[(b)]TJ /F5 11.955 Tf 13.15 8.09 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2b2w 2bww+rwbw)]TJ /F5 11.955 Tf 11.95 0 Td[(1+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2b2w 22bww=0=)log1 bw)]TJ /F3 11.955 Tf 11.95 0 Td[(b+()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2b2w 22bww+rwbw)]TJ /F5 11.955 Tf 11.95 0 Td[(1=0 takingnecessaryderivativesandpluggingintoEquation( 3 )gives (1)]TJ /F3 11.955 Tf 11.95 0 Td[(Kl0)logw+)]TJ /F5 11.955 Tf 11.29 0 Td[(log(Kl0))]TJ /F3 11.955 Tf 11.96 0 Td[(Kl1+Kl0()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22+rKl0)]TJ /F5 11.955 Tf 11.96 0 Td[(1=0 Notingthattheterminparenthesesisconstantinwgivesthatthesumiszeroonlyifthecoefcientonthelogwiszeroaswell(Sincetheconstanttermandthelogtermareindependentofeachother).Theconstantofthelogtermbeingzerogives 1)]TJ /F3 11.955 Tf 11.95 0 Td[(Kl0=0=)Kl0=1 whilesettingtheconstanttermequaltozeroandsolvingforKl1gives Kl1=1 )]TJ /F5 11.955 Tf 11.29 0 Td[(log(Kl0)+Kl0()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22+rKl0)]TJ /F5 11.955 Tf 11.96 0 Td[(1 UsingthefactthatKl0=1 wemayndtheotherconstantis Kl1=1 2log()+()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2 22+r)]TJ /F5 11.955 Tf 11.96 0 Td[(1 46

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Hencethesolutiontotheoptimalcontrolprobleminthiscaseisgivenby (s,w)=e)]TJ /F16 7.97 Tf 6.59 0 Td[(s1 logw+1 2log()+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22+r)]TJ /F5 11.955 Tf 11.95 0 Td[(1 (3) Thissolutionisonlyvalidifthewealthprocessisstrictlypositivei.e.Wt>0foralltimet,inparticularwemusthavethattheinitialwealthsatisesw>0.Usingthesameparametersasthepowerutilitycaseandplottingthevaluefunctionoverthesamerangeasbefore,weenduptakingonnegativevaluesof(s,w).ThiscanbeseeninFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(3 below,thisisaresultofthemodelsdependenceon.InthepowerutilitycasetheparameterswererestrictedinsuchawaythattheconditioninEquation( 3 )issatised.Inthelogutilitycasewemaychooseanydeltawelikesolongas>0.Thechoiceof=1cancelsoneofthelogtermsinEquation( 3 ),whichleadstothevaluefunctiontakingonnegativevaluesonthegivenrange.Theconditionontoallowthevaluefunctiontoonlytakepositivevaluesisdifcultifnotimpossibletondexplicitlyduetothelog()terminthevaluefunction.Thistermrestrictsourabilitytosolveforexplicitly(coulddomorehere).Usinganiterativenumericalargumentwendthatforbiggerthanavaluebetween1.7and1.8wehavepositivevaluefunction.Giventhe Figure3-3. Valuefunctionforlogutilitywithnojumpswithparameters,=.16,r=.06,=1,=.5,=.3 47

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solution(s,w)asabovewendtheoptimalcontrols c=w Kl0=w=)]TJ /F6 11.955 Tf 11.96 0 Td[(r 2 (3) Hencewehavethatct=Wt,whichmeansthatweagainhavethattheconsumptionprocessislinearfunctionofthewealthprocess.Alsosince>0wehavethesamepositivelycorrelatedrelationshipbetweenthewealthandconsumptionprocessasinthepowerlawcase.Thatis,anincreaseinthewealthoftheinvestorleadstoanincreaseintheinvestorsconsumption,whichisaresultthatmatchesrealworldintuition.Thefractionofwealthintheriskyassetisgivenbyt=)]TJ /F9 7.97 Tf 6.58 0 Td[(r 2whichisagainconstantintime.Tocompletetheanalysisofthisproblemweusethatfactthat ct=Wtt=)]TJ /F6 11.955 Tf 11.95 0 Td[(r 2 (3) toshowthatthewealthprocessbecomes dWt=()]TJ /F6 11.955 Tf 11.95 0 Td[(r)tWtdt+[rWt)]TJ /F6 11.955 Tf 11.95 0 Td[(ct]dt+tWtdBt=()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2 2Wtdt+rWtdt)]TJ /F3 11.955 Tf 11.96 0 Td[(Wtdt+)]TJ /F6 11.955 Tf 11.96 0 Td[(r WtdBt=()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2 2+r)]TJ /F3 11.955 Tf 11.96 0 Td[(Wtdt+)]TJ /F6 11.955 Tf 11.96 0 Td[(r WtdBt hencewehavethewealthprocessisaGeometricBrownianmotionoftheform dWt Wt=()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 2+r)]TJ /F3 11.955 Tf 11.95 0 Td[(dt+)]TJ /F6 11.955 Tf 11.95 0 Td[(r dBtWs=w (3) whosesolutionexistsanduniquegiventheinitialconditionWs=w.TheexplicitsolutiontothisGeometricBrownianmotionisgivenby Wt=we()]TJ /F13 5.978 Tf 5.76 0 Td[(r)2 2+r)]TJ /F16 7.97 Tf 6.58 0 Td[()]TJ /F12 5.978 Tf 7.78 3.26 Td[(1 2()]TJ /F13 5.978 Tf 5.76 0 Td[(r)2 2(t)]TJ /F9 7.97 Tf 6.59 0 Td[(s)+)]TJ /F13 5.978 Tf 5.76 0 Td[(r p t)]TJ /F9 7.97 Tf 6.59 0 Td[(sN(0,1) (3) whereN(0,1)isastandardnormalrandomvariable.AsimulationofonepathofthewealthprocessusingthesameparametersasbeforeispresentedinFigure 3)]TJ /F5 11.955 Tf 11.96 0 Td[(4 .Toshowthatthesolution(s,w)isoptimalweneedtoshowthattheconditionsofthe 48

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Figure3-4. WealthprocessforLogutilitywithnojumpswithparameters,w=1,=.16,r=.06,=1,=.3 vericationtheoremaresatisedsothatthemethodologyislegitimateforthisproblem.Werstneedtoshowthatthecontrolprocessut=(ct,t)isanelementofA[s,1).Thepreciseformofthecontrolprocessisut=Wt,)]TJ /F6 11.955 Tf 11.96 0 Td[(r 2 whichismeasurableandFt-adapted.Wenextcheckthatthecontrolprocesssatisesthefollowingintegrabilitycondition Es,wZ1sjjutjj2dt=Es,w"Z1s(Wt)2+)]TJ /F6 11.955 Tf 11.95 0 Td[(r 22)dt#=2Es,wZ1sW2tdt+)]TJ /F6 11.955 Tf 11.96 0 Td[(r 22Ps,w[s,1)<1 ThersttermisnitesinceWtisauniquestrongsolution,andthesecondtermsisnitesinceweareworkingonaprobabilityspace.Thelastconditionforadmissibilityistocheckthenitenessofthecostfunctionalusingthecontrolwehavefound, J,c(s,w)=Es,wZ1se)]TJ /F16 7.97 Tf 6.58 0 Td[(tlog(Wt)dt 49

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Es,wZ1s(log+logWt)dtEs,wZ1slogdt+Es,wZ1s(1+W2t)dt(1+log)Ps,w[s,1)+Es,wZ1sW2tdt<1 Hencethecontrolprocessut2A[s,1),tonishthevericationofthetechnicalconditionswemustverifythatthehypothesesofthevericationtheoremaresatised.Firstwenotethat(s,w)2C1,20,iscontinuous,andsatisestheHamiltonJacobiBellmanequationbyconstructionoftheconstant.Thegrowthconditionalsoholdssince j(s,w)j=jeslogwjjlogwj(1+jwj2) (3) Asimilarcalculationshowsthattheconditionsforfu(s,w)=e)]TJ /F16 7.97 Tf 6.59 0 Td[(slogcandu(s,w)=walsohold,sothatthecandidate(s,w)isactuallytheoptimalsolutiontotheproblemi.e.(s,w)=(s,w).Althoughthelogarithmicutilityprovidesaclosedformsolutioninthissetting,itwillturnoutthataclosedformsolutioninthenitehorizoncasecannotbefoundusingthemethodologyusedinthispaper.Wewillprovidetheclosedformsolutioninthecasewithjumpsforaninnitehorizoninalatersection,andwewillshowwhyaclosedformsolutionisnottractableinalatersection.Beforemovingonthethecaseswithjumpsweconsiderthepowerutilitycaseoverthenitehorizon[0,T].OncendingaclosedformsolutionthetheoptimalcontrolproblemweshowthatinthelimitasT!1thesolutionwendwillconvergetothesolutionfortheinnitehorizon.Thiswillbeshownexplicitlyintheformulaandalsousingnumericalarguments,toshowthisweproceedinasimilarmannerasbefore. 3.4FiniteHorizonPowerUtility Inthissectionwewouldliketoconsidertheamoregeneralresultofthatwhichwehavealreadysolved.Inparticular,wewouldliketoconsiderthecaseofpowerutilityoveranitehorizon[0,T].Thenitehorizoncasedoesnotallowustousethe 50

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calculationwhichshowedthat(s,w)=e)]TJ /F16 7.97 Tf 6.58 0 Td[(s(0,w)sincewedonotretainthesameformafterasubstitution.Thecostfunctionalhasthefollowingform J,c(s,w)=Es,wZTse)]TJ /F16 7.97 Tf 6.58 0 Td[(tct dt=EyZT)]TJ /F9 7.97 Tf 6.58 0 Td[(s0e)]TJ /F16 7.97 Tf 6.59 0 Td[((s+t)cs+t dt (3) Weseethatinoursolutionwewillbeabletofactoroute)]TJ /F16 7.97 Tf 6.59 0 Td[(sasbefore,howevertheexpectationisnolongerjustafunctionofw.TheexpectationwillnowinvolvebothwandthenewstoppingtimeT)]TJ /F6 11.955 Tf 12.46 0 Td[(s,wewillexplicitlyndthisexpectationandshowthatthissolutiongeneralizesthecasewhenT!1.TheHamiltonJacobiBellmanequationforthisproblemisthesameastheinnitehorizoncaseandisgivenbyEquation( 3 ),hencetheoptimalcontrolsaregivenbyEquation( 3 ).InthenitehorizoncasetheHamiltonJacobiBellmantheoremaddsaninitialconditionfromthefactthat(y)=g(y)forally2@G.Forthisparticularproblemthisequationtranslatesto(T,w)=0,pluggingtheoptimalcontrolsintotheHamiltonJacobiBellmanequationasbeforeweendupwiththepartialdifferentialequationwithboundaryconditiongivenby 1)]TJ /F3 11.955 Tf 11.95 0 Td[( (es)1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1w+s)]TJ /F5 11.955 Tf 20.45 8.09 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(r)22w 22ww+rww=0(T,w)=0 Tondasolutionwemayusetheideaofseparationofvariables,thefunctionofwwillretainthesameformastheinnitecasebutthetimecomponentwillbedifferentasitnowinvolvesT)]TJ /F6 11.955 Tf 12.17 0 Td[(s.Assumingthesolutionhastheform(s,w)=f(s)wwemaytakethenecessaryderivativesandplugintotheHamiltonJacobiBellmanequationtoget 1)]TJ /F3 11.955 Tf 11.96 0 Td[( (es)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1f(s)w)]TJ /F10 7.97 Tf 6.59 0 Td[(1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1+f0(s)w)]TJ /F5 11.955 Tf 13.15 8.09 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2f2(s)2w2)]TJ /F10 7.97 Tf 6.59 0 Td[(2 22f(s)()]TJ /F5 11.955 Tf 11.95 0 Td[(1)w)]TJ /F10 7.97 Tf 6.58 0 Td[(2+rwf(s)w)]TJ /F10 7.97 Tf 6.58 0 Td[(1=0 SimplifyinganddividingoutbywweendupwiththefollowingBernoulliordinarydifferentialequation f0(s)+r)]TJ /F5 11.955 Tf 16.45 8.09 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22()]TJ /F5 11.955 Tf 11.95 0 Td[(1)f(s)=()]TJ /F5 11.955 Tf 11.95 0 Td[(1)(es)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1f(s) )]TJ /F12 5.978 Tf 5.76 0 Td[(1 (3) 51

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f(T)=0 (3) TorecognizethisequationasaBernoulliequationandthenwritedownthegeneralsolutionlet=:r)]TJ /F10 7.97 Tf 15.53 5.48 Td[(()]TJ /F9 7.97 Tf 6.58 0 Td[(r)2 22()]TJ /F10 7.97 Tf 6.59 0 Td[(1)=P(s),q(s)=()]TJ /F5 11.955 Tf 11.98 0 Td[(1)(es)1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1andn= )]TJ /F10 7.97 Tf 6.59 0 Td[(1thenEquation( 3 )becomes f0(s)+p(s)f(s)=q(s)fn(s) Thegeneralsolutionofthistypeofequationisgivenby f(s)="(1)]TJ /F6 11.955 Tf 11.96 0 Td[(n)Re(1)]TJ /F9 7.97 Tf 6.59 0 Td[(n)Rp(s)dsq(s)ds+C e(1)]TJ /F9 7.97 Tf 6.59 0 Td[(n)Rp(s)ds#1 1)]TJ /F13 5.978 Tf 5.76 0 Td[(n (3) aslongasn6=1.Usingthisgeneralsolutionwendthatthesolutionforourproblemisgivenby f(s)=")]TJ /F5 11.955 Tf 10.49 8.09 Td[(()1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1 e )]TJ /F12 5.978 Tf 5.76 0 Td[(1s+Ce )]TJ /F12 5.978 Tf 5.75 0 Td[(1s#1)]TJ /F16 7.97 Tf 6.59 0 Td[( where=)]TJ /F16 7.97 Tf 6.58 0 Td[( )]TJ /F10 7.97 Tf 6.59 0 Td[(1.Usingtheboundaryconditionf(T)=0wendthattheconstantisC=()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.76 0 Td[(1Twhichleadstothegeneralsolution f(s)=")]TJ /F5 11.955 Tf 9.3 0 Td[(()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 e )]TJ /F12 5.978 Tf 5.76 0 Td[(1s+()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 e )]TJ /F12 5.978 Tf 5.75 0 Td[(1T#1)]TJ /F16 7.97 Tf 6.59 0 Td[(="()1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1e )]TJ /F12 5.978 Tf 5.75 0 Td[(1s )]TJ /F5 11.955 Tf 9.3 0 Td[(1+e )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)#1)]TJ /F16 7.97 Tf 6.59 0 Td[(=)]TJ /F10 7.97 Tf 6.59 0 Td[(1 e)]TJ /F16 7.97 Tf 6.59 0 Td[(sh)]TJ /F5 11.955 Tf 9.3 0 Td[(1+e )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.58 0 Td[(s)i1)]TJ /F16 7.97 Tf 6.59 0 Td[(=1 1 1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(r)]TJ /F5 11.955 Tf 16.46 8.09 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22(1)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sh1)]TJ /F6 11.955 Tf 11.96 0 Td[(e )]TJ /F12 5.978 Tf 5.75 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)i1)]TJ /F16 7.97 Tf 6.59 0 Td[( Hencewehavethatthevaluefunctioninthiscaseisgivenby (s,w)=1 1 1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(r)]TJ /F5 11.955 Tf 16.46 8.08 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22(1)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sh1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.75 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)i1)]TJ /F16 7.97 Tf 6.58 0 Td[(w (3) 52

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Asintheinnitehorizoncaseisonlyvalidif>r+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22(1)]TJ /F3 11.955 Tf 11.96 0 Td[() sothatweavoidthevaluefunctionbeingbeinginnite.LettingT!1inEquation( 3 )sinceT)]TJ /F6 11.955 Tf 12.92 0 Td[(s0and )]TJ /F10 7.97 Tf 6.59 0 Td[(1<0weseethattheterme )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)!0andwegetthesolutionfortheinnitehorizoncase.Thiscanserveasacheckoftheoreticalconsistencyfortheformulawehavederivedinthenitehorizoncase.IhavealsousedthefactthatthenitehorizoncaseshouldrecovertheinnitehorizoncaseasT!1tocheckmynumericalresultsandthevalidityoftheprogramsIhavewritten.TherstofthesechecksofconsistencycomesfromtheplotsofthevaluefunctionforincreasingvaluesofT.AsTgetslargeweshouldhavethattheplotsofthevaluefunctioninthenitehorizoncaseshouldapproachthatoftheinnitehorizoncase,theresultsoftheplotsforT=1,2,100arepresentedinFigure 5)]TJ /F5 11.955 Tf 11.95 0 Td[(1 .ForT=1weclearlyseeadifferencebetweentheplots,howeverassoonasT=2itbecomesdifculttodifferentiatethatplotfromtheinnitehorizoncase.ToshowthereisinfactadifferencebetweentheplotsIhaveprovidedthemaximumandminimumvaluesof(s,w)forcomparison.ForthevaluesofT=1andT=2weseethemax/minvaluesdiffer,implyingthegraphsandvaluesof(s,w)aredifferent.OnceT=100(actuallysooner)weseethatthevaluesforthatcaseandtheinnitehorizoncasearethesame,whichmeanstheplotsarethesameandhencethemodelsareindistinguishableforallintentsandpurposes.Wealsopresentmorenumericalvalidationofthenitehorizonmodelbyconsideringthemaximumdifferencesbetweenthemodelforeachnitetimeandtheinnitetimehorizon.WeexpectthatasTgetslargerthedifferencesshouldtendtozero,sincetheterm(1)]TJ /F6 11.955 Tf 12.65 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s))tendstooneandwerecovertheinnitehorizoncase.OurexpectedresultsareconrmedinTable 3)]TJ /F5 11.955 Tf 11.96 0 Td[(1 ,againonceT=100weseethatthemaximumdifferenceofthiscasewiththeinnitehorizoncaseiszerowhichmeansthevaluesof(s,w)arethesameinbothcases.Tondtheconsumptionprocesswe 53

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Figure3-5. PlotsofthevaluefunctionforvaluesofT=1,2,100comparedtotheplotofthevaluefunctionforinnitehorizon Table3-1. MaximumdifferencesbetweennitehorizonandinnitehorizonfordifferentvaluesofT T=1T=2T=3T=5T=100 Maxdifference1.67045492290.11714316540.01536842040.00028021110.0000000000 rewriteEquation( 3 )inasimplerformsothatwemaytakethederivative,combiningthetermsinthebracketswegetEquation( 3 ).UponcloserinspectionweseethatEquation( 3 )isalmostthesameasEquation( 3 ),theonlydifferencecomesfromtheterm(1)]TJ /F6 11.955 Tf 12.01 0 Td[(e )]TJ /F12 5.978 Tf 5.75 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)).Intuitivelythissaysthatthenitehorizoncaseisessentiallytheinnitehorizoncasediscountedbyanappropriatefactor. (s,w)=1 24)]TJ /F3 11.955 Tf 11.95 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.75 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)35)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw (3) 54

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GiventhecompactforminEquation( 3 )wemaycalculatewandplugintotheequationforc=(e)]TJ /F16 7.97 Tf 6.58 0 Td[(sw)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1tondthat c=24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.95 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.58 0 Td[(s)35w (3) Henceforgeneraltimewehavethattheoptimalcontrolsare ct=24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1T35Wtt=()]TJ /F6 11.955 Tf 11.96 0 Td[(r) 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[() PluggingtheseoptimalcontrolsintoEquation( 3 )wendthatthewealthprocessisaGeometricBrownianmotionoftheform dWt Wt=24()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[()+r)]TJ /F3 11.955 Tf 52.09 8.09 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.75 0 Td[(1T35dt+)]TJ /F6 11.955 Tf 11.95 0 Td[(r (1)]TJ /F3 11.955 Tf 11.96 0 Td[()dBtWs=w Checkingtheconditionsofthevericationtheoreminthenitehorizoncaseisslightlymoredifcultthantheinnitehorizoncase.Theaddedcomplicationcomesfromthefactthefunction(s,w)=f(s)wsothatwecannotjustpullouttheconstantwhencheckingtheboundednessconditions.WerstshowthatthecontrolprocessisanelementofA[s,T],thecontrolprocessinthiscaseisgivenbyut=0@24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.75 0 Td[(1T35Wt,()]TJ /F6 11.955 Tf 11.95 0 Td[(r) 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[()1A Itturnsoutthattheconstantinbracketsispositive,wealreadyknowthat>and1)]TJ /F3 11.955 Tf 12.46 0 Td[(>0.Theonlytermthatneedscheckingistheterminvolvingtheexponential.Ifwerewritethistermas1)]TJ /F6 11.955 Tf 12.44 0 Td[(e)]TJ /F17 5.978 Tf 12.67 3.25 Td[( 1)]TJ /F17 5.978 Tf 5.76 0 Td[(T,thenforallT0wehaveanegativeexponentialfunctionhencewehavethate)]TJ /F17 5.978 Tf 12.66 3.26 Td[( 1)]TJ /F17 5.978 Tf 5.76 0 Td[(TisadecreasingfunctionforT2[0,1)sothemaximumoccursatT=0i.e.e)]TJ /F17 5.978 Tf 12.67 3.26 Td[( 1)]TJ /F17 5.978 Tf 5.76 0 Td[(T1asneeded.Followingthesameargumentas 55

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beforewecanshowthat Es,wZTsjjutjj2dt=0@)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1T1A2Es,wZTsW2tdt (3) +)]TJ /F6 11.955 Tf 11.96 0 Td[(r 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[()2Ps,w[s,T]<1 Wenextcheckthenitenessofthecostfunctionalusingthecontrolprocesswehavefoundforthisproblem.Thiscostfunctionalonlydiffersfromtheinnitecaseinthattheintegralisoveraniteintervalandtheconstantbeingmultipliedbythewealthprocessisdifferent.Hencethisgivesthattheargumentisessentiallythesamewithonlyaslightmodication,Ipresentitforcompleteness J,c(s,w)=Es,w241 ZTse)]TJ /F16 7.97 Tf 6.59 0 Td[(t24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.75 0 Td[(1TWt35dt351 24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.95 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1T35Es,wZTsWtdt1 24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.95 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1T35Es,wZTs(1+W2t)dt1 24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.95 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1T35Ps,w[s,T)+Es,wZTsW2tdt<1 SincethecontrolprocessutisFtadaptedwehavethatut2A[s,T],giventhisadmissiblestrategyweneedonlychecktheremainingconditionsofthevericationtheorem.Thefunction2C1,20iscontinuousandsatisestheconditionsoftheHamiltonJacobiBellmanequation.Toshowthatthegrowthconditionissatisedwerstneedtoperformsomeelementaryanalysisontheexponentialfunction.Considerthefunctionf(x)=e)]TJ /F17 5.978 Tf 10.92 3.26 Td[(x 1)]TJ /F17 5.978 Tf 5.76 0 Td[(,thenthederivativef0(x)=)]TJ /F16 7.97 Tf 16.4 4.71 Td[( 1)]TJ /F16 7.97 Tf 6.58 0 Td[(e)]TJ /F17 5.978 Tf 10.92 3.26 Td[(x 1)]TJ /F17 5.978 Tf 5.76 0 Td[(<0forallxhencethefunctionfisdecreasing.InparticularsinceT)]TJ /F6 11.955 Tf 11.96 0 Td[(sTforalls2[0,1)wehavethate)]TJ /F17 5.978 Tf 12.67 3.26 Td[( 1)]TJ /F17 5.978 Tf 5.75 0 Td[((T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)e)]TJ /F17 5.978 Tf 12.67 3.26 Td[( 1)]TJ /F17 5.978 Tf 5.76 0 Td[(T 56

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hencewehavethefollowingrelationthatwillhelpusndaboundfortheterminparentheses1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.75 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)1)]TJ /F6 11.955 Tf 11.96 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1T Withthispreliminaryresultandwritingapositivepowerintheexponentwemaynowshowthatthecandidateforthevaluefunctionisbounded j(s,w)j=1 24(1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e )]TJ /F12 5.978 Tf 5.75 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s) )]TJ /F3 11.955 Tf 11.95 0 Td[(351)]TJ /F16 7.97 Tf 6.58 0 Td[(e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw1 24(1)]TJ /F3 11.955 Tf 11.95 0 Td[()1)]TJ /F6 11.955 Tf 11.96 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1T )]TJ /F3 11.955 Tf 11.96 0 Td[(351)]TJ /F16 7.97 Tf 6.58 0 Td[(jwj1 24(1)]TJ /F3 11.955 Tf 11.95 0 Td[()1)]TJ /F6 11.955 Tf 11.96 0 Td[(e )]TJ /F12 5.978 Tf 5.76 0 Td[(1T )]TJ /F3 11.955 Tf 11.96 0 Td[(351)]TJ /F16 7.97 Tf 6.58 0 Td[((1+jwj2) Theremainingconditionsofthevericationtheoremhavealreadybeenveriedhencewehavethatthecandidateforthesolution(s,w)isactuallythesolutionoftheoptimalcontrolproblemi.e.(s,w)=(s,w).WenotethatinanyoftheequationsaboveacheckofconsistencybetweenthenitehorizoncaseandtheinnitehorizoncasebylettingT!1inanyoftheaboveequationsgivesthecorrectequation.HencewehavegeneralizedtheinnitehorizoncaseofthehyperbolicabsoluteriskaversionutilityfunctionU(c)=c sothatthenitehorizoncaseincludesbothcases.Anattemptatgeneralizingthelogarithmiccasewasmadebutaresultwasnotfoundduetoanequationarisingwhichdoesnothaveananalyticsolution.Henceaclosedformresultwasnotpossibleasinthecaseofthelogarithmicfunction,howeveronecaneasilyusenumericalmethodstondthesolutionofthenonanalyticequationwhicharises.BeforemovingontothecaseofLevyprocesseswithjumpsweprovidetheattemptedanalysisforthelogarithmiccaseforcompleteness. 57

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3.5FiniteHorizonLogUtility Inthissectionweexplorethecaseoflogarithmicutilityoveranitetimehorizon,itturnsoutthatmyattemptatasolutioninthiscasewasunsuccessful.Thereasonforthisisthatweendupwithanequationwhichhasnoanalyticsolution,howeverIprovidetheattemptforcompleteness.Sinceweareinthenitecasewearenotabletocompletelyremovethetimedependencefromthevaluefunction.Toseethisweconsiderthecostfunctionaloftheform J,c(s,w)=Es,wZTse)]TJ /F16 7.97 Tf 6.59 0 Td[(tlogctdt=EyZT)]TJ /F9 7.97 Tf 6.59 0 Td[(s0e)]TJ /F16 7.97 Tf 6.58 0 Td[((s+t)logcs+tdt (3) Weseethatevenafterfactoringoutthee)]TJ /F16 7.97 Tf 6.58 0 Td[(stermtheremainingexpectationstillinvolvesthetimeparameter,inparticularwehavethat(s,w)=e)]TJ /F16 7.97 Tf 6.58 0 Td[(sb(s,w).TheHamiltonJacobiBellmanequationwithboundaryconditionforthisproblemisthesameastheinnitehorizoncase sup,ce)]TJ /F16 7.97 Tf 6.58 0 Td[(slogc+s+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)ww+(rw)]TJ /F6 11.955 Tf 11.96 0 Td[(c)w+1 22w22ww=0(T,w)=0 Usingthefactthatwemayremovethee)]TJ /F16 7.97 Tf 6.59 0 Td[(sfromthevaluefunctionwemayrewritetheHamiltonJacobiBellmanequationinthefollowingsimpliedform sup,clogc)]TJ /F3 11.955 Tf 11.96 0 Td[(b+bw+()]TJ /F6 11.955 Tf 11.96 0 Td[(r)wbw+(rw)]TJ /F6 11.955 Tf 11.95 0 Td[(c)bw+1 22w22bww=0b(T,w)=0 Assumingthevaluefunctionbhastheformb(s,w)=g(s)logw,takingnecessaryderivativesandpluggingintotheHamiltonJacobiBellmanweget sup,clogc)]TJ /F3 11.955 Tf 11.96 0 Td[(g(s)logw+g0(s)logw+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)g(s)+rg(s))]TJ /F6 11.955 Tf 14.66 8.09 Td[(c wg(s))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 222g(s)=0 Theboundaryconditionnowstatesthatb(T,w)=e)]TJ /F16 7.97 Tf 6.59 0 Td[(Tg(T)logw=0,wenotethattherearetwopossiblecasesforthisproducttobezero.Eitherw=1andg(T)isallowedtobeanyrealnumberorw6=1andg(T)=0.Ineithercasetheoptimal 58

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controlsarec=w g(s)=()]TJ /F6 11.955 Tf 11.96 0 Td[(r) 2 WiththeseoptimalcontrolstheHamiltonJacobiBellmanequationbecomesthefollowingordinarydifferentialequation logw g(s))]TJ /F3 11.955 Tf 11.96 0 Td[(g(s)logw+g0(s)logw+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 2g(s)+rg(s))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22g(s)=0 Simplifyingthisexpressionandcombiningliketermsweendupwiththefollowingequation [1)]TJ /F3 11.955 Tf 11.96 0 Td[(g(s)+g0(s)]logw+()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2 22+rg(s))]TJ /F5 11.955 Tf 11.96 0 Td[(log(g(s)))]TJ /F5 11.955 Tf 11.96 0 Td[(1=0 (3) Ineithercasewewillhavetheproblemofnotbeingabletosolvethisproblemanalytically,howeverweshowthatbothcasesreducetosolvingthesameequation. case1:Werstconsiderthecasewherew6=1thenfromEquation( 3 )thecoefcientoflogwmustbeequaltozerowhichgivesthefollowingequationwithboundaryconditiong0(s))]TJ /F3 11.955 Tf 11.96 0 Td[(g(s)=)]TJ /F5 11.955 Tf 9.3 0 Td[(1g(T)=0 ThisisalinearODEwhosesolutionisgivenbyg(s)=1 +Ces TheproblemofsolvingthisequationisthatweneedtosatisfyEquation( 3 )sowemustalsohavethattheremainingconstanttermwhichdoesnotdependonwmustalsobeequaltozeroi.e.()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22+rg(s))]TJ /F6 11.955 Tf 11.95 0 Td[(log(g(s))=1 case2:Thiscaseessentiallyhasthesameproblemasthelastcase,ifw=1thelogtermfromEquation( 3 )disappearsandweendupwiththesameequationasbefore()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22+rg(s))]TJ /F6 11.955 Tf 11.95 0 Td[(log(g(s))=1g(T)2R 59

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IneithercaseweendupwithanequationwhichisnotabletobesolvedanalyticallysothemethodologyIusedintheinnitehorizoncaseisnotabletoprovideasolution.Howevertheremayexistsanothermethodologywhichmaybeabletoprovidearesultwhichgeneralizestheinnitehorizoncase.ThesamedifcultyarisesinlatersectionswhenweassumetheunderlyingstockpricefollowsaLevyprocesssoIwillnotpresentthatcaseexplicitly.Ifwewereabletondasolutioninthiscasethegeneralizationtothejumpcasewouldfollowthesamemethodologythatwillbeusedinfuturesections. 60

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CHAPTER4LEVYPROCESS 4.1WhyuseaLevyProcesswithJumps? InthecasesaboveweassumetheriskyassetfollowsageometricBrownianmotion,sothattheassetreturnsarelog-normallydistributed.TheassumptionofassetpricesfollowingaGeometricBrownianmotionhasseveralproblemswhencomparetorealworldphenomenon.TherstoftheseisthefactthatGeometricBrownianmotionassumesthatstockpricesfollowacontinuouspath,howevermanyequitypricesexhibitlargecorrectionswhichbehavelikediscontinuousjumpsintheprice.Muchoftheriskassociatedwithequitypricemovementsoccursduringtheperiodsofthecorrection,soinordertoaccuratelymodelthepriceswemayrequirejumpdiffusionmodels.FurtherevidencethatsuggestsJumpdiffusionmodelsmaybeusefulinmodelingthebehaviorofstockpricescomesfromconsideringthereturnsonassetprices.GiventheassumptionofGeometricBrownianmotionwehavethatlogofassetreturnsarenormallydistributed,thisdoesnotmatchtheempiricallycollecteddatafromrealworlddata.Manyassetpricesexhibitadistributionofreturnswhichhavesocalledheavytails.Thismeansthatthedistributionsdecayslowlytozeroattheextremes,theresultofthisisthatrare(tail)eventsareassignedlargerprobabilitiesofoccurringthananormaldistribution.Thisleadstothemodelunderestimatingtheriskoflargemovementsintheassetpricesthatistypicallyobservedinarealworldsetting.HenceamoreaccuratemodelofrealworldassetpricesshouldexhibithigherprobabilityoftaileventsthantheBrownianmotionmodel.Thelibraryofavailablediffusionmodelsisveryextensive,andtherearesomediffusionmodelswhichmaybeabletoreplicateaheavytaileddistribution,forexamplewemayusestochasticvolatilitymodelstoallowforgreatervarianceintheassetpricesleadinghigherprobabilityoftailevents.However,nomatterhowaccuratelyweareabletomodelreturnsofassetprices,itismuchmoredifculttouseBrownianmotiontomodelsuddenjumpsinthepricesonthetimescaleweareinterestedin. 61

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4.2PreliminariesofLevyProcess WewouldnowliketoconsidertheoptimalcontrolproblemwhenthestockpriceisdrivenbyamoregeneralprocessthangeometricBrownianmotion.Inordertodothisweneedtostudyaprocesswhoseunderlyingdistributionmaynothaveanormaldistribution.AsinthecaseofaprocessdrivenbyBrownianmotion,thisnewprocessshouldhaveindependentandstationaryincrements.Hencewewilluseessentiallythesameassumptionsforournewprocesswiththemajordifferencelyinginthefactthatincrementsarenolongernormallydistributed.ThesesocalledLevyprocessesmayhaveamuchmoregeneralunderlyingdistributionwhichincludethenormaldistributioncaseasbefore.ThestudyofLevyprocessesisextensive,andwewillonlyprovideashortintroductionofthematerialnecessarytosolveourproblem.AnextensivepresentationofLevyprocessesandthetheoreticalframeworkcanbefoundinbookssuchas[ 1 ],[ 4 ],[ 25 ].TheapplicationofLevyprocessestothenancialmarketsisstillatanelementarystageandmoreapplieddescriptionsofLevyprocessinnancecanbefoundin[ 8 ]and[ 26 ]. Denition4.1. Let(,F,fFtgt0,P)bealteredprobabilityspace.AnFtadaptedstochasticprocess(Xt)t0iscalledaLevyprocessifthefollowingconditionshold (i) X0=0a.s (ii) Givenapartitiont00andh0limt!hP(jXt)]TJ /F6 11.955 Tf 11.95 0 Td[(Xhj>)=0 JustasinthecaseofstockpricesdrivenbyBrownianmotionwemustextendourideaofNewtonianCalculustoincludethesemoregeneralLevyprocesses.TheextensionisaidedbywhatwealreadynowabouttheItoCalculus,butmustbeextendedtoinclude 62

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jumpprocesseswhichareabletobemodeledintheLevyprocesscase.InordertoperformanalysisonastockpricedrivenbyaLevyprocesswemustdeneintegralswithrespecttoaLevyprocess,themainextensionwillbeduetointegralswithrespecttoadiscontinuousprocess.Togettothepointofdeningintegralsagainstadiscontinuousmeasureweneedthefollowingbackgroundonjumpprocesses. Denition4.2. Let(Xt)t0beaLevyprocess,wedenethejumpprocessbytheprocess(Xt)t0whereXt=Xt)]TJ /F6 11.955 Tf 11.95 0 Td[(Xt)]TJ ET BT /F1 11.955 Tf 0 -179.31 Td[(IfweconsiderthejumpsofaLevyprocessdirectlywendthatadifcultyarisesfromthefactthatwemayhaveP0stjXsj=1a.s.Toavoidthisproblemwewouldliketoconsiderjumpsofaspeciedsizesothatwemayhavea-nitemeasuretointegratewithrespectto.Thefollowingdenitionallowsustoavoidthisdifculty. Denition4.3. Lett2[0,1)andU2B(Rd)-280(f0g)wedenethePoissonrandommeasureN(t,U)(!)=#f0st;Xs(!)2Ug=X0stU(Xs(!)) Ifwext0and!2weseethatN(t,)isacountingmeasureonB(Rd)-279(f0g)whichcountsthenumberofjumpsontheinterval[0,t]ofsizeXsinagivenBorelsetU.AlternativelyifwextandUwehavethatN(t,U)isarandomvariabledenedonsothatthequantityE[N(t,U)]=ZN(t,U)(!)dP(!)iswelldened,thisleadstothefollowingdenition Denition4.4. Let(Xt)t0beaLevyprocessandN(t,)betheassociatedPoissonrandommeasurethen()=E[N(t,)] isaBorelmeasureonB(Rd)-221(f0g)calledtheLevymeasure. Itcanbeshownthatif0=2 UthenN(t,U)<1forallt0,sothatthemap!7!N(t,U)!isa-nitemeasure,andimplicationofthisresultisthat(U)<1sothatis 63

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a-nitemeasurealso.AsarandomvariablethePoissonrandommeasureN(t,U)hasaparticularlyfamiliarformthatleadstomanyusefulproperties. Theorem4.1. ForeveryU2B(R)with0=2 Uwehavethat(N(t,U))t0isaPoissonprocesswithintensity=(U),inparticularP[N(t,U)=n]=e)]TJ /F16 7.97 Tf 6.58 0 Td[(t(t)n n!n=1,2,3, Proof. Aproofofcanbefoundin[ 1 ]orotherintroductorybooksonLevyprocess. ThispropertyisusefultokeepinmindwhenperformingcalculationsinvolvingPoissonintegrals,aparticularlyusefulresultisthatintegralswithrespecttothisPoissonrandommeasurewillturnouttobeacompoundPoissonprocess.ToseethisandtheotherusefulpropertiesweprovideaformaldenitionoftheintegralwithrespecttoN(t,U), Denition4.5. LetfbeaBorelmeasurablefunctiononUwith0=2 UwedenethePoissonintegralofftobeZUf(z)N(t,dz)=Xz2Uf(z)N(t,fzg)=X0
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satisesthefollowingproperties Forallt0,therandomvariableZthasacompoundPoissondistributionwithcharacteristicfunctionZt(u)=E[eiuZt]=etRU(eiuz)]TJ /F10 7.97 Tf 6.58 0 Td[(1)f(dz)forallu2R wheref=f)]TJ /F10 7.97 Tf 6.58 0 Td[(1. Iff2L1(U,jU)thenE[Zt]=tZUf(z)(dz) Iff2L2(U,jU)thenVar[Zt]=tZUjf(z)j2(dz) (Zt)t0isacompoundPoissonprocess. (eZt)t0isamartingale,whereeZt=Zt)]TJ /F6 11.955 Tf 11.95 0 Td[(tZUf(z)(dz) ThemostusefulofthethesepropertiesisthefactthateZtisamartingale,thisgivesthatforareasonablefunctionf(z)theexpectationofeZtwillbeequaltozero.Forouranalysiswewillbeconsideringtheparticularlysimplefunctionf(z)=z,whoseintegralwithrespecttoPoissonmeasurehasthefollowingusefulinterpretation.ThePoissonintegralZUzN(t,dz)=X0
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Theorem4.3. (Levy-ItoDecomposition)Let(Xt)t0beaLevyprocess,thenXthasthedecompositionXt=t+Bt+Zt0Zjzj
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=+ZjzjRz(dz)t+Bt+Zt0Zjzj
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Theorem4.6. ConsidertheIto-Levyprocessstochasticdifferentialequationon[s,T]oftheformdXt=(t,w))]TJ /F15 11.955 Tf 11.95 16.28 Td[(Zjzj0suchthatforallx,y2Rj(t,x))]TJ /F3 11.955 Tf 11.95 0 Td[((t,y)j2+j(t,x))]TJ /F3 11.955 Tf 11.95 0 Td[((t,y)j2+ZRj(t,x,z))]TJ /F3 11.955 Tf 11.95 0 Td[((t,y,z)j2(dz)C1jx)]TJ /F6 11.955 Tf 11.95 0 Td[(yj2 ThereexistsconstantC2>0suchthatforallx2R j(t,x)j2+j(t,x)j2+ZRj(t,x,z)j2(dz)C2(1+jxj2) ThenthereexistsauniquecadlagadaptedsolutionXtsuchthatEs,xZTsjXtj2<1 ThroughoutthepaperwewillbedealingwiththesocalledGeometricLevyprocesswhichsatisestheconditionsofthefortheexistenceanduniquenessforasolutionofthecorrespondingstochasticdifferentialequation.ThegeneralformofaGeometricLevyprocessindifferentialformisgivenby dXt=Xtdt+XtdBt+Xt)]TJ /F15 11.955 Tf 9.08 18.06 Td[(ZR(t,z)eN(dt,dz)Xs=x (4) whoseuniquesolutioncanbefoundbyapplyingtheIto-Levytheoremtoln(Xt).TheIto-LevytheoremisanextensionofIto'slemmatoincludeprocesseswithjumps,henceitwillbeusefultoconsideraLevyprocessasasumofcontinuousanddiscontinuousparts.TheLevy-Itodecomposition 4.3 allowsustobreakeveryLevyprocessintotwopieces,oncethisisdonewemayapplytheclassicalIto'slemmatothecontinuouspart.Oncewehavefoundthedifferentialforthecontinuouspartonemayapplyan 68

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argumenttogureouthowthediscontinuouspartchangesbeforeandafterjumps.Thisinterestedreadercanndthisargumentin[ 21 ],weprovidetheresultofthatanalysisintheIto-Levytheorem Theorem4.7. (OnedimensionalIto-Levytheorem)SupposeXt2RisanIto-LevyprocessoftheformdXt=(t,w))]TJ /F15 11.955 Tf 11.95 16.27 Td[(Zjzj0sothatthelogarithmictermiswelldened.FortheIto-Levyprocesswewillbeconsideringwewillhavethat(s,z)=z,thisimpliesthatwewillneedtoconsiderjumpsizeslargerthan)]TJ /F5 11.955 Tf 9.3 0 Td[(1sothatthisequationiswelldened.ToperformtheanalysisinthejumpcaseusingdiffusiontheoryweneedtobeabletouseanequivalentversionoftheHamiltonJacobiBellmantheorem.InordertowritedowntheHamiltonJacobiBellmanequation 69

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forthecasewithjumpsweneedtocomputetheinnitesimalgeneratorofanIto-Levyprocess,morespecicallyweneedthegeneratorofanIto-Levydiffusion.ThegeneralversionoftheIto-LevyprocesspresentedinTheorem 4.7 canbesimpliedbyassumingthatEjXtj<1asbefore.GiventhisassumptionwehavethatthemostgeneralformreducesandItoprocessoftheformfoundinTheorem 4.4 ,thiswillbetheversionweusethroughoutthepaper.TheinnitesimalgeneratorforaLevyprocesswithjumpsisfoundbyusingtheIto-LevytheoremwiththefactthateNisamartingale,thestatementandproofareprovidedinthefollowingtheorem. Theorem4.8. LetXtbeanIto-LevydiffusionoftheformdXt=(t,Xt)dt+(t,Xt)dBt+ZR(t,Xt)]TJ /F5 11.955 Tf 6.75 -.3 Td[(,z)eN(dt,dz)Xs=x if2C1,20([0,1),R)thenA(s,x)existsforallx2Rands2[0,1)andisgivenby A(s,x)=@ @t(s,x)+(s,x)@ @x(s,x)+1 2(s,x)@2 @x2(s,x)+ZR(s,x+(s,x,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((s,x))]TJ /F3 11.955 Tf 13.15 8.09 Td[(@ @x(s,x,z)(dz) (4) Proof. Since2C1,20([0,1),R)wemayapplyIto'sformulatocomputed(t,Xt)=@ @t(t,Xt)+@ @x(t,Xt)dXct+1 2@2 @x2(t,Xt)dXctdXct+ZR[(t,Xt)]TJ /F5 11.955 Tf 9.74 1.8 Td[(+(t,Xt)]TJ /F5 11.955 Tf 7.09 1.8 Td[(,z)))]TJ /F3 11.955 Tf 11.96 0 Td[((t,Xt)]TJ /F5 11.955 Tf 7.08 1.8 Td[()]N(dt,dz)=@ @t(t,Xt)+(t,Xt)@ @x(t,Xt)+1 22(t,Xt)@2 @x2(t,Xt)dt+(t,Xt)@ @x(t,Xt)dBt+ZR[(t,Xt)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+(t,Xt)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((t,Xt)]TJ /F5 11.955 Tf 7.09 1.79 Td[()]N(dt,dz))]TJ /F15 11.955 Tf 19.26 16.28 Td[(ZR@ @x(t,Xt)(t,Xt)]TJ /F5 11.955 Tf 7.08 1.79 Td[(,z)(dz)dt IfwenowaddandsubtractthetermZR[(t,Xt)]TJ /F5 11.955 Tf 9.74 1.8 Td[(+(t,Xt)]TJ /F5 11.955 Tf 7.08 1.8 Td[(,z)))]TJ /F3 11.955 Tf 11.96 0 Td[((t,Xt)]TJ /F5 11.955 Tf 7.08 1.8 Td[()](dz)dt 70

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fromtherighthadsideoftheequationford(t,Xt)weendupwithamartingaleterminvolvingeN(dt,dz)whichwemayeliminatewhenwetakeexpectations.Beforetakingexpectationswehavethat d(t,Xt)=@ @t(t,Xt)+(t,Xt)@ @x(t,Xt)+1 22(t,Xt)@2 @x2(t,Xt)dt+ZR(t,Xt)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+(t,Xt)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((t,Xt)]TJ /F5 11.955 Tf 7.09 1.79 Td[())]TJ /F3 11.955 Tf 13.15 8.09 Td[(@ @x(t,Xt)(t,Xt)]TJ /F5 11.955 Tf 7.08 1.79 Td[(,z)(dz)dt+ZR[(t,Xt)]TJ /F5 11.955 Tf 9.74 1.8 Td[(+(t,Xt)]TJ /F5 11.955 Tf 7.09 1.8 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((t,Xt)]TJ /F5 11.955 Tf 7.09 1.8 Td[()]eN(dt,dz)+(t,Xt)@ @x(t,Xt)dBt=:bA(t,Xt)dt+ZR[(t,Xt)]TJ /F5 11.955 Tf 9.75 1.8 Td[(+(t,Xt)]TJ /F5 11.955 Tf 7.09 1.8 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((t,Xt)]TJ /F5 11.955 Tf 7.08 1.8 Td[()]eN(dt,dz)+(t,Xt)@ @x(t,Xt)dBt Integrationofbothsidesoftheequationwithrespecttothepropermeasureswendthat (t,Xt))]TJ /F3 11.955 Tf 11.95 0 Td[((s,Xs)=ZtsbA(r,Xr)dr+ZtsZR[(r,Xr)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+(r,Xr)]TJ /F5 11.955 Tf 7.08 1.79 Td[(,z)))]TJ /F3 11.955 Tf 9.3 0 Td[((r,Xr)]TJ /F5 11.955 Tf 7.09 1.79 Td[()]eN(dr,dz)+Zts@ @x(r,Xr)(r,Xr)dBr usingthefactthatXs=xandtakingtheexpectationEs,xofbothsidesoftheequationweareabletocomputethenumeratorofthelimit Es,x[(t,Xt)])]TJ /F3 11.955 Tf 11.96 0 Td[((s,x)=Es,xZtsbA(r,Xr)dr+Es,xZts@ @x(r,Xr)(r,Xr)dBr+Es,xZtsZR[(r,Xr)]TJ /F5 11.955 Tf 9.74 1.8 Td[(+(r,Xr)]TJ /F5 11.955 Tf 7.08 1.8 Td[(,z)))]TJ /F3 11.955 Tf 11.96 0 Td[((r,Xr)]TJ /F5 11.955 Tf 7.09 1.8 Td[()]eN(dr,dz) Giventhatthefunction@ @x(r,Xr)(r,Xr)isBF-measurable,Fr-adapted, andEs,xhRts)]TJ /F16 7.97 Tf 6.68 -4.43 Td[(@ @x(r,Xr)(r,Xr)2dri<1astandardresultfromstochasticcalculusgivesthatEs,xZts@ @x(r,Xr)(r,Xr)dBr=0 71

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GiventhatthemeasureeN(dr,dz)isamartingalewehaveasimilarintegrabilityconditionforPoissonintegralsEs,xZtsZR[(r,Xr)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+(r,Xr)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((r,Xr)]TJ /F5 11.955 Tf 7.08 1.79 Td[()]2(dz)dr<1 oncethisconditionissatisedwemayusethatfactthatthemartingaleprocessisconstantonaveragewhichallowsustoconcludethatEs,xZtsZR[(r,Xr)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+(r,Xr)]TJ /F5 11.955 Tf 7.08 1.79 Td[(,z)))]TJ /F3 11.955 Tf 11.96 0 Td[((r,Xr)]TJ /F5 11.955 Tf 7.09 1.79 Td[()]eN(dr,dz)=0 hencewehavethatbothmartingaletermsareremovedwhenwetakeexpectationssothattheequationfortheinnitesimalgeneratorsimpliestoEs,x[(t,Xt)])]TJ /F3 11.955 Tf 11.95 0 Td[((s,x)=Es,xZtsbA(r,Xr)dr dividingbothsidesbyt)]TJ /F6 11.955 Tf 11.95 0 Td[(sandtakinglimitsgivesA(s,x)=:limt!sEs,x[(t,Xt)])]TJ /F3 11.955 Tf 11.95 0 Td[((s,x) t)]TJ /F6 11.955 Tf 11.96 0 Td[(s=limt!sEs,x1 t)]TJ /F6 11.955 Tf 11.96 0 Td[(sZtsbA(r,Xr)dr=Es,xlimt!s1 t)]TJ /F6 11.955 Tf 11.96 0 Td[(sZtsbA(r,Xr)dr=Es,xd dtZtsbA(r,Xr)dr=Es,xhbA(t,Xt)i=@ @t(s,x)+(s,x)@ @x(s,x)+1 2(s,x)@2 @x2(s,x)+ZR(s,x+(s,x,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((s,x))]TJ /F3 11.955 Tf 13.15 8.09 Td[(@ @x(s,x,z)(dz) TheargumentisessentiallythesameasthecasewithoutjumpsoncewehaveappliedtheIto-Levyformula,foradifferentargumentinvolvingFouriertransformsthereadermayseetheorem3.3.3of[ 1 ]. 72

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WemaynowaversionoftheHamiltonJacobiBellmanequationforaLevyprocesswithjumps,thetheoremisessentiallythesameasthecasewithoutjumps.Theonlymajordifferencecomesfromachangeintheintegrabilitycondition,theremaininghypothesesandtheconclusionisthesameasthecasewithoutjumps. Theorem4.9. Let2C2(G)\C( G),supposethefollowingconditionshold Au(y)+fu(y)0forally2G,u2U f)]TJ /F5 11.955 Tf 7.09 -4.34 Td[((Y)gGisuniformlyintegrableforallu2A[0,G]andy2G. Ey"j(Y)j+ZG0 jAu(Yt)j+(Yt)@ @y(Yt)2+ZRj(Yt+(Yt,ut,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((Yt)j2(dz)dt<1 YG2@Sa.s.onfS<1gandlimt!S)]TJ /F3 11.955 Tf 8.25 5.81 Td[((Yt)=g(YS)fS<1ga.sforallu2A[0,G] then(y)Ju(y)forallMarkovcontrolsu2Aandy2G.Moreoverifforally2GwendaMarkovcontrolu=u0(y)suchthat fu0(y)(y)+Au0(y)(y)=0 thenut=u0(Yt)isoptimaland(y)=(y)=Ju(y). Proof. TheproofisthesameasTheorem 2.2 thecasewithnojumps,theonlydifferenceshowsupinthegeneratorAuwhichdoesnotaffecttheargumentfromTheorem 2.2 .TheextraintegrabilityconditionistomakesuretheperformancefunctionJu(y)isnite,forthenojumpcasethiswastakencareofsincethecontroluwasadmissible. 4.4VericationTheoremforLevyProcesswithJumps AsbeforewewillusetheHamiltonJacobiBellmanequationindirectlytosolvetheoptimalcontrolproblem,thesolutionwillcomefromanequivalentvericationtheoremforLevyjumpprocesses.Thestatementandproofofthevericationtheoreminthe 73

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jumpcaseisverysimilartothenonjumpcaseinTheorem 2.3 ,sowewillfocusmainlyonthedifferences.Thestatementofthetheoremonlydiffersbytheadditionofaconditionforthefunction(r,x,z),andisgivenby Theorem4.10. Letu2A[s,G]and(s,x)2Gandsupposethefollowingconditionsaresatisedforalls2[0,G]andx2R 2C1,2([0,G)R)iscontinuouson[0,G]Randsatisesthequadraticgrowthconditionj(s,x)jC(1+jxj2) satisestheHamiltonJacobiBellmanequationsupu2A[s,G][fu(s,x)+Au(s,x)]=0s2[0,G)(G,x)=g(G,x) fuiscontinuouswithjfu(s,x)jCf(1+jxj2+jjujj2)forsomeconstantCf>0. ju(s,x)j2C(1+jxj2+jjujj2)forsomeconstantC>0. ZRj(s,x,z)j2(dz)C(1+jxj2+jjujj2) then(s,x)(s,x)forall(s,x)2G.Moreoverifuo(s,x)isthemaxofu7!fu(s,x)+Au(s,x)andu=u0(s,Xs)isadmissiblethen(s,x)=(s,x)forall(s,x)2Ganduisandoptimalstrategyi.e.(s,x)=Ju(s,x). Proof. Fortheproofweproceedasbeforeuntilwendadifference,wethenadjustaccordinglyandreferthereadertotheproofofTheorem 2.3 sincetheremainingjusticationwillbethesame.Fixs2[0,G]andx2R,togetthattheprocessXisboundedwedenethestoppingtimen=G^infft>s:jXt)]TJ /F6 11.955 Tf 11.95 0 Td[(Xsjng Letu2A[s,G]beanadmissiblecontrolandXs=xthenbytheIto-Levytheoremwehavethat(n,Xn)=(s,x)+ZnsAur(r,Xr)dr+Znsx(r,Xr)ur(r,Xr)dBr 74

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+ZnsZR[(r,Xr)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+(r,Xr)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((r,Xr)]TJ /F5 11.955 Tf 7.08 1.79 Td[()]eN(dr,dz) (4) Wewouldnowliketoshowthatthetwomartingaleintegralsgotozeroinexpectation,thetermwiththeintegralwithrespecttoBrownianmotioncanbetreatedinthesamemannerasbefore.HenceourfocuswillbeontheterminvolvingtheintegralwithrespecttothemartingaleeN,wewouldliketoshowthistermgoestozeroaswell.TodothiswemustshowthatthefollowingintegrabilityconditionissatisedEs,xZnsZR[(r,Xr)]TJ /F5 11.955 Tf 9.74 1.8 Td[(+(r,Xr)]TJ /F5 11.955 Tf 7.09 1.8 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((r,Xr)]TJ /F5 11.955 Tf 7.08 1.8 Td[()]2(dz)dr<1 TodothiswemustusethefactthatisC1on[s,n]henceisLipschitzcontinuoussothatthereexistsaconstantK>0suchthatj(r,Xr)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+(r,Xr)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((r,Xr)]TJ /F5 11.955 Tf 7.08 1.79 Td[()jKj(r,Xr)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,z)j UsingthisfactandtheadditionalhypothesiswemaynowshowthattheintegrabilityconditionissatisedEs,xZnsZR[(r,Xr)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+(r,Xr)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((r,Xr)]TJ /F5 11.955 Tf 7.08 1.79 Td[()]2(dz)drK2Es,xZnsZRj(r,Xr)]TJ /F5 11.955 Tf 7.09 1.79 Td[(,z)j2(dz)drK2Es,xCZns(1+jXr)]TJ /F2 11.955 Tf 7.08 1.79 Td[(j2+jjujj2)dr<1 WiththisintegrabilityconditionsatisedwehavebyastandardresultofmartingalesthatEs,xZnsZR[(r,Xr)]TJ /F5 11.955 Tf 9.74 1.8 Td[(+(r,Xr)]TJ /F5 11.955 Tf 7.09 1.8 Td[(,z)))]TJ /F3 11.955 Tf 11.95 0 Td[((r,Xr)]TJ /F5 11.955 Tf 7.08 1.8 Td[()]eN(dr,dz)=0 HencetakingexpectationsinEquation( 4 )wegetthat Es,x[(n,Xn)]=Es,x(s,x)+ZnsAur(r,Xr)dr FromthispointontheproofisthesameastheproofofTheorem 2.3 ,thereadershouldconsultthistheoremforfurtherdetails. 75

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ArmedwithavericationtheoreminthecaseforLevyprocesseswithjumpswemaynowproceedwiththesamealgorithmasbeforetosolvetheoptimalcontrolproblem.Thebasicmethodologyofthealgorithmisthesameasthecasewithoutjumps,howevertheintroductionofadiscontinuouspartoftheLevyprocesswillmaketheanalysismoreinteresting.Theadditionalcomplicationofajumpprocesswillnothinderaclosedformsolutioninthepowerutilitycase,hencewewillbeabletodirectlycomparethetwocases.Thecomparisonbetweenthetwocaseswillbethemainfocusfortheremainderofthepaper,wedeveloptheremainingstructureneededforasolutionthenproceedwiththemainresults. 76

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CHAPTER5OPTIMALCONTROLPROBLEMINTHEJUMPCASE 5.1ClassicMertonProblemwithJumps InthissectionweextendtheclassicalsolutionoftheoptimalcontrolproblemfromtheMertoncasetoincludeapriceprocesswithjumps.Wewillinheritmuchofthesamenotationasthenojumpcase,asmuchoftheworktofollowisageneralizationofthatcasetoincludeprocesseswithjumps.Wewillconsiderboththeniteandinnitehorizoncasesandshowthatthenitehorizoncaseisjustageneralizationoftheinnitehorizoncase.However,werstbeginouranalysisnaivelyassumingwedonotknowthisforcertain,hencewewillconsidertheinnitehorizoncaseandthenworkourwaytomoregeneralcases.Let(,F,F[0,1),P)bealteredprobabilityspace.Wewillagainassumethatwemayapplythemutualfundtheoremsothatourmarketconsistsofariskfreeassetandariskyasset.TheriskfreeRtassetevolvesaccordingtothedifferentialequation dRt=rRtdt;R0=1 wherer0isaconstantwhichrepresentstheriskfreerateofinterest.TheriskyassetStevolvesaccordingtoaGeometricLevyprocesswhichisessentiallyaGeometricBrownianmotionmodelwithanaddedintegralforthejump/discontinuouspart.Thedifferentialformofthethisprocessisgivenby dSt=Stdt+StdBt+St)]TJ /F15 11.955 Tf 9.08 18.06 Td[(Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1zeN(dt,dz);S0=s0 where0and0representtherateofthereturnandvolatilityoftheassetStrespectively.WiththisassumedformoftheGeometricLevyprocessfortheriskyassetwehavethat(s,z)=z,tohaveauniquesolutiontothisstochasticdifferentialequationwemusthavethat1+(s,z)>0whichimpliesthatz>)]TJ /F5 11.955 Tf 9.3 0 Td[(1.Thismeanswemayonlyassumejumpsizesthatarelargerthan)]TJ /F5 11.955 Tf 9.3 0 Td[(1,sowemayhavejumpsinbothdirectionsbutnegativejumpscannotbetoolarge.Toincludelargernegativejumpsonemaychange 77

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themodeloftheriskyasset,wedonotaddressthatinthispaperandwillrestricttojumpslargerthan)]TJ /F5 11.955 Tf 9.29 0 Td[(1.Assumingwehaveaselfnancingtradingstrategyasbeforewehavethatthewealthoftheportfolioevolvesaccordingto dWt=()]TJ /F6 11.955 Tf 11.96 0 Td[(r)tWtdt+[rWt)]TJ /F6 11.955 Tf 11.95 0 Td[(ct]dt+tWtdBt+tWt)]TJ /F15 11.955 Tf 9.07 18.07 Td[(Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1zeN(dt,dz) (5) withinitialconditionW0)]TJ /F5 11.955 Tf 13.98 1.79 Td[(=w.InorderforEquation( 5 )tosatisfythepropermeasurabilityconditionswenotethatweareusingWt)]TJ /F1 11.955 Tf 10.41 1.8 Td[(ratherthanWt,howeversinceWtisacadlagprocesswehavethatWtisthesameasWt)]TJ /F1 11.955 Tf 7.09 1.8 Td[(.Oncewehavewrittendownthewealthprocessexplicitlywhichservesastheconstraintinouroptimizationproblem,weproceedtorestatethecostfunctionalinthepowercasethatwillbeusedfortheoptimization. J,c(s,w)=Es,wZ1se)]TJ /F16 7.97 Tf 6.59 0 Td[(tct dt=e)]TJ /F16 7.97 Tf 6.59 0 Td[(sEyZ10e)]TJ /F16 7.97 Tf 6.59 0 Td[(tcs+t dt Giventhiswealthprocessandcostfunctionalwewouldliketondoptimalcontrolsc,,and(s,w)suchthat(s,w)=sup,cJ,c(s,w)=J,c Tosolvetheoptimizationproblemweproceedinthesamefashionasthenojumpcase,inparticularwerstwritedowntheHamiltonJacobiBellmanequationusingTheorem 4.9 .TheHamiltonJacobiBellmanequationinthejumpcaseisgivenbyEquation( 5 ),thisisessentiallythesameasthecasewithoutjumpswiththeadditionofanintegralterm.Theintegraltermisindependentoftheconsumptionprocessc,howeverthistermcontainsanexplicitdependenceon.Theexplicitdependenceonthetaaddsalayerofcomplexitynotseeninthecasewithoutjumps,itisthisaddedcomplicationthatmakesthejumpscaseinterestingforourresearch.WeproceedinndingtheoptimalcontrolscandgiventheHamiltonJacobiBellman,oncefoundwewillwritedowntheexplicitvaluefunction(s,w)andusethevericationtheoremtoshowthattheoptimalcontrol 78

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problemissolved.sup,ce)]TJ /F16 7.97 Tf 6.59 0 Td[(sc +s+()]TJ /F6 11.955 Tf 11.96 0 Td[(r)ww+(rw)]TJ /F6 11.955 Tf 11.96 0 Td[(c)w+1 22w22ww (5)+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(s,w(1+z)))]TJ /F3 11.955 Tf 11.95 0 Td[((s,w))]TJ /F3 11.955 Tf 11.95 0 Td[(w(s,w)wz](dz)=0 Theoptimalsolutionfortheconsumptionrateisthesameasthecasewithoutjumpsandisgivenbyc=)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F16 7.97 Tf 6.58 0 Td[(sw1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1 Howeverthetheoptimalcontrolforpresentsmuchmoreofachallengeinthiscase,inparticularwearenotabletondexplicitlyunlesswehavemoreinformationaboutthegivenLevyprocess.ThesolutionforinthejumpcaseisasolutionofthenonlinearEquation( 5 ),wherethenonlinearitycomesfromtheextraintegraltermnotpresentinthecasewithoutjumps.()]TJ /F6 11.955 Tf 11.96 0 Td[(r)ww+2w2ww+@ @Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(s,w(1+z)))]TJ /F3 11.955 Tf 11.95 0 Td[((s,w))]TJ /F3 11.955 Tf 11.95 0 Td[(w(s,w)wz](dz)=0 (5) SincethisterminvolvesanintegralwithrespecttoLevymeasuretheexplicitsolutionrequiresknowingthespecicLevymeasure(dz)tondanexplicitsolutionfor.However,wemaymakeconsiderableprogressinouranalysisbeforechoosingaLevymeasureifweassumethevaluefunctionhasthesameformasbefore.Inparticular,assumingthevaluefunctionhastheform(s,w)=Kjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(swwemaywritetheequationinvolvingas()]TJ /F6 11.955 Tf 11.96 0 Td[(r)wKjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw)]TJ /F10 7.97 Tf 6.59 0 Td[(1+2w2Kjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(s()]TJ /F5 11.955 Tf 11.96 0 Td[(1)w)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (5)+@ @Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1hKjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw(1+z))]TJ /F6 11.955 Tf 11.95 0 Td[(Kjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw)]TJ /F3 11.955 Tf 11.96 0 Td[(wzKjp0e)]TJ /F16 7.97 Tf 6.58 0 Td[(sw)]TJ /F10 7.97 Tf 6.58 0 Td[(1i(dz)=0 79

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WemayfactoroutthecommontermsKjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(swfromthisequationtoget Kjp0e)]TJ /F16 7.97 Tf 6.58 0 Td[(s()]TJ /F6 11.955 Tf 11.96 0 Td[(r)+2()]TJ /F5 11.955 Tf 11.95 0 Td[(1)+@ @Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+z))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(z](dz)w=0 Sinceweareassumingthatw>0andKjp06=0wemaydivideoutthefactorKjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(swtogetasimpliedformofthisequation.Movingthederivativeinsideoftheintegraltheconditionforbecomes ()]TJ /F6 11.955 Tf 11.95 0 Td[(r)+2()]TJ /F5 11.955 Tf 11.96 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1(1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1z)]TJ /F3 11.955 Tf 11.96 0 Td[(z(dz)=0 since2(0,1)wemayfactoroutaanddivideitouttoget ()]TJ /F6 11.955 Tf 11.95 0 Td[(r)+2()]TJ /F5 11.955 Tf 11.95 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1(1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(1z(dz)=0 (5) AlthoughanexplicitsolutiontoEquation( 5 )isnotpossibleuntilwespecifyaLevymeasurewemayderiveaconditionunderwhichasolutionofthisequationexistsbyusingtheMeanvaluetheorem.InfactalthoughndingasolutiontoEquation( 5 )ispossibleoncegivenaLevymeasure,itturnsoutthatmanyoftheseequationswillbeanonlinearinwhichleadstoclosedformsolutionsinonlyasmallnumberofcases.Althoughsomeclosedformsolutionsforwerefoundwewillnotprovidethoseresultshere,insteadweusenumericalalgorithmstondthesolutionstothisequationformanyofthecaseslistedbelow.ToproceedwithndingaconditionthatallowsasolutiontoEquation( 5 )wedenethefunction()=()]TJ /F6 11.955 Tf 11.96 0 Td[(r))]TJ /F3 11.955 Tf 11.96 0 Td[(2(1)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(11)]TJ /F5 11.955 Tf 11.95 0 Td[((1+z))]TJ /F10 7.97 Tf 6.58 0 Td[(1z(dz) Thenwehavethat(0)=)]TJ /F6 11.955 Tf 12.04 0 Td[(r>0,usingthefactthat()iscontinuousinasolution2(0,1]existsif(1)0whichhappensexactlywhen )]TJ /F6 11.955 Tf 11.96 0 Td[(r2(1)]TJ /F3 11.955 Tf 11.96 0 Td[()+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(11)]TJ /F5 11.955 Tf 11.96 0 Td[((1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1z(dz) (5) 80

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SogivenaLevymeasurewemayusethisconditiontocheckfortheexistenceofanoptimalcontrolinthejumpcase,noteif=0wedorecoverintheclassicalcaseasshouldbethecase.WealsoseethatEquation( 5 )doesnotdependondependonthewealthwsothatthereisnoexplicittimedependencejustasintheclassicalcase.Assumingtheoptimalsolutionfor=1wemayexplicitlyndtheconstantKjp0,muchofthisworkhasalreadybeendoneweneedonlymakeminorchangestotheresultsabove.Puttingtheoptimalcontrolsc=(Kjp0)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1wand=1intotheHamiltonJacobiBellmanEquation( 5 )weget1 e)]TJ /F16 7.97 Tf 6.58 0 Td[(s(Kjp0) )]TJ /F12 5.978 Tf 5.75 0 Td[(1w)]TJ /F3 11.955 Tf 11.96 0 Td[(Kjp0e)]TJ /F16 7.97 Tf 6.58 0 Td[(sw+1()]TJ /F6 11.955 Tf 11.96 0 Td[(r)wKjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw)]TJ /F10 7.97 Tf 6.59 0 Td[(1+rwKjp0e)]TJ /F16 7.97 Tf 6.58 0 Td[(sw)]TJ /F10 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 9.3 0 Td[((Kjp0)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1wKjp0e)]TJ /F16 7.97 Tf 6.58 0 Td[(sw)]TJ /F10 7.97 Tf 6.58 0 Td[(1+1 22w221Kjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(s()]TJ /F5 11.955 Tf 11.95 0 Td[(1)w)]TJ /F10 7.97 Tf 6.59 0 Td[(2+Kjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(s()]TJ /F6 11.955 Tf 11.96 0 Td[(r)+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz)w=0 (5) CombiningliketermsandfactoringoutthecommontermKjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(swwehaveKjp0e)]TJ /F16 7.97 Tf 6.59 0 Td[(s(Kjp0)1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1(1)]TJ /F3 11.955 Tf 11.95 0 Td[())]TJ /F3 11.955 Tf 11.96 0 Td[(+1()]TJ /F6 11.955 Tf 11.95 0 Td[(r)+r+1 2221()]TJ /F5 11.955 Tf 11.95 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(1z](dz)w=0 whichgivesthat(Kjp0)1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1(1)]TJ /F3 11.955 Tf 11.95 0 Td[()=)]TJ /F3 11.955 Tf 11.96 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(r(1)]TJ /F3 11.955 Tf 11.95 0 Td[(1))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2221()]TJ /F5 11.955 Tf 11.95 0 Td[(1))]TJ /F15 11.955 Tf 11.29 16.27 Td[(Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(1z](dz) solvingforKjp0wegetKjp0=1 1 1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(r(1)]TJ /F3 11.955 Tf 11.95 0 Td[(1))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2221()]TJ /F5 11.955 Tf 11.96 0 Td[(1))]TJ /F15 11.955 Tf 11.3 16.27 Td[(Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz))]TJ /F10 7.97 Tf 6.59 0 Td[(1 81

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hencewehavethevaluefunctioninthiscaseisgivenby(s,w)=1 1 1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(r)]TJ /F5 11.955 Tf 11.95 0 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(r)1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2221()]TJ /F5 11.955 Tf 11.95 0 Td[(1))]TJ /F15 11.955 Tf 11.29 16.27 Td[(Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz))]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw (5) Asinthenonjumpcaseswehavethattheconstantisbeingraisedtothe)]TJ /F5 11.955 Tf 12.37 0 Td[(1powersothesolutionisonlyvalidiftheterminbracketsofEquation( 5 )ispositive.Forthistermtobepositivewemusthavethat >r+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)1+1 2221()]TJ /F5 11.955 Tf 11.96 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz) (5) BeforewritingthevaluefunctioninamorecompactnotationweperformaquickcheckofconsistencyofEquation( 5 )byletting=0sothattheintegraltermisremovedand1=)]TJ /F9 7.97 Tf 6.59 0 Td[(r 2(1)]TJ /F16 7.97 Tf 6.59 0 Td[()sowemaycombinethetermsinvolving1toget )]TJ /F5 11.955 Tf 11.96 0 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(r)1)]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 2221()]TJ /F5 11.955 Tf 11.96 0 Td[(1)=)]TJ /F5 11.955 Tf 13.8 8.08 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2 22(1)]TJ /F3 11.955 Tf 11.96 0 Td[() sothatthevaluefunctioninthejumpcaseEquation( 5 )isthesameasthenojumpcaseEquation( 3 ).Tosimplifythenotationwemaytakeallthetermsinparenthesesthatinvolve1andthetermrwhichareallconstantins,wandset =r+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)1+1 2221()]TJ /F5 11.955 Tf 11.95 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz) (5) hencethevaluefunctionbecomes (s,w)=1 )]TJ /F3 11.955 Tf 11.96 0 Td[( )]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw (5) Giventhissimpliedformofthevaluefunctionwemaytakethenecessaryderivativewandplugintotheequationforctondthat c=)]TJ /F3 11.955 Tf 11.96 0 Td[( )]TJ /F5 11.955 Tf 11.95 0 Td[(1w 82

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sothattheoptimalcontrolsforthisversionoftheproblemare ct=)]TJ /F3 11.955 Tf 11.96 0 Td[( )]TJ /F5 11.955 Tf 11.96 0 Td[(1Wtt=1 (5) PluggingtheseoptimalcontrolsintoEquation( 3 )wendthatthewealthprocessisaGeometricLevyprocessoftheform dWt Wt=()]TJ /F6 11.955 Tf 11.96 0 Td[(r)1+r)]TJ /F3 11.955 Tf 18.05 8.09 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.95 0 Td[()dt+1dBt+1Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1zeN(dt,dz)W0=w Thisequationhasauniquestrongsolutionjustasinthenonjumpcase,toshowthatthevaluefunctioninEquation( 5 )istooptimalsolutionweneedtoshowtheotherconditionsofthevericationtheoremaresatised.Werstbeginbyshowingthatthecontrolut=)]TJ /F3 11.955 Tf 11.95 0 Td[( )]TJ /F5 11.955 Tf 11.96 0 Td[(1Wt,1 isadmissiblei.e.ut2A[s,1).ThecontrolutismeasurableandFtadapted,thecheckforintegrabilityofthecontrolprocessissimilartopreviouscases Es,wZ1sjjutjj2dt=Es,w"Z1s()]TJ /F3 11.955 Tf 11.96 0 Td[( )]TJ /F5 11.955 Tf 11.96 0 Td[(1Wt2+21)dt#=)]TJ /F3 11.955 Tf 11.95 0 Td[( )]TJ /F5 11.955 Tf 11.95 0 Td[(12Es,wZ1sW2tdt+21Ps,w[s,1)<1 Theintegrabilityconditionforthecostfunctionalissimilartothenonjumpcasesothecompletedetailsareleftout,thesummaryisasfollows Jc,(s,w)1 )]TJ /F3 11.955 Tf 11.95 0 Td[( )]TJ /F5 11.955 Tf 11.96 0 Td[(1Ps,w[s,1)+Es,wZ1sW2tdt<1 (5) Hencealltheconditionsofadmissibilityhavebeenmetinthiscasesothecontrolut2A[s,1),wenowproceedtochecktheremainingconditionsofthevericationtheorem.Wehavethatthefunction(s,w)2C1,2iscontinuous,andsatisestheHamiltonJacobiBellmanequationbyconstructionsoweonlyneedtocheckthegrowth 83

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conditionwhichcanbedoneusingthesameargumentasbefore j(s,w)j=1 )]TJ /F3 11.955 Tf 11.95 0 Td[( )]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F10 7.97 Tf 6.58 0 Td[(1esw1 )]TJ /F3 11.955 Tf 11.96 0 Td[( )]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F10 7.97 Tf 6.58 0 Td[(1jwj1 )]TJ /F3 11.955 Tf 11.95 0 Td[( )]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F10 7.97 Tf 6.59 0 Td[(1(1+jwj2) Theremainingconditionsonfc,(s,w)andc,(s,w)havealreadybeencheckedorcanbecheckedwithsimilararguments,hencethefunction(s,w)satisesthevericationtheoremsothat(s,w)=(s,w).Thismeansthefunction(s,w)inEquation( 5 )isthesolutiontotheoptimizationproblem,thesolutionpresentedisindependentoftheLevymeasure.Weprovidenumericalanalysisofthissolutioninthenumericalresultssectionofthepaper,wewilltrytowritedowntheexplicitvaluefunctionsandprovidethecorrespondingplots. 5.2JumpCasewithLogUtility Wenowconsiderthecasewheretheutilityfunctionisoflogarithmicform,itturnsoutthatthiscaseisjustanextremecaseofapowerutility.However,thelogmodelpresentssomeroadblocksthathinderacompleteanalysisasperformedinthepowerutilitycase.Inparticular,Ihavenotbeenabletosolvethenitehorizoncaseforlogarithmicutility.Italsoturnsoutthatthenumericalresultsarenotasrobustasthepowerutilitycase,hencewewillnotperformasextensiveofananalysis.ThissectionismostlypresentedforcompletenessoftheworkthatIhavedoneuptothispointandthissectionmayserveasaprecursortofutureworkalongtheselines.Webegintheanalysiswiththecostfunctionalandproceedalongthesamelinesasbefore,hencetheexplanationswillbesparse.J,c(s,w)=Es,wZ1se)]TJ /F16 7.97 Tf 6.58 0 Td[(tlogctdt=e)]TJ /F16 7.97 Tf 6.59 0 Td[(sEyZ10e)]TJ /F16 7.97 Tf 6.59 0 Td[(tlogcs+tdt (5) Wewillconsiderthesameportfolioasinthepowerfunctioncaseabove,inparticulartheriskyassetisdrivingbythesameGeometricLevyprocesssothatthewealthoftheportfolioevolvesaccordingtoEquation( 5 ).GiventhenewformoftheutilityfunctiontheHamiltonJacobiBellmanchangesonlyslightly(theonlydifferenceisintherst 84

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term)andnowhasthefollowingform sup,ce)]TJ /F16 7.97 Tf 6.59 0 Td[(slogc+s+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)ww+(rw)]TJ /F6 11.955 Tf 11.96 0 Td[(c)w+1 22w22ww (5) +Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(s,w(1+z)))]TJ /F3 11.955 Tf 11.95 0 Td[((s,w))]TJ /F3 11.955 Tf 11.95 0 Td[(w(s,w)wz](dz)=0 Tosimplifytheanalysislaterwewouldliketoremovethee)]TJ /F16 7.97 Tf 6.58 0 Td[(sbeingmultipliedbythelogcterm,todothiswedeneanauxiliaryfunctionbandsolveit'sHamiltonJacobiBellmanequation.Sinceweareworkingoveraninnitehorizonwemayagainwrite(s,w)=e)]TJ /F16 7.97 Tf 6.58 0 Td[(sb(0,w),thisgivesusanewformoftheHamiltonJacobiBellmanequationintermsofbgivenbysup,clogc)]TJ /F3 11.955 Tf 11.95 0 Td[(b+()]TJ /F6 11.955 Tf 11.95 0 Td[(r)wbw+(rw)]TJ /F6 11.955 Tf 11.96 0 Td[(c)bw+1 22w22bww (5)+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1hb(w(1+z)))]TJ /F15 11.955 Tf 13.07 3.16 Td[(b(w))]TJ /F15 11.955 Tf 13.07 3.16 Td[(bw(w)wzi(dz)=0 sothattherstorderconditionfortheconsumptionleadsto 1 c)]TJ /F15 11.955 Tf 13.08 3.15 Td[(bw=0=)c=1 bw (5) Theseconddifferenceisfoundwhenlookingattheoptimalfractionofwealthintheriskyasset.TheoptimalstillsatisesEquation( 5 )andasbeforetoperformfurtheranalysisontheexistenceofanoptimalcontrolwemustassumetheformofthevaluefunction.Inthiscasethevaluefunctionisassumedtohavetheformb(w)=(Kjl0logw+Kjl1).GiventhisfunctionalformofthevaluefunctionwemaytakeacloserlookatEquation( 5 ),plugginginbweget ()]TJ /F6 11.955 Tf 11.96 0 Td[(r)wKjl01 w+2w2Kjl0)]TJ /F5 11.955 Tf 9.3 0 Td[(1 w2+@ @Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1Kjl0logw(1+z))]TJ /F6 11.955 Tf 11.96 0 Td[(Kjl0logw)]TJ /F3 11.955 Tf 11.96 0 Td[(wzKjl01 w(dz)=0 FactoringoutKjl0andsimplifytheexpressionwendthat Kjl0()]TJ /F6 11.955 Tf 11.95 0 Td[(r))]TJ /F3 11.955 Tf 11.96 0 Td[(2+@ @Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[log(1+z))]TJ /F3 11.955 Tf 11.96 0 Td[(z](dz)=0 85

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Bringingthepartialderivativeintotheintegralweget 1()=:()]TJ /F6 11.955 Tf 11.95 0 Td[(r))]TJ /F3 11.955 Tf 11.95 0 Td[(2)]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(11)]TJ /F5 11.955 Tf 26.63 8.09 Td[(1 1+zz(dz)=0 (5) Asbeforewehavethat1(0)=)]TJ /F6 11.955 Tf 12.29 0 Td[(r>0soif1(1)0usingthecontinuityof1andtheMeanvaluetheoremwehavethata2(0,1]exists,i.e ()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(11)]TJ /F5 11.955 Tf 23.74 8.09 Td[(1 1+zz(dz) (5) Lettheoptimalcontrolfortheportfolioweightbegivenby=2,thenweareabletondtheconstantsKjl0andKjl1explicitly.Usingthefactthatc=w Kjl0and=2wemayplugthesebackintotheHamiltonJacobiBellmanEquation( 5 )logw Kjl0)]TJ /F3 11.955 Tf 11.95 0 Td[((Kjl0logw+Kjl1)+2()]TJ /F6 11.955 Tf 11.96 0 Td[(r)wKjl01 w+rwKjl01 w)]TJ /F6 11.955 Tf 15.91 8.08 Td[(w Kjl0Kjl01 w+1 22w222Kjl0)]TJ /F5 11.955 Tf 9.3 0 Td[(1 w2+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1Kjl0log(w(1+2z)))]TJ /F6 11.955 Tf 11.96 0 Td[(Kjl0logw)]TJ /F3 11.955 Tf 11.96 0 Td[(2wzKjl01 w(dz)=0 Simplifyingtheexpressionwegetthatlogw Kjl0)]TJ /F3 11.955 Tf 11.95 0 Td[(Kjl0logw)]TJ /F3 11.955 Tf 11.95 0 Td[(Kjl1+2()]TJ /F6 11.955 Tf 11.96 0 Td[(r)Kjl0+rKjl0)]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2222Kjl0+Kjl0Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[log(1+2z))]TJ /F3 11.955 Tf 11.96 0 Td[(2z](dz)=0 whichwemayrewriteas(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Kjl0)logw+)]TJ /F5 11.955 Tf 11.29 0 Td[(logKjl0)]TJ /F3 11.955 Tf 11.96 0 Td[(Kjl1+2()]TJ /F6 11.955 Tf 11.96 0 Td[(r)Kjl0+rKjl0)]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2222Kjl0 (5)+Kjl0Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[log(1+2z))]TJ /F3 11.955 Tf 11.95 0 Td[(2z](dz)=0 TheterminparenthesesinEquation( 5 )isconstantinw,sincelogwisindependentoftheconstanttermwemusthavethatthescalarmultipleofthelogarithmtermmustbezero.ThisgivesusthatKjl0=1 ,sincetheconstanttermmustalsobezerotogetazerosumwecanalsosolvefortheconstantKjl1.IsolatingtheconstantKjl1andfactoringouta 86

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1 wend Kjl1=1 2log+2()]TJ /F6 11.955 Tf 11.96 0 Td[(r)+r)]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F5 11.955 Tf 13.16 8.09 Td[(1 2222+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[log(1+2z))]TJ /F3 11.955 Tf 11.95 0 Td[(2z](dz) (5) Hencethevaluefunctionisjustanextensionofthecasewithnojumpsandhastheform(s,w)=e)]TJ /F16 7.97 Tf 6.58 0 Td[(s1 logw+1 2log+2()]TJ /F6 11.955 Tf 11.96 0 Td[(r)+r)]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F5 11.955 Tf 13.16 8.08 Td[(1 2222 (5)+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[log(1+2z))]TJ /F3 11.955 Tf 11.96 0 Td[(2z](dz) Theoptimalconsumptionprocessisthesameasthecasewithnojumpsc=wwhiletheoptimalchoiceforthefractionofwealthoftheriskyassetis=2.Henceforgeneraltimeswehavethattheoptimalcontrolsare ct=Wtt=2 hencethewealthequationisaGeometricBrownianmotionoftheform dWt Wt=[()]TJ /F6 11.955 Tf 11.96 0 Td[(r)2+r)]TJ /F3 11.955 Tf 11.96 0 Td[(]dt+2dBt+2Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1zeN(dt,dz)W0=w (5) Asacheckforconsistencybetweenthejumpandnojumpcasewenotethatif=0and2=()]TJ /F9 7.97 Tf 6.59 0 Td[(r) 2thenwehavethat 2()]TJ /F6 11.955 Tf 11.96 0 Td[(r))]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 2222=()]TJ /F6 11.955 Tf 11.95 0 Td[(r)2 22 sothatthevaluefunctionforthejumpcasereducestothenojumpcaseEquation( 3 ).Thiswillbetheextentofouranalysisinthelogutilityinnitehorizoncase,thereasonforthisisthatusingsameparametersasbeforetheconditionsfortheexistenceofanoptimalarenotsatised.Itturnsoutthatforthecondition(1)0tobesatisedwemustusevaluesof,r,thatarenotfeasibleinpractice.InparticularifweconsiderthePoissoncasewithajumpofsizeoneatz=1thentheequationforEquation( 5 )givesthat()]TJ /F6 11.955 Tf 12.13 0 Td[(r))]TJ /F3 11.955 Tf 12.13 0 Td[(2+1 20.Since()]TJ /F6 11.955 Tf 12.13 0 Td[(r)>0theonlywayforthisequationtoholdisif2)]TJ /F10 7.97 Tf 13.39 4.71 Td[(1 2>0butthisonlyholdswhen>1=p 2i.e.when=2[0,1]. 87

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Ifonewereconsideringamoregeneralproblemoutsideofthenancialsettingwecouldusevaluesofsigmaoutsideofthisrangeandpresenttheresults,howeverwearenotcurrentlyconcernedwiththismoregeneralsettingsoweomitanyresultsinthisdirection. 5.3FiniteHorizonPowerUtilitywithJumps Havingcompletedtheanalysisonthelogutilitycasewemovebacktothecaseofpowerutility,andperformtheanalysisthatisthecentralresultsofthepaper.InparticularwederiveaclosedformsolutiontotheoptimizationproblemoveranitehorizonandshowthatwemayrecoverthesolutiontotheinnitehorizoncaseasT!1.Oncewehavefoundtheclosedformsolutioninthiscasewewillthenmoveontotoamoregeneralcostfunctionalpresentedinthissectionandderivedaclosedformsolutioninthatcaseaswell.Aftergeneralizingtheresultsinthissectionwewillshowthatthenewresultsareconsistentwithwhatwendinthissection.Thecaseofpowerutilitywithjumpsoveranitehorizon,willfollowmanyofthesameproceduresasbeforesoweagainwillbesparsewiththeexplanationsofthegeneralprocedure.Thecostfunctionalisofthefollowingform J,c(s,w)=Es,wZTse)]TJ /F16 7.97 Tf 6.58 0 Td[(tct dt=EyZT)]TJ /F9 7.97 Tf 6.58 0 Td[(s0e)]TJ /F16 7.97 Tf 6.59 0 Td[((s+t)cs+t dt (5) WiththiscostfunctionaltheHamiltonJacobiBellmanequationbecomes sup,ce)]TJ /F16 7.97 Tf 6.58 0 Td[(sc +s+()]TJ /F6 11.955 Tf 11.96 0 Td[(r)ww+(rw)]TJ /F6 11.955 Tf 11.95 0 Td[(c)w+1 22w22ww (5) +Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(s,w(1+z)))]TJ /F3 11.955 Tf 11.96 0 Td[((s,w))]TJ /F3 11.955 Tf 11.96 0 Td[(w(s,w)wz](dz)=0 withboundarycondition(T,w)=0.WenotethattheHamiltonJacobiBellmanequationisthesameastheinnitehorizoncase,howeverthemajordifferencecomesinthefactthatthePDEproblemnowconsistsofaboundaryconditionattheendofthetimehorizon.ThisboundaryconditionwillbeusedtosolvefortheconstanttoarriveataparticularsolutionamongtheinnitelymanysolutionsoftheHamiltonJacobiBellman 88

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equation,beforewearriveatthePDEwemustndtheoptimalcontrolstoremovethesupremum.Theoptimalcontrolsarethesameasbefore,c=(esw)1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1andisasolutionoftheequation ()]TJ /F6 11.955 Tf 11.96 0 Td[(r)ww+2w2ww+@ @Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(s,w(1+z)))]TJ /F3 11.955 Tf 11.96 0 Td[((s,w))]TJ /F3 11.955 Tf 11.96 0 Td[(w(s,w)wz](dz)=0 Sincethecostfunctionalisoveranitehorizonweassumethevaluefunctionisoftheform(s,w)=h(s)w,inparticularwearenotabletofactoroutthedependencecompletelybyremovinge)]TJ /F16 7.97 Tf 6.59 0 Td[(sfromthecostfunctional.ThiswillmakethePDEwearetryingtosolvemorecomplicatedthantheinnitehorizoncase,butanexplicitsolutionisstillpossible.Toproceedinthatdirectionweusethefactthattheoptimalcontrolformustsatisfytheequation h(s)()]TJ /F6 11.955 Tf 11.96 0 Td[(r)+2()]TJ /F5 11.955 Tf 11.95 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1(1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(1z(dz)w=0 henceanoptimalcontrol=1existsif )]TJ /F6 11.955 Tf 11.96 0 Td[(r2(1)]TJ /F3 11.955 Tf 11.96 0 Td[()+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(11)]TJ /F5 11.955 Tf 11.96 0 Td[((1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1z(dz) (5) Giventhisconditionissatisedsothatexistsandthatc=(esw)1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1wemaywritetheHamiltonJacobiBellmanequationasthefollowingintegro-partialdifferentialequation1)]TJ /F3 11.955 Tf 11.96 0 Td[( (es)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1w+s+1()]TJ /F6 11.955 Tf 11.95 0 Td[(r)ww+rww+1 22w221ww+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(s,w(1+1z)))]TJ /F3 11.955 Tf 11.95 0 Td[((s,w))]TJ /F3 11.955 Tf 11.96 0 Td[(w(s,w)1wz](dz)=0 (5) Giventheformofthevaluefunctionwemayndthefunctionh(s)explicitlyasbeforebytakingthenecessaryderivativesofonpluggingintoEquation( 5 )togetthefollowingordinarydifferentialequationforh(s),1)]TJ /F3 11.955 Tf 11.96 0 Td[( (es)1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1(h(s)) )]TJ /F12 5.978 Tf 5.75 0 Td[(1w+1 22w221()]TJ /F5 11.955 Tf 11.95 0 Td[(1)w)]TJ /F10 7.97 Tf 6.59 0 Td[(2h(s)+1()]TJ /F6 11.955 Tf 11.95 0 Td[(r)wh(s)w)]TJ /F10 7.97 Tf 6.59 0 Td[(1 89

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+h0(s)w+rwh(s)w)]TJ /F10 7.97 Tf 6.58 0 Td[(1+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1h(s)w(1+1z))]TJ /F6 11.955 Tf 11.95 0 Td[(h(s)w)]TJ /F3 11.955 Tf 11.95 0 Td[(1wzh(s)w)]TJ /F10 7.97 Tf 6.59 0 Td[(1(dz)=0 Wemaysimplifythisequationbyfactoringoutanddividingbywtoendupwithh0(s)+1()]TJ /F6 11.955 Tf 11.96 0 Td[(r)+r+1 2221()]TJ /F5 11.955 Tf 11.96 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz)h(s)=)]TJ /F5 11.955 Tf 11.96 0 Td[(1 (es)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1(h(s)) )]TJ /F12 5.978 Tf 5.76 0 Td[(1 Itturnsoutthatthisequationhasaparticularlyniceformoncewelookpastallthecomplicationsinvolvedwiththeterm,anditendsupbeinganBernoulliordinarydifferentialequationinh(s)oftheform h0(s)+p(s)h(s)=q(s)h(s) )]TJ /F12 5.978 Tf 5.76 0 Td[(1 (5) wherewehavethat=1()]TJ /F6 11.955 Tf 11.96 0 Td[(r)+r+1 2221()]TJ /F5 11.955 Tf 11.96 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz)=:p(s) andq(s)=()]TJ /F5 11.955 Tf 11.96 0 Td[(1)(es)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 Wenotethatthetermisindependentofthetimeparameterandisinfactconstant,whichmakessolvingtheBernoullidifferentialequationslightlyeasier.TosolveEquation( 5 )weusethegeneralformofthesolutionforBernoullitypeequations.UsingEquation( 3 )wehavethatthesolution h(s)=")]TJ /F5 11.955 Tf 10.5 8.09 Td[(()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 e )]TJ /F12 5.978 Tf 5.75 0 Td[(1s+Ce )]TJ /F12 5.978 Tf 5.76 0 Td[(1s#1)]TJ /F16 7.97 Tf 6.58 0 Td[( where=)]TJ /F16 7.97 Tf 6.58 0 Td[( )]TJ /F10 7.97 Tf 6.59 0 Td[(1.Usingtheboundaryconditionh(T)=0wendthattheconstantisC=()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.76 0 Td[(1Twhichleadstothegeneralsolutionh(s)=")]TJ /F5 11.955 Tf 9.29 0 Td[(()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 e )]TJ /F12 5.978 Tf 5.75 0 Td[(1s+()1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1 e )]TJ /F12 5.978 Tf 5.76 0 Td[(1Te)]TJ /F17 5.978 Tf 7.78 3.86 Td[((T)]TJ /F13 5.978 Tf 5.76 0 Td[(s) )]TJ /F12 5.978 Tf 5.76 0 Td[(1#1)]TJ /F16 7.97 Tf 6.59 0 Td[( 90

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=)]TJ /F10 7.97 Tf 6.59 0 Td[(1 e)]TJ /F16 7.97 Tf 6.59 0 Td[(sh)]TJ /F5 11.955 Tf 9.3 0 Td[(1+e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)i1)]TJ /F16 7.97 Tf 6.59 0 Td[(=1 )]TJ /F3 11.955 Tf 11.95 0 Td[( 1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F16 7.97 Tf 6.58 0 Td[(sh1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F17 5.978 Tf 5.75 0 Td[( )]TJ /F12 5.978 Tf 5.75 0 Td[(1(T)]TJ /F9 7.97 Tf 6.58 0 Td[(s)i1)]TJ /F16 7.97 Tf 6.59 0 Td[( pluggingthefunctionh(s)intotheequationfor(s,w)wendthatthevaluefunctioninthiscaseisgivenby (s,w)=1 )]TJ /F3 11.955 Tf 11.96 0 Td[( 1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sh1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)i1)]TJ /F16 7.97 Tf 6.59 0 Td[(w Wemayrewritethevaluefunctionbycombiningthetermsinparenthesestoget (s,w)=1 24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.95 0 Td[()1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.75 0 Td[(1(T)]TJ /F9 7.97 Tf 6.58 0 Td[(s)35)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F16 7.97 Tf 6.58 0 Td[(sw (5) Thissolutionisonlyvalidifthecondition>issatised,inparticularwemusthavethat >1()]TJ /F6 11.955 Tf 11.96 0 Td[(r)+r+1 2221()]TJ /F5 11.955 Tf 11.96 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz) Aninterestingresulttonotehereistheweendupfactoringoute)]TJ /F16 7.97 Tf 6.58 0 Td[(sanywayasintheinnitehorizoncase,butthisdoesnotremovealltimedependence.Thereisstillatimedependenceintheterm1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)whichactslikeadiscountfactorforthiscasewhencomparedwiththeinnitehorizoncase.TocheckconsistencywiththeinnitehorizoncasewenotethatifweletT!1inEquation( 5 )weshouldrecovertheinnitehorizoncasewithjumpsinEquation( 5 ).Justasinthenonjumpcaseweseethatthisisinfactwhathappens,wemayalsoprovideanumericaljusticationofthisresultbytakinglargervaluesofTandpresentingtheplots.ThisisdoneinFigure 5)]TJ /F5 11.955 Tf 11.96 0 Td[(1 ,weseethatasthevaluesofTgetlargerthemaximum/minimumvaluesof(s,w)approachthevaluesoftheinnitehorizoncase.TheplotsprovidedinFigure 5)]TJ /F5 11.955 Tf 11.95 0 Td[(1 areforthexedLevymeasure(dz)=(1),asthisisthesimplestLevymeasureweconsider.OnecaneasilyperformthesameanalysisusingadifferentLevymeasureas 91

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longastherequiredintegralscanbefound.Giventhat(s,w)existsandiswrittenin Figure5-1. PlotsofthevaluefunctionforvaluesofT=1,2.5,5,7.5,100comparedtotheplotofthevaluefunctionintheinnitehorizoncase. thesimpliedformofEquation( 5 )wendthattheoptimalcontrolsaregivenby ct=24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.95 0 Td[()1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F17 5.978 Tf 5.75 0 Td[( )]TJ /F12 5.978 Tf 5.75 0 Td[(1T35Wtt=3 hencethewealthEquation( 3 )isaGeometricLevyprocessoftheform dWt Wt=24()]TJ /F6 11.955 Tf 11.95 0 Td[(r)1+r)]TJ /F3 11.955 Tf 51.81 8.08 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.76 0 Td[(1T35dt+1dBt+1Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1zeN(dt,dz)W0=w Wemaysimplifytheremaininganalysisthatinvolvesshowingthatthecontrolprocessutisadmissibleforthiscase,specicallywehavealreadyperformedananalysiswhichonlydiffersslightlywiththiscase.InSection 3.4 wehavethattheoptimalconsumptionprocessctonlydiffersfromthiscasebytheconstantbeingsubtractedinthenumerator, 92

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whilethefractionofwealthinbothcasesareconstants.ThismeansthattheanalysisusedtoshowthatthetechnicalconditionsofthevericationtheoreminSection 3.4 canbecarriedovertoshowthatallconditionsaresatisedinthiscase. 5.4GeneralizationofBequestFunctionintheJumpCase Wemaysolvetheproblemfromtheprevioussectioninamoregeneralsetting,byextendingthebequestfunctiontobenonzero.Theproblemwejustsolvedoverthenitehorizonwasforthecasewherethebequestfunctionwaszeroinourassumedcostfunctional.Anonzerobequestfunctiononlychangestheprobleminthesensethattheboundaryconditionforthefunctionh(s)changessothattheconstantCismoregeneral.Let>0beaconstant,wewillnowconsideracostfunctionaloftheform J,c(s,w)=Es,wZTse)]TJ /F16 7.97 Tf 6.59 0 Td[(tct dt+WT=EyZT)]TJ /F9 7.97 Tf 6.58 0 Td[(s0e)]TJ /F16 7.97 Tf 6.59 0 Td[((s+t)cs+t dt+WT)]TJ /F9 7.97 Tf 6.59 0 Td[(s Giventhiscostfunctiontheoptimalcontrolproblemremainsthesameastheprevioussectioninthatthevaluefunctionhastheform(s,w)=h(s)wwherehsatisestheBernoulliequation h0(s)+p(s)h(s)=q(s)h(s) )]TJ /F12 5.978 Tf 5.76 0 Td[(1 (5) wherewehavethat=1()]TJ /F6 11.955 Tf 11.96 0 Td[(r)+r+1 2221()]TJ /F5 11.955 Tf 11.96 0 Td[(1)+Z1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz)=:p(s) andq(s)=()]TJ /F5 11.955 Tf 11.96 0 Td[(1)(es)1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 andwhosegeneralsolutionisgivenby h(s)=")]TJ /F5 11.955 Tf 10.5 8.08 Td[(()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 es )]TJ /F12 5.978 Tf 5.76 0 Td[(1+Ces )]TJ /F12 5.978 Tf 5.76 0 Td[(1#1)]TJ /F16 7.97 Tf 6.59 0 Td[( Themaindifferencebetweenthegeneralcaseisthattheboundaryconditionnowsaysthat(T,w)=w,whichimpliesthath(T)=.Hencewemaybeginournew 93

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analysisofthisproblemformthispoint,themainimpactofthischangewillbethenewvalueoftheconstantC.Wewillndthenewconstantinthiscaseandthenshowthatthegeneralcasereducestocasefromtheprevioussectionwhen=0.Applyingtheboundaryconditionwehavethat =")]TJ /F5 11.955 Tf 10.5 8.09 Td[(()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 e )]TJ /F12 5.978 Tf 5.76 0 Td[(1T+Ce )]TJ /F12 5.978 Tf 5.76 0 Td[(1T#1)]TJ /F16 7.97 Tf 6.58 0 Td[( SolvingfortheconstantCbyperformingtherequiredalgebraleadsto C=e)]TJ /F17 5.978 Tf 9.48 3.7 Td[(T )]TJ /F12 5.978 Tf 5.75 0 Td[(1"()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(+()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 eT )]TJ /F12 5.978 Tf 5.75 0 Td[(1# (5) Pluggingthisintotheequationforh(s)wemayndtheexplicitsolutionwiththismoregeneralizedboundaryconditionh(s)=")]TJ /F5 11.955 Tf 10.5 8.09 Td[(()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 es )]TJ /F12 5.978 Tf 5.75 0 Td[(1+e)]TJ /F17 5.978 Tf 9.47 3.7 Td[(T )]TJ /F12 5.978 Tf 5.75 0 Td[(1"()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(+()1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1 eT )]TJ /F12 5.978 Tf 5.76 0 Td[(1#es )]TJ /F12 5.978 Tf 5.76 0 Td[(1#1)]TJ /F16 7.97 Tf 6.59 0 Td[(="()1 )]TJ /F12 5.978 Tf 5.76 0 Td[(1 )]TJ /F6 11.955 Tf 9.3 0 Td[(es )]TJ /F12 5.978 Tf 5.76 0 Td[(1+e)]TJ /F17 5.978 Tf 7.78 3.86 Td[((T)]TJ /F13 5.978 Tf 5.75 0 Td[(s) )]TJ /F12 5.978 Tf 5.76 0 Td[(1" ()1 )]TJ /F12 5.978 Tf 5.75 0 Td[(1()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(+eT )]TJ /F12 5.978 Tf 5.76 0 Td[(1#!#1)]TJ /F16 7.97 Tf 6.59 0 Td[(=)]TJ /F10 7.97 Tf 6.58 0 Td[(1 h)]TJ /F6 11.955 Tf 9.3 0 Td[(es )]TJ /F12 5.978 Tf 5.76 0 Td[(1+e)]TJ /F17 5.978 Tf 7.79 3.86 Td[((T)]TJ /F13 5.978 Tf 5.76 0 Td[(s) )]TJ /F12 5.978 Tf 5.76 0 Td[(1h()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(+eT )]TJ /F12 5.978 Tf 5.76 0 Td[(1ii1)]TJ /F16 7.97 Tf 6.59 0 Td[(=1 )]TJ /F3 11.955 Tf 11.95 0 Td[( 1)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F10 7.97 Tf 6.59 0 Td[(1hes )]TJ /F12 5.978 Tf 5.76 0 Td[(1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F17 5.978 Tf 7.79 3.86 Td[((T)]TJ /F13 5.978 Tf 5.76 0 Td[(s) )]TJ /F12 5.978 Tf 5.76 0 Td[(1h()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(+eT )]TJ /F12 5.978 Tf 5.76 0 Td[(1ii1)]TJ /F16 7.97 Tf 6.59 0 Td[(=1 )]TJ /F3 11.955 Tf 11.96 0 Td[( 1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sh1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F17 5.978 Tf 7.78 3.86 Td[((T)]TJ /F13 5.978 Tf 5.76 0 Td[(s) )]TJ /F12 5.978 Tf 5.76 0 Td[(1e)]TJ /F17 5.978 Tf 5.76 0 Td[(s )]TJ /F12 5.978 Tf 5.76 0 Td[(1h()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(+eT )]TJ /F12 5.978 Tf 5.75 0 Td[(1ii1)]TJ /F16 7.97 Tf 6.59 0 Td[(=1 )]TJ /F3 11.955 Tf 11.96 0 Td[( 1)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sh1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)h()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(e)]TJ /F17 5.978 Tf 5.75 0 Td[(T )]TJ /F12 5.978 Tf 5.76 0 Td[(1+1ii1)]TJ /F16 7.97 Tf 6.59 0 Td[( Ifwenowcombinethetermsinbracketstoasinglepowerwendthat h(s)=1 24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)h)]TJ /F16 7.97 Tf 6.58 0 Td[( 1)]TJ /F16 7.97 Tf 6.59 0 Td[(()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[(T )]TJ /F12 5.978 Tf 5.76 0 Td[(1+1i35)]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(s 94

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Giventhat>sothath(s)iswelldenedwehavethatthevaluefunctionisgivenby (s,w)=1 24)]TJ /F3 11.955 Tf 11.96 0 Td[( (1)]TJ /F3 11.955 Tf 11.96 0 Td[()1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.76 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)h)]TJ /F16 7.97 Tf 6.59 0 Td[( 1)]TJ /F16 7.97 Tf 6.59 0 Td[(()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[(T )]TJ /F12 5.978 Tf 5.76 0 Td[(1+1i35)]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F16 7.97 Tf 6.59 0 Td[(sw (5) Equation( 5 )isthemostgeneralvaluefunctioninthepowerutilitycasewehavederived,eachofthetwopreviouscasescanberecoveredfromEquation( 5 ).First,letting=0wemayrecoverEquation( 5 )sincethetermh)]TJ /F16 7.97 Tf 6.59 0 Td[( 1)]TJ /F16 7.97 Tf 6.58 0 Td[(()1 1)]TJ /F17 5.978 Tf 5.75 0 Td[(e)]TJ /F17 5.978 Tf 5.75 0 Td[(T )]TJ /F12 5.978 Tf 5.75 0 Td[(1+1i=1,aresultthatprovidesomereassurancetothecorrectnessoftheequations.Furthermore,letting=0andT!1inEquation( 5 )werecoverEquation( 5 )since1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F17 5.978 Tf 5.76 0 Td[( )]TJ /F12 5.978 Tf 5.75 0 Td[(1(T)]TJ /F9 7.97 Tf 6.59 0 Td[(s)h)]TJ /F16 7.97 Tf 6.59 0 Td[( 1)]TJ /F16 7.97 Tf 6.59 0 Td[(()1 1)]TJ /F17 5.978 Tf 5.76 0 Td[(e)]TJ /F17 5.978 Tf 5.75 0 Td[(T )]TJ /F12 5.978 Tf 5.76 0 Td[(1+1i!1,againareassuringresult.FromamorenumericalstandpointwegetconsistencyintheclosedformsolutionpresentedabovebycomparingtheplotsofEquation( 5 )withthecasewhere=0.Inparticularwebeginwithpositivevaluesofandapproachzerofromabove,fromatheoreticalstandpointweshouldhavethatplotswithpositivelambdashouldbecomemoreliketheplotwith=0whenwearecloseenoughtozero.TheresultsoftheplotsofthisprocessispresentedinFigure 5)]TJ /F5 11.955 Tf 11.96 0 Td[(2 (againthisplotisforaLevymeasureoftheform(dz)=(1)forsimplicity)andweseethatthenumericalresultsareconsistentwithwhatwewouldexpecttheoretically.Inparticular,one=1e)]TJ /F5 11.955 Tf 12.3 0 Td[(9thevaluesofthemaximum/minimumarethesameasthecasewhen=0,sothatFigure 5)]TJ /F5 11.955 Tf 11.96 0 Td[(2 modelsthebehavioroftheinnitehorizoncasecorrectly.Thisprovidesyetanothercheckofconsistencytotheclosedformsolutionwehavederivedinthissectionandputsthevalidityoftheisformulaonsolidground.WenotethatthisequationisaproductoftheassumptionofgeometricLevymarketmodel,onemayconsidermorecomplexmodelsfortheriskyassetwhichmayleadtodifferentformulas.TheassumptionofageometricLevymodelconsistentlyleadstoaformulaofthisformwhichhintingthatacompletelygeneralformulamaybepossible,thisisleftforfutureresearch. 95

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Figure5-2. Plotsofthevaluefunctionforvaluesof=1,.5,.25,.125,1e-09comparedtotheplotofthevaluefunctionfor=0 96

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CHAPTER6NUMERICALRESULTSANDCONCLUSION 6.1JumpdiffusionandLevytriplets ThefourdiffusionmodelsweconsiderthroughoutthepaperareallexamplesofthegeneralMertonjumpdiffusionmodelpresentedin[ 19 ].TheyprovidearangeofjumpcharacteristicsthatcanbefoundbyconsideringthecharacteristicfunctionsoftheLevyprocessassociatedwiththejumppartofthethediffusionmodel.Inthissectionwewillpresentthedifferentjumpcharacteristicsofeachofthemodelsandthenprovideexplicitnumericalschemestocomputethevaluesofineachcase.WerstconsiderthePoissonprocesswhichhasexponentiallydistributedjumpsandonlyhasjumpsofsizeone.ToconstructthePoissonprocessletfkg1k=1beindependentexponentialrandomvariableswithparameteri.e.P(ky)=e)]TJ /F16 7.97 Tf 6.59 0 Td[(y LetTn=Pnk=1kthentheprocesst=P1n=11(tTn)iscalledthePoissonprocesswithparameter.ThejumpsofthePoissonprocessoccuratthetimesTnandthewaitingtimesbetweenjumpsisexponentialdistributed.ThepathsofthePoissonprocesstarecadlagandforeacht>0hasPoissondistributionwithintensityti.e.P(t=n)=(t)n n!e)]TJ /F16 7.97 Tf 6.58 0 Td[(tn=0,1,2 AsimplecalculationshowsthatthecharacteristicfunctionofthePoissonprocessisgivenbyE[eiut]=et(eiu)]TJ /F10 7.97 Tf 6.58 0 Td[(1) usingthenotationofTheorem 4.5 theLevyexponentis(u)=(eiu)]TJ /F5 11.955 Tf 11.89 0 Td[(1)whichhasLevytriplet(0,0,(dz)=(1)).ThePoissonprocesscasegreatlysimpliesallnumericalcalculationsinvolving(dz),howeveraprocesswithjumpsizerestrictedtounityisnotarobustmodelofstockprices.Hence,wewouldliketogeneralizethisideatoallowforprocesseswithvaryingjumpsizes,inparticularwewanthaveadistributionofjump 97

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sizes.ThegeneralizationofthePoissonprocessisgivenbythecompoundPoissonprocessdenedby Denition6.1. Let(Zn)n2Nbeasequenceofi.i.drandomvariablestakingvaluesinRwithcommondistributionZ1=ZandtaPoissonprocessofintensityindependentofeachZnthethecompoundPoissonprocess(Yt)t0isdenedbyYt=tXi=1Zi ThecompoundPoissonprocessretainstheexponentialwaitingtimesbetweenjumpshoweverwenowhavethatthejumpsizeshavethedistributionZ.ThecharacteristicfunctionofthecompoundPoissonprocesshasLevyexponent(u)=ZR(eiuz)]TJ /F5 11.955 Tf 11.95 0 Td[(1)(dz) hencetheLevytripletis(0,0,(dz)).ThecompoundPoissonprocessnowallowsforjumpsofarbitrarysizechoosefromagivendistribution,wenotethatwhen(dz)=(1)wedoinfactrecoverthePoissonprocess.ItisalsothebasisoftheMertonjumpdiffusionmodelthatweareusingtomodeltheriskyasset,it'sgeneralformisXt=t+Bt+ZRzeN(t,dz)=t+Bt+Yt sinceRRzeN(t,dz)hasacompoundPoissondistribution.ThecharacteristicfunctionforXthasLevyexponent(u)=iu)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 22u2+ZR(eiuz)]TJ /F5 11.955 Tf 11.96 0 Td[(1)(dz) andLevytripletgivenby(,2,(dz)).IfXtisacompoundPoissonprocessthenwehavethatthereareanitenumberofjumpsoneachtimeintervalwhichgivesthat(R)<1andwemayshow(dz)=f(z)dzforsomedensityfunctionofjumpsizes.However,theLevymeasuredoesnotnecessarilyneedtohaveanitenumberofjumpsoneachtimeinterval,asalegitimateLevymeasureneedonlysatisfythe 98

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weakerconditionRR(z2^1)(dz)<1.Inparticularwewillconsiderthemeasure(dz)=z)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(z1(z>0)dzwhichhasinniteactivity.ALevyprocesswithinniteactivitystillprovidesforatractablemodel,sincemanyofthejumpsizesaresmallandonlyanitenumberhaveabsolutevaluelargerthananygivenpositivenumber. 6.2OptimalforPowerUtility Wewouldliketoprovidesomenumericalresultsforthecaseswesolvedabove.InparticularwewouldliketocomparethevalueoftheoptimalportfolioweightsfordifferentLevymeasures.GeneralLevyprocessesmayhaveacomplicatedLevymeasureassociatedwhichmakesnddifcultifnotimpossible.HoweverifwerestrictourstudytothecasewheretheLevymeasureisabsolutelycontinuouswithrespecttoLebesguemeasureweareabletoprovidetangibleresults.Throughoutthissectionwewillrestrictourattentiontothecaseofthepowerutility,sowewouldliketosolveforinthefollowingtheequation ()]TJ /F6 11.955 Tf 11.96 0 Td[(r))]TJ /F3 11.955 Tf 11.95 0 Td[(2(1)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F15 11.955 Tf 11.95 16.28 Td[(ZR[1)]TJ /F5 11.955 Tf 11.96 0 Td[((1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1]z(dz)=0 (6) Thisequationisnonlinearinandthedifcultycomesfromtheintegralterm,theoreticallyweareabletosolvethisequationforanyLevymeasure.WerecallaLevymeasureisanynonnegativemeasure(dz)onRsatisfying(f0g)=0andZR(z2^1)(dz)<1 IwillpresentafewLevymeasureswhichallowustorewriteEquation( 6 )intermsofknownstatisticaldistributionssothatthenumericalworkisgreatlysimplied.TheGamma(a,b)processisastochasticprocess()]TJ /F9 7.97 Tf 11.66 -1.79 Td[(t)t0suchthatforeachxedttherandomvariable)]TJ /F9 7.97 Tf 6.78 -1.79 Td[(thasaGamma(a,b)distributionwithdensityf)]TJ /F13 5.978 Tf 4.82 -1 Td[(t(z)=bat \(at)zat)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.58 0 Td[(bz1(z>0) 99

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TheGamma(a,b)processisapurejumpnondecreasing(a.s.)LevyprocesswithLevymeasure(dz)=az)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(bz1(z>0)wherea,b>0.Thismeansthatjumpswhosesizeareintheinterval[z,z+z]occurasaPoissonprocesswithintensity(dz),wheretheparameteracontrolstherateofjumparrivalsandbcontrolsthejumpsizes.GiventhisLevymeasurewemaysimplifyEquation( 6 )sothatwemaycarryoutthenumericalalgorithmtondtheoptimal. 6.2.1\(1,1)Process Weconsiderthecaseofthe\(1,1)LevyprocesswhoseLevymeasureisgivenby(dz)=z)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.58 0 Td[(z1(z>0) ThisisinfactaLevymeasuresinceitsatisestheconverseoftheLevy-KhintchinetheoremfortheexistenceofaLevyprocess,weneedonlyshowthatZRmin(1,z2)(dz)=Z10min(1,z2)z)]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz=Z10ze)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz+Z11z)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.58 0 Td[(zdz<1 (6) TherstintegralinEquation( 6 )isnitebyjustasimpleintegrationbyparts,thesecondintegralisaknowintegralcalledtheexponentialintegralwhosevalueisgivenbyEi(1)=1.89511781...ThenitenessofthisquantityguaranteestheexistenceofaLevyprocesscorrespondingtothisLevymeasure,thatLevyprocessisthe\(1,1).WiththisLevymeasuretheintegralequationforbecomes ZR1)]TJ /F5 11.955 Tf 11.96 0 Td[((1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1z(dz)=Z101)]TJ /F5 11.955 Tf 11.95 0 Td[((1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz=Z10e)]TJ /F9 7.97 Tf 6.58 0 Td[(zdz)]TJ /F15 11.955 Tf 11.96 16.28 Td[(Z10(1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz=1)]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z10(1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz Wemaynowrewritetheremainingintegralusingthesubstitutionu=(1+z)toget Z10(1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz=e1= Z11u)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.58 0 Td[(u=du 100

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=e1= Z10u)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.58 0 Td[(u=du)]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z10u)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.58 0 Td[(u=du=e1= \())]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z10u)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(u=du=e1= "\())]TJ /F3 11.955 Tf 11.95 0 Td[(Z1=0t)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(tdt#(t=u=)=e1= [\())]TJ /F3 11.955 Tf 11.96 0 Td[(\()P(1=,)]=)]TJ /F10 7.97 Tf 6.58 0 Td[(1e1=\()[1)]TJ /F6 11.955 Tf 11.95 0 Td[(P(1=,)] wherewehaveusedapredenedstatisticalfunctionfromtheprogramRtorewritetheintegral,thefunctioniscalledthepgammafunctionwhichisdenedbyP(a,x)=1=\(a)Zx1ta)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.58 0 Td[(tdt CombiningtheseresultswehavethatEquation( 6 )becomes()]TJ /F6 11.955 Tf 11.95 0 Td[(r))]TJ /F3 11.955 Tf 11.95 0 Td[(2(1)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F5 11.955 Tf 11.95 0 Td[(1+)]TJ /F10 7.97 Tf 6.59 0 Td[(1e1=\()[1)]TJ /F6 11.955 Tf 11.96 0 Td[(P(1=,)]=0 Thisequationishighlynonlinearandcannotbesolvedexplicitly,howeverwritteninthisformwemayeasilyimplementanumericalalgorithmusingtheprogramR. 6.2.2CompoundPoissonProcesswithExponentialDensity WewillconsiderthespecialcasewheneachZnisexponentiallydistributed,inparticularweconsidertheexp(1)distributionwhosedensityfunctionisgivenbyfZ(z)=e)]TJ /F9 7.97 Tf 6.58 0 Td[(z1(z>0) InthiscasetheLevymeasureforthecompoundPoissonprocessYtisgivenby(dz)=fZ(z)dz=e)]TJ /F9 7.97 Tf 6.58 0 Td[(z1(z>0)dz 101

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GiventheLevymeasurewemayproceedinasimilarfashionasbeforetocalculatetheintegralterminEquation( 6 )ZR[1)]TJ /F5 11.955 Tf 11.95 0 Td[((1+z)]z(dz)=Z10ze)]TJ /F9 7.97 Tf 6.58 0 Td[(zdz)]TJ /F15 11.955 Tf 11.96 16.28 Td[(Z10(1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1ze)]TJ /F9 7.97 Tf 6.58 0 Td[(zdz=1)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z10(1+z))]TJ /F10 7.97 Tf 6.58 0 Td[(1ze)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz Beforemovingontotheevaluationoftheintegralwewriteageneralformulaforthetypeofintegralfromthepreviousexamplesincewewillencounterittwicemoreinthisexample.Leta,b>0beconstants,then Z11ua)]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(u=bdu=Z10ua)]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(u=bdu)]TJ /F15 11.955 Tf 11.96 16.28 Td[(Z10ua)]TJ /F10 7.97 Tf 6.58 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(u=bdu=ba\(a)[1)]TJ /F6 11.955 Tf 11.95 0 Td[(P(1=b,a)]=:DP(1=b,a) (6) Wemayagainrewritethisintegralinaformthatmakesiteasiertoevaluatenumerically,todothisweagainmakethesubstitutionu=1+zZ10(1+z))]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(zzdz=e1= 2Z11[u)]TJ /F6 11.955 Tf 11.96 0 Td[(u)]TJ /F10 7.97 Tf 6.59 0 Td[(1]e)]TJ /F9 7.97 Tf 6.59 0 Td[(u=du=e1= 2Z11ue)]TJ /F9 7.97 Tf 6.58 0 Td[(u=du)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z11u)]TJ /F10 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(u=du=e1= 2+1\(+1)(1)]TJ /F6 11.955 Tf 11.96 0 Td[(P(1=,+1)))]TJ /F3 11.955 Tf 11.96 0 Td[(\()(1)]TJ /F6 11.955 Tf 11.96 0 Td[(P(1=,))=)]TJ /F10 7.97 Tf 6.59 0 Td[(2e1=[\()DP(1=,+1))]TJ /F5 11.955 Tf 11.96 0 Td[(\()DP(1=,)] Usingthissimpliedformoftheintegralwemaywritetheequationfortheoptimalcontrolwherewewillassume=1forexplicitcomputation()]TJ /F6 11.955 Tf 11.95 0 Td[(r))]TJ /F3 11.955 Tf 11.95 0 Td[(2(1)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F5 11.955 Tf 11.95 0 Td[(1+)]TJ /F10 7.97 Tf 6.59 0 Td[(2e1=\()[DP(1=,+1))]TJ /F6 11.955 Tf 11.95 0 Td[(DP(1=,)]=0 OncewehaveformulatedtheeachofthetheLevyintegralsthatallowsustorunnumericalsimulationswecomparethevaluesofwhilevaryingdifferentparametervaluesandpresenttheresultsbelow.Theanalysisweperforminthissectionwillbeforthecaseovertheinnitehorizonbutcouldbeperformedintheothercasesaswell.We 102

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usethiscasesothatweareonlyvaryingthecontrolandnotthevaluesoforT.WepresenttheresultsofthisanalysisinTables 6)]TJ /F5 11.955 Tf 11.96 0 Td[(1 and 6)]TJ /F5 11.955 Tf 11.96 0 Td[(2 .EachtablecontainstheclassicalnojumpcaseandfourcaseswithjumpswheretheLevyprocessesareacompoundPoissonprocess,a\(1,1),aPoissonprocesswithonejump,andaPoissonprocesswithtwojumps.Table 6)]TJ /F5 11.955 Tf 11.95 0 Td[(1 presentthevaluesoftheoptimalaswevarythevolatilityparameter.Therearefewresultsfromthetablewhichprovideacheckofconsistencyofouranalysis,therstisthatforagivenprocessanincreasingvolatilityleadstoadecrease.Thisresultisconsistentwithwhatwewouldexpect,asthevolatilityincreasetheriskyassetinvolvesmoreuncertaintyhenceweshouldreducethefractionofwealthinvestedintheriskyassetandincreasethefractioninthesafeasset.Thesecondresultisthatthefractionofwealthintheriskyassetinalwayslargerinthenojumpcasewhencomparedtoanyofthevaluesinthejumpcases.Thisisconsistentwithourintuitionasthejumpsleadtogreateruncertaintyinthedynamicsoftheriskyasset,thisuncertaintyleadstotheinvestorchoosingasmallerfractionoftheriskyassetinhisconstructionofanoptimalportfolio.Finallywenotethatforxedthevaluesofdecreaseacrosseachrowofthetable,thisisalittlelessintuitiveanddeservessomefurtherexplanation.Webeginwiththetwocolumnsthatintuitivelymakesense,goingfromthePoissoncasewithonejumptothecasewithtwojumpsweexpectthevalueoftodecrease.ThisintuitivelymakessenseasweareaddinganextrajumpofthesamesizetotheLevyprocesswhichleadstogreateruncertainty.Furthermore,wewouldexpectthatgoingfromthe\(1,1)processtothePoissoncasesthatsincethe\(1,1)casehasalargernumberofexpectedjumpsperintervalthanthePoissoncasesthevalueofshouldbelargerinthatcase.Thisintuitionisveriedfromthetablewhichshowsthatourmodelviewsthe\(1,1)caseasmoreriskythanthePoissoncasesduetothelowerfractionofwealthintheriskyassetforthe\(1,1)casewhencomparedtothePoissoncases.Thereasoningbehindthisbehavioristhatthe\(1,1)containsalargernumberofexpectedjumpsoneachintervalhencethis 103

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leadstoagreaterdealofuncertaintyinthetrajectoryofthepriceprocess.ThePoissonprocesscontainsonlyonejumpofknownsizesothevariationinthetrajectoryofthepriceprocessismuchsmaller,hencetheinvestoriswillingtoputalargerfractionoftheriskyassetinthePoissonprocesswhencomparedtothe\(1,1)process.Finally,wenotethatthecompoundPoissonprocessofdensity(dz)=e)]TJ /F9 7.97 Tf 6.58 0 Td[(z1(z>0)istheleastriskyoftheLevyprocesseswithjumpsinourcomparison.AnexplanationofthisbehaviorcouldbethatthisLevyprocessalwayshasjumpssizelessthanoneonz>0sincee)]TJ /F9 7.97 Tf 6.59 0 Td[(z<1onthisinterval.ThismeansthattheLevyprocessassociatedwiththisLevymeasureiscontainsmanyjumpsofsmallsizeoneachtimeinterval,thisbehaviormorecloselymodelsacontinuouspriceprocessthananyoftheotherLevyprocessesunderconsideration.SincethisisclosettoacontinuousLevyprocesswewouldexpectthatthefractionofwealthintheriskyassetwouldbeclosesttothenojumpcase,thisexpectationisveriedineachofthetablesbelow.Asanothercheckofconsistencyof Table6-1. Tableforasafunctionof,where=.16,r=.06,=.5 NoJumpse)]TJ /F21 6.974 Tf 6.22 0 Td[(z1(z>0)Poisson1Poisson2\(1,1) 0.50.800000.190620.176260.143790.1057460.510.768940.188510.174560.142670.1050900.520.739640.186420.172860.141560.1044300.530.712000.184340.171160.140440.1037670.540.685870.182270.169470.139310.1031020.550.661160.180220.167780.138190.1024330.560.637760.178180.166100.137070.1017620.570.615570.176150.164420.135940.1010900.580.594530.174140.162750.134820.1004150.590.574550.172150.161090.133690.0997390.60.555560.170170.159440.132570.099062 Tables 6)]TJ /F5 11.955 Tf 11.96 0 Td[(1 and 6)]TJ /F5 11.955 Tf 11.96 0 Td[(2 wenotethattherstrowofTable 6)]TJ /F5 11.955 Tf 11.95 0 Td[(1 hasthesameparametersasthelastrowofTable 6)]TJ /F5 11.955 Tf 11.96 0 Td[(2 hencethevaluesshouldbethesame,aquickglanceatthesetworowsshowsthatthereissomeconsistencywiththealgorithmsusedtogeneratethesetables. 6.3ExplicitValueFunctions OncewehavefoundtheoptimalvaluesofthecontroltwewritedowntheexplicitformofthevaluefunctionfortheinnitehorizoncaseusingEquation( 5 ).Toproceed 104

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Table6-2. Tableforasafunctionof,where=.16,r=.06,=.5 NoJumpse)]TJ /F21 6.974 Tf 6.22 0 Td[(z1(z>0)Poisson1Poisson2\(1,1) 0.40.666670.156170.145340.118830.0865450.410.677970.159050.147940.120930.0881480.420.689660.162030.150630.123110.0898100.430.701750.165130.153420.125360.0915360.440.714290.168350.156310.127700.0933290.450.727270.171700.159320.130130.0951940.460.740740.175180.162440.132650.0971340.470.754720.178800.165690.135270.0991540.480.769230.182580.169070.137990.1012590.490.784310.186510.172590.140830.1034540.50.800000.190620.176260.143790.105746 ndingtheexplicitvaluefunctionswewillneedtocomputetheconstantwhichinvolvescomputingtheintegralZ1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz) (6) where1istheoptimalcontrolfoundinthesectionabove.ThisintegralinthecaseofthePoissonpointmeasuresisnotdifcultandhasbeenusedthroughoutthepaper,butwewillpresentthemforcompleteness.Thecaseofasinglejumpatz=1hasLevymeasure(dz)=(1)sotheintegralinEquation( 6 )becomesZ1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz)=(1+1))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1 whichgivesrisetoanexplicitvaluefunction.ThecaseofthePoissonprocesswithtwojumpshasLevymeasure(dz)=(.5)+(1)whichgivestheintegralZ1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz)=(1+1)+(1+.51))]TJ /F5 11.955 Tf 11.96 0 Td[(2)]TJ /F5 11.955 Tf 11.96 0 Td[(1.51 whichagaingivesrisetoasimpleformforthevaluefunction.ThemoreinterestingcasesarisewiththeothertwoLevymeasures,forexamplefortheLevymeasure(dz)=e)]TJ /F9 7.97 Tf 6.59 0 Td[(z1(z>0)theintegralbecomesZ1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz)=Z10[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z]e)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz 105

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Toevaluatethisintegralbybreakingintothreeintegral,wherethesecondtwointegralareeasilycomputedusingthefactthatR10zne)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz=n!foralln2N.Therstintegralwemaycomputeusingtheanalysisintheprevioussectionforintegralinvolvingthegammadistribution,bymakingthesubstitutionu=1+1z.Z10(1+1z)e)]TJ /F9 7.97 Tf 6.59 0 Td[(zdz=e1=1 1DP(1,+1) HenceintegralinEquation( 6 )becomesZ1)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(1z](dz)=e1=1 1DP(1,+1))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(1 ThenalLevymeasureweconsideristhatofthe\(1,1)process,werecallthisLevymeasureisgivenby(dz)=e)]TJ /F13 5.978 Tf 5.75 0 Td[(z z1(z>0),soweneedtocomputetheintegralZ1)]TJ /F10 7.97 Tf 6.58 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(1z](dz)=Z10z)]TJ /F10 7.97 Tf 6.59 0 Td[(1[(1+1z))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(1z]e)]TJ /F9 7.97 Tf 6.58 0 Td[(zdz Thisintegralpresentsafewchallengesintermsofndingasimplenumericalmethodologyatthistime,howeverfurtherworkinthisdirectionmayprovideaworkablesolution.Inparticulartheremaybeavailablenumericalalgorithmsthatallowforefcientcomputation,howeverthatisnotthemainfocusofthepapersothisisleftforfuturework.ThediscretetimeLevymeasuresallowforanadequateanalysisofthemethodologiespresentedinthispaperandcanbeusedtochecktheconsistencyofthevaluefunctions. 6.4Conclusion WendthattheinvestmentandconsumptioncontrolprobleminthecaseofhyperbolicabsoluteriskaversionpowerutilityU(c)=c withstockpricesdrivenbyLevyprocesseswithjumpsadmitsaclosedformsolution.TheclosedformsolutionisfoundinboththeniteandinnitehorizoncasesandcomparedwiththeMertonsolutionwithoutjumps.TheformulasfortheoptimalportfolioweightsandthethevaluefunctiondependontheLevyprocessusedtomodelthepriceoftheriskyassetintheportfolio,inparticulartheydependontheLevydensityoftheassociatedLevymeasure.Given 106

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fourdifferentLevymeasureswewereabletonumericallycomputethevaluesofandsothatacomparisoncouldbemadetothecasewithoutjumps.WealsoconsiderthedifferencesbetweenthefourdifferentLevymeasuresandshowthatresultsareconsistentwithexpectations.ThenumericalresultsshowedthattheoptimalweightislargerinthecasewithoutjumpswhencomparedtoanyoftheLevyjumpcases,thisbehaviormodelsaninvestorsriskpreferencewhenthereisgreateruncertaintyinvolvedintheunderlyingpriceprocess.Theinvestoriswillingtoputalargerfractionoftheportfoliointheriskyassetwhentherearenojumpspresent,andthisfractionisreducedoncejumpsarefactoredintothemodel.Asthethesizeandfrequencyofthejumpsincreasethefractionofwealthintheriskyassetdecreases,thisinverserelationshipiswhatonewouldexpectinpractice.ToquantifythisresultwepresenttablescomparingthedifferencesinbetweeneachoftheLevyprocesses,theresultsshowthatanincreaseinthefrequencyofjumpsleadstodecreaseinthefractionofwealthaninvestoriswillingtohaveintheriskyasset.Anotherinterestingresultinouranalysiswasthatthejumpdiffusionmodelswerelesssensitivethanthenojumpmodelstochangesintheparametervalues.Inparticularthecasewithoutjumpswouldfrequentlygivevaluesofoutsideoftheregion[0,1]whilethejumpdiffusionmodelconsistentlyprovidedvalueswithinthisrange.Avalueofin[0,1]isdesirableforthisparticularproblemasthisvaluerepresentafractionoftotalwealthoftheportfolio,andthejumpdiffusionmodelsseemtocapturethisfeaturebetterthanthenonjumpmodel.Acheckofconsistencyisprovidedforeachoftheclosedformsolutionsofthatwepresent.Therstcheckofconsistencyisthecomparisonoftheformulaforinthenitecasewithoutjumps,wecomparethenitehorizoncaseforvaluesofincreasingTandshowthatthegraphsofforthenitecaseapproachtheinnitehorizoncaseasTgetslarger.Thisanalysisisalsodoneinthecasewithjumpsandprovideaconsistencycheckoftheformulaforthenitehorizoncase.Thereisalsoacheckofthenalformulapresentedinwhichthecostfunctiondependsontheparameter.Thecasefor=0issolvedinaprevioussection 107

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andthisisthencomparedwithdifferentcasesfor>0butapproachingzerofromtheright.Theresultsshowthatthatforsmallenoughthevaluesofbecomeidenticalandthemodelsprovidethesamenumericalvalues.Inconclusion,itseemsthattheremaybesomevaliditytousingajumpdiffusionmodeltosolvethisparticularoptimalportfoliochoiceproblem.Thepresentationofaclosedformsolutionavoidsmanyoftheaddedtechnicaldifcultiesthatmaytypicallyarisefromusingjumpdiffusionmodels.Henceitseemsthattheaddedbenetofusingajumpdiffusiontobettermodelvolatilityskewandtheheavytailofassetsreturnsmaybeworththeextraeffortforthesolutionofthisparticularproblem.FutureresearchinthisdirectioncouldinvolveusingmorecomplicatedLevyprocessesandhenceLevymeasuretocomputethevaluesof,thiscouldentailsolvingtheprobleminmorethanonedimension.Onemayalsobeabletondageneralclosedformsolutionforthenonlinearequationforforthepowerutilitycase. 108

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REFERENCES [1] D.Applebaum,Levyprocessesandstochasticcalculus,volume116,CambridgeUniversityPress,Cambridge,secondedition,2009. [2] S.Asmussen,P.Glynn,Stochasticsimulation:Algorithmsandanalysis,SpringerVerlag,2007. [3] R.Bellman,R.Kalaba,Dynamicprogrammingandmoderncontroltheory,AcademicPressNewYork,1965. [4] J.Bertoin,Levyprocesses,volume121,CambridgeUniversityPress,Cambridge,1996. [5] P.Billingsley,Probabilityandmeasure,Wiley-India,2008. [6] T.Chan,PricingcontingentclaimsonstocksdrivenbyLevyprocesses,AnnalsofAppliedProbability9(1999)504. [7] G.Chen,G.Chen,S.Hsu,Linearstochasticcontrolsystems,CRC,1995. [8] R.Cont,P.Tankov,Financialmodellingwithjumpprocesses,Chapman&Hall/CRC,BocaRaton,FL,2004. [9] J.Cox,C.Huang,Optimalconsumptionandportfoliopolicieswhenassetpricesfollowadiffusionprocess,Journalofeconomictheory49(1989)33. [10] M.H.A.Davis,A.R.Norman,Portfolioselectionwithtransactioncosts,Math.Oper.Res.15(1990)676. [11] W.H.Fleming,T.Pang,Anapplicationofstochasticcontroltheorytonancialeconomics,SIAMJ.ControlOptim.43(2004)502. [12] N.Framstad,B.Oksendal,A.Sulem,Optimalconsumptionandportfolioinajumpdiffusionmarketwithproportionaltransactioncosts,JournalofMathematicalEconomics35(2001)233. [13] P.Glasserman,MonteCarlomethodsinnancialengineering,SpringerVerlag,2004. [14] D.Kannan,V.Lakshmikantham,Handbookofstochasticanalysisandapplications,CRC,2002. [15] I.Karatzas,S.Shreve,Brownianmotionandstochasticcalculus,Springer,1991. [16] F.Longstaff,Optimalportfoliochoiceandthevaluationofilliquidsecurities,Reviewofnancialstudies14(2001)407. [17] H.Markowitz,Portfolioselection,JournalofFinance7(1952)77. 109

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[18] R.C.Merton,Optimumconsumptionandportfoliorulesinacontinuous-timemodel,J.Econom.Theory3(1971)373. [19] R.C.Merton,Optionpricingwhenunderlyingstockreturnsarediscontinuous,Journalofnancialeconomics3(1976)125. [20] B.Oksendal,Stochasticdifferentialequations,Springer-Verlag,Berlin,sixthedition,2003.Anintroductionwithapplications. [21] B.Oksendal,A.Sulem,Appliedstochasticcontrolofjumpdiffusions,Springer,Berlin,secondedition,2007. [22] P.E.Protter,Stochasticintegrationanddifferentialequations,volume21,Springer-Verlag,Berlin,secondedition,2004.StochasticModellingandAppliedProbability. [23] L.Rogers,D.Williams,Diffusions,Markovprocesses,andmartingales,Cambridgeuniversitypress,2000. [24] P.Samuelson,Lifetimeportfolioselectionbydynamicstochasticprogramming,TheReviewofEconomicsandStatistics51(1969)239. [25] K.i.Sato,Levyprocessesandinnitelydivisibledistributions,volume68,CambridgeUniversityPress,Cambridge,1999.Translatedfromthe1990Japaneseoriginal,Revisedbytheauthor. [26] W.Schoutens,Levyprocessesinnance:pricingnancialderivatives,Wiley,2003. [27] E.Schwartz,C.Tebaldi,Illiquidassetsandoptimalportfoliochoice,NBERWorkingpaper(2007). [28] C.Tapiero,Appliedstochasticmodelsandcontrolfornanceandinsurance,KluwerAcademicPub,1998. [29] J.Yong,X.Zhou,Stochasticcontrols:HamiltoniansystemsandHJBequations,SpringerVerlag,1999. 110

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BIOGRAPHICALSKETCH RyanSankarpersadwasbornintherepublicofTrinidadandTobagoin1981.HemovedtoFloridaintheearly90'sandhasresidedthereeversince.HegraduatedtheUniversityofFloridawithaBachelorofScienceinmathematicsandaBachelorofArtsinphysicsin2005.HethenbegangraduateworkattheUniversityofFloridaintheFallof2005andreceivedhisMastersofScienceinmathematicsin2007.Hecompletedhisdoctorateofphilosophyintheareaofmathematicalnanceinthespringof2011. 111