Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications: BSDEs with Jumps

285 Pages·2013·2.1 MB·English

by  Łukasz Delong (auth.)

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
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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