Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications: BSDEs with Jumps PDF
Preview Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications: BSDEs with Jumps
EAA Series Łukasz Delong Backward Stochastic Diff erential Equations with Jumps and Their Actuarial and Financial Applications BSDEs with Jumps EAA Series Editors-in-chief HansjoergAlbrecher UniversityofLausanne,Lausanne,Switzerland UlrichOrbanz UniversitySalzburg,Salzburg,Austria Editors MichaelKoller ETHZurich,Zurich,Switzerland ErmannoPitacco UniversitàdiTrieste,Trieste,Italy ChristianHipp UniversitätKarlsruhe,Karlsruhe,Germany AntoonPelsser MaastrichtUniversity,Maastricht,TheNetherlands AlexanderJ.McNeil Heriot-WattUniversity,Edinburgh,UK EAAseriesissuccessoroftheEAALectureNotesandsupportedbytheEuropean Actuarial Academy (EAA GmbH), founded on the 29 August, 2005 in Cologne (Germany)bytheActuarialAssociationsofAustria,Germany,theNetherlandsand Switzerland.EAAoffersactuarialeducationincludingexamination,permanented- ucationforcertifiedactuariesandconsultingonactuarialeducation. actuarial-academy.com Forfurthertitlespublishedinthisseries,pleasegoto http://www.springer.com/series/7879 Łukasz Delong Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications BSDEs with Jumps ŁukaszDelong InstituteofEconometrics,Division ofProbabilisticMethods WarsawSchoolofEconomics Warsaw,Poland ISSN1869-6929 ISSN1869-6937(electronic) EAASeries ISBN978-1-4471-5330-6 ISBN978-1-4471-5331-3(eBook) DOI10.1007/978-1-4471-5331-3 SpringerLondonHeidelbergNewYorkDordrecht LibraryofCongressControlNumber:2013942002 MathematicsSubjectClassification: 60G51,60G57,60H05,60H30 ©Springer-VerlagLondon2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) The fartherbackward youcanlook,the fartherforward youare likelytosee. Winston Churchill Preface AlinearbackwardstochasticdifferentialequationwasintroducedbyBismut(1973) inanattempttosolveanoptimalstochasticcontrolproblembythemaximumprin- ciple. The general theory of nonlinear backward stochastic differential equations with Lipschitz generators was first presented by Pardoux and Peng (1990). Since then,BSDEshavebeenthoroughlystudiedandfoundnumerousapplications.Back- wardstochasticdifferentialequationscanbeusedtosolvestochasticoptimalcontrol problems, establish probabilistic representations of solutions to partial differential equationsanddefinenonlinearexpectations.Sincemanyfinancialproblemscanbe relatedtostochasticoptimizationproblemsandnonlinearexpectations,itisnotsur- prising that BSDEs have become a very important tool in financial mathematics. Nowadays,backwardstochasticdifferentialequationsareanactivefieldofresearch whichisstimulatedbynewfinancialandactuarialapplications. The first motivation for this book is to provide a self-contained overview of the theory of backward stochastic differential equations with jumps and their ap- plications to insurance and finance. Two classical books on BSDEs: “Backward StochasticDifferentialEquations”byElKarouiandMazliak(1997)and“Forward- Backward Stochastic Differential Equations and Their Applications” by Ma and Yong (2000) target theory-oriented readers and miss some important applications which were developed in financial mathematics in recent years. Possible insur- ance applications are not mentioned at all in these books. The recent mono- graph “Some Advances on Quadratic BSDE: Theory–Numerics–Applications” by Dos Reis (2011) points out an actuarial and financial application but the author focuses on advanced theory of quadratic BSDEs, which definitely is not the first stepinthestudyofBSDEs.AllthreebooksdealwithBSDEsdrivenbyBrownian motions and omit BSDEs with jumps which are very important for actuarial and financialmodelling.Thereexistsaconsiderablevolumeofmathematicalpaperson BSDEsandBSDEswithjumps.However,thesepapersarequitedifficulttoaccess byabeginnerinthefieldofBSDEsandstochasticprocesses.Ourgoalistopresent a book on BSDEs with jumps which covers key theoretical results and focuses on applicationsandwhichcanbefollowedbynonspecialistsinstochasticmethods. vii viii Preface Thesecondmotivationforthisbookistopromotebackwardstochasticdifferen- tialequationsintheactuarialcommunity.BSDEsseemnottobewell-knowninin- surancemathematics,despitetheirrecognizedadvantagesinfinancialmathematics and optimal control theory. This state should be changed as many actuarial prob- lems are closely related to financial problems, hence they can be approached with BSDEs.Sinceoptimizationproblemsaregainingimportanceinactuarialmathemat- ics, efficient and modern solution methods for stochastic control problems should bepresented.Whilethemonograph“StochasticControlinInsurance”bySchmidli (2007)dealswithHamilton-Jacobi-Bellmanequations,ourgoalistoshowhowto applyBSDEstosolveoptimizationproblems. Jump processes play a leading role in actuarial modelling. Following Mikosch (2009),wecansaythatmodellingofclaimnumbersbypointprocessesisbreadand butter for the actuary. Jump processes are also used in financial mathematics. Let usremarkthatLévyprocesseshavebeenintroducedwithsuccesstofinancialmod- els, see the monographby Cont and Tankov (2004) and the textbookby Øksendal andSulem(2004)whereHJBequationsareappliedtosolvefinancialoptimization problems for Lévy-driven processes. Due to the importance of jump processes in actuarial and financial applications, we investigate BSDEs driven by a Brownian motionandacompensatedrandommeasure(calledBSDEswithjumps).SinceBS- DEscanbeusedinageneralstochasticframework,weconsidergeneral(quasi-left continuous)jumpprocesses.Consequently,wealsoextendtheactuary’stoolboxfor stochasticmodelling. WehopethatthisbookwillmakeBSDEsmoreaccessibletothosewhoarein- terested in applying these equations to actuarial and financial problems. Our book should be beneficial to students and researchers in applied probability and practi- tioners.Studentsandresearchersinappliedprobabilityshouldgetastrongmathe- maticalintroductiontothetheoryandapplicationsofBSDEs.Practitionersshould learnhowtoderiveasset-liabilitystrategiesinsophisticatedinternalmodels(advo- catedbySolvencyIIDirective),setuphedgingstrategiesandpricecomplexinsur- anceproductswithfinancialguarantees.Thisbookmayalsobeusefulinactuarial educationsinceitcoversappliedstochasticcalculusandstochasticoptimalcontrol theory,whichareincludedintheeducationalsyllabusesoftheGroupeConsultatif andtheInternationalActuarialAssociation. Warsaw,Poland ŁukaszDelong April2013 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 PartI BackwardStochasticDifferentialEquations—TheTheory 2 StochasticCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 BrownianMotionandRandomMeasures . . . . . . . . . . . . . . 13 2.2 ClassesofFunctions,RandomVariablesandStochasticProcesses . 18 2.3 StochasticIntegration . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 ThePropertyofPredictableRepresentation . . . . . . . . . . . . . 25 2.5 EquivalentProbabilityMeasures . . . . . . . . . . . . . . . . . . 27 2.6 TheMalliavinCalculus . . . . . . . . . . . . . . . . . . . . . . . 31 3 BackwardStochasticDifferentialEquations—TheGeneralCase . . 37 3.1 ExistenceandUniquenessofSolution. . . . . . . . . . . . . . . . 37 3.2 ComparisonPrinciples . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 ExamplesofLinearandNonlinearBSDEsWithoutJumps . . . . . 58 3.4 ExamplesofLinearandNonlinearBSDEswithJumps . . . . . . . 61 3.5 MalliavinDifferentiabilityofSolution . . . . . . . . . . . . . . . 64 4 Forward-BackwardStochasticDifferentialEquations . . . . . . . . 79 4.1 TheMarkovianStructureofFBSDEs . . . . . . . . . . . . . . . . 79 4.2 The Feynman-KacFormula and the Connection with Partial Integro-DifferentialEquations . . . . . . . . . . . . . . . . . . . . 87 4.3 CoupledFBSDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5 NumericalMethodsforFBSDEs . . . . . . . . . . . . . . . . . . . . 101 5.1 Discrete-TimeApproximationandLeastSquaresMonteCarlo. . . 101 5.2 Discrete-TimeandDiscrete-SpaceMartingaleApproximation . . . 106 5.3 FiniteDifferenceMethod . . . . . . . . . . . . . . . . . . . . . . 109 6 NonlinearExpectationsandg-Expectations . . . . . . . . . . . . . . 113 6.1 ChoquetExpectations . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Filtration-ConsistentNonlinearExpectationsandg-Expectations . 114 ix x Contents Part II Backward Stochastic Differential Equations—The Applications 7 CombinedFinancialandInsuranceModel . . . . . . . . . . . . . . . 125 7.1 TheFinancialMarket . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2 TheInsurancePaymentProcess . . . . . . . . . . . . . . . . . . . 127 7.3 AdmissibleInvestmentStrategies . . . . . . . . . . . . . . . . . . 132 8 Linear BSDEs and Predictable Representations of Insurance PaymentProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.1 TheApplicationoftheItô’sFormula . . . . . . . . . . . . . . . . 135 8.2 TheApplicationoftheMalliavinCalculus . . . . . . . . . . . . . 142 9 Arbitrage-FreePricing,PerfectHedgingandSuperhedging . . . . . 151 9.1 Arbitrage-FreePricingandMarket-ConsistentValuation . . . . . . 151 9.2 PerfectHedgingintheFinancialMarket . . . . . . . . . . . . . . 152 9.3 SuperhedgingintheFinancialandInsuranceMarket . . . . . . . . 158 9.4 PerfectHedgingintheFinancialandInsuranceMarketCompleted withaMortalityBond . . . . . . . . . . . . . . . . . . . . . . . . 164 10 QuadraticPricingandHedging . . . . . . . . . . . . . . . . . . . . . 173 10.1 QuadraticPricingandHedgingUnderanEquivalentMartingale Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 10.2 QuadraticPricingandHedgingUndertheReal-WorldMeasure . . 177 10.3 QuadraticPricingandHedgingUnderLocalRisk-Minimization . . 189 10.4 Quadratic Pricing and Hedging Under an Instantaneous Mean-VarianceRiskMeasure . . . . . . . . . . . . . . . . . . . . 195 11 UtilityMaximizationandIndifferencePricingandHedging . . . . . 205 11.1 ExponentialUtilityMaximization . . . . . . . . . . . . . . . . . . 205 11.2 ExponentialIndifferencePricingandHedging . . . . . . . . . . . 213 12 PricingandHedgingUnderaLeastFavorableMeasure . . . . . . . 221 12.1 PricingandHedgingUnderModelAmbiguity . . . . . . . . . . . 221 12.2 No-Good-DealPricing . . . . . . . . . . . . . . . . . . . . . . . . 231 13 DynamicRiskMeasures . . . . . . . . . . . . . . . . . . . . . . . . . 235 13.1 DynamicRiskMeasuresbyg-Expectations . . . . . . . . . . . . . 235 13.2 GeneratorsofDynamicRiskMeasures . . . . . . . . . . . . . . . 240 13.3 OptimalRiskSharing . . . . . . . . . . . . . . . . . . . . . . . . 243 PartIII OtherClassesofBackwardStochasticDifferentialEquations 14 OtherClassesofBSDEs . . . . . . . . . . . . . . . . . . . . . . . . . 253 14.1 Time-DelayedBackwardStochasticDifferentialEquations . . . . 253 14.2 ReflectedBackwardStochasticDifferentialEquations . . . . . . . 260 14.3 ConstrainedBackwardStochasticDifferentialEquations . . . . . . 268 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Chapter 1 Introduction Abstract Wediscussadvantagesofsolvingoptimalcontrolproblemsanddefining nonlinearexpectationsbybackwardstochasticdifferentialequations.Wecomment onapplicationsofbackwardstochasticdifferentialequationstopricingandhedging ofliabilitiesandmodellingofdynamicriskmeasures. A backward stochastic differential equation (BSDE) with jumps is an equation of theform (cid:2) T (cid:3) (cid:4) Y(t)=ξ + f s,Y(s),Z(s),U(s,.) ds t (cid:2) (cid:2) (cid:2) T T − Z(s)dW(s)− U(s,z)N˜(ds,dz), 0≤t≤T, (1.1) t t R ˜ where W is a Brownian motion and N is a compensated random measure. Given a terminal condition ξ and a generator f, we are interested in finding a triple (Y,Z,U) which solves (1.1). More precisely, we aim to find an adapted process Y :=(Y(t),0≤t≤T)whichismodelledbythedynamics (cid:2) (cid:3) (cid:4) dY(t)=−f t,Y(t),Z(t),U(t,.) dt+Z(t)dW(t)+ U(t,z)N˜(dt,dz), (1.2) R and satisfies Y(T)=ξ where ξ is an F -measurable random variable. At first T sight it seems to be a hopeless task to construct such a process. However, the dy- namics (1.2) is driven by two predictable processes Z :=(Z(t),0≤t ≤T) and U :=(U(t,z),0≤t ≤T,z∈R)whichareallowedtobechosenasthepartofthe solution to the BSDE (1.1). The processes Z and U are called control processes. TheycontroltheprocessY sothatY satisfiestheterminalcondition. It should be pointed out that we would not be able to find an adapted solution to an equation with a random terminal condition if we did not introduce control processes.Letusconsidertheequation dY(t)=0, Y(T)=ξ, (1.3) Ł.Delong,BackwardStochasticDifferentialEquationswithJumpsandTheirActuarial 1 andFinancialApplications,EAASeries,DOI10.1007/978-1-4471-5331-3_1, ©Springer-VerlagLondon2013
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