Backward stochastic differential equations with ra

arXiv:0707.4387v1[math.PR]30Jul2007TheAnnalsofProbability2007,Vol.35,No.3,1071–1117DOI:10.1214/009117906000000746cInstituteofMathematicalStatistics,2007BACKWARDSTOCHASTICDIFFERENTIALEQUATIONSWITHRANDOMSTOPPINGTIMEANDSINGULARFINALCONDITIONByA.PopierUniversit´edeProvenceInthispaperweareconcernedwithone-dimensionalbackwardstochasticdifferentialequations(BSDEinshort)ofthefollowingtype:Yt=ξ−Zτt∧τYr|Yr|qdr−Zτt∧τZrdBr,t≥0,whereτisastoppingtime,qisapositiveconstantandξisaFτ-measurablerandomvariablesuchthatP(ξ=+∞)0.WestudythelinkbetweentheseBSDEandtheDirichletproblemonadomainD⊂Rdandwithboundaryconditiong,withg=+∞onasetofpositiveLebesguemeasure.WealsoextendourresultsformoregeneralBSDE.Introduction.Let(Ω,F,P)beaprobabilityspace,B=(Bt)t≥0aBrow-nianmotiondefinedonthisspace,withvaluesinRd.(Ft)t≥0isthestandardfiltrationoftheBrownianmotion.Alsogivenareτa{Ft}-stoppingtime,ξareal,Fτ-measurablerandomvariable,calledthefinalcondition,andf:Ω×R+×R×Rd→Rthegenerator.Wewishtofindaprogressivelymeasurablesolution(Y,Z),withvaluesinR×Rd,oftheBSDEYt=ξ+Zτt∧τf(r,Yr,Zr)dr−Zτt∧τZrdBr,t≥0.(1)Suchequations,inthenonlinearcase,havebeenintroducedbyPardouxandPengin1990in[19],whenτisreplacedbyaconstanttimeT0.Theygavethefirstexistenceanduniquenessresult.Sincethen,BSDEhavebeenstudiedwithgreatinterest(seethereferencesin[18]).Inparticular,Peng[20]describeshowthesolutionYof(1)foranunboundedrandomterminaltimeisrelatedtoasemilinearellipticPDE.ViscositysolutionsforsuchReceivedApril2005;revisedMay2006.AMS2000subjectclassifications.60H10,60G40,35J60,49L25,35J65.Keywordsandphrases.BackwardSDE,nonintegrabledata.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofProbability,2007,Vol.35,No.3,1071–1117.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12A.POPIERequationswillbeconstructedbystochasticmethods(seeTheorem8below).ThisgeneralizationoftheFeynman–Kacformulaisareasonforstudyingrandomterminaltimes.Letusrecallthedefinitionofasolutionof(1)whichcanbefoundin[4].Definition1.AsolutionoftheBSDE(1)isapair{(Yt,Zt),t≥0}ofprogressivelymeasurableprocesseswithvaluesinR×Rdsuchthat,P-a.s.:•ontheset{t≥τ},Yt=ξandZt=0,•t7→1t≤τf(t,Yt,Zt)belongstoL1loc(0,∞),t7→ZtbelongstoL2loc(0,∞),•andforall0≤t≤T,Yt∧τ=YT∧τ+ZT∧τt∧τf(r,Yr,Zr)dr−ZT∧τt∧τZrdBr.AsolutionissaidtobeanLp-solutionforsomep1if,moreover,forsomeλ∈R,Esup0≤t≤τepλt|Yt|p+Zτ0epλt|Yt|pdt+Zτ0epλt|Yt|p−2kZtk2dt+∞.Weassumethatthegeneratorf:Ω×R+×R×Rd→Rissuchthat:(H0)f(·,y,z)isprogressivelymeasurable,forally,z;(H1)∃K≥0,suchthata.s.∀t,y,z,z′,|f(t,y,z)−f(t,y,z′)|≤Kkz−z′k;(H2)∃μ∈R,suchthata.s.∀t,y,y′,z,(y−y′)(f(t,y,z)−f(t,y′,z))≤μ|y−y′|2;(H3)y7→f(t,y,z)iscontinuous,∀t,z,a.s.(H4)forallr0andalln∈N∗,ψr(t)=sup|y|≤r|f(t,y,0)−f(t,0,0)|be-longstoL1((0,n)×Ω).Nowforsomep1wesupposethatthereexistsλνp=μ+K22(p−1),suchthatEZτ0epλt|f(t,0,0)|pdt+∞(H5)andEepλτ|ξ|p+Zτ0epλt|f(t,e−νptξt,e−νptηt)|pdt+∞,(H6)whereξ=eνpτξ,ξt=E(ξ|Ft)andηispredictableandsuchthatξ=E(ξ)+Z+∞0ηtdBt,EZ∞0|ηt|2dtp/2∞.LetusrecallTheorem5.2of[4].BSDEWITHSINGULARFINALCONDITION3Theorem1.Undertheconditions(H0)–(H6),thereexistsauniquesolution(Y,Z)oftheBSDE(1),which,moreover,satisfies,forλνpsuchthat(H5)and(H6)hold:Esup0≤t≤τepλt|Yt|p+Zτ0epλr|Yr|p−2(|Yr|2+kZrk2)dr(2)≤cEepλτ|ξ|p+Zτ0epλr|f(t,0,0)|pdr,forsomeconstantc=c(p,λ,K,μ).Remark1.TheprevioustheoremisageneralizationoftheresultofDarlingandPardoux(Theorem3.4in[6])orofPardoux(Theorem4.1in[18]).In[6]or[18]theresultisgiveninthecasep=2.Herewehaveexpressedthetheoremforthedimensionone(ξandYtbelongtoR).Butitisstilltrueinhigherdimensions(see[4];theproductin(H2)mustbereplacedbythescalarproductinRm).NotethatiffisaLipschitzfunction,thecondition(H2)holds.FromnowandintherestofthepaperweareconcernedwiththeBSDEYt=ξ−Zτt∧τYr|Yr|qdr−Zτt∧τZrdBrwithq0.(3)Herethefunctionfisdeterministicandequaltof(t,y,z)=−y|y|q.fsatisfiesallconditions(H0)–(H4)ofTheorem1,withK=μ=0(whichimpliesνp=0forallp1).Indeed,fisanonincreasingfunction,thereby,−(y−y′)(y|y|q−y′|y′|q)≤0.Sincef(t,0,0)≡0,(H5)isalwayssatisfied.Thestoppingtimeτisdefinedasfollows.LetDbeanopenboundedsubsetofRd,whoseboundaryisatleastofclassC2(see[12]forthedefinitionofaregularboundary).Forallx∈Rd,letXxdenotethesolutionoftheSDE:Xxt=x+Zt0b(Xxr)dr+Zt0σ(Xxr)dBrfort≥0.(4)ThefunctionsbandσaredefinedonRd,withvaluesrespectivelyinRdandRd×d,andaremeasurablesuchthat:•Lipschitzcondition:thereexistsK≥0suchthat∀(x,y)∈Rd×Rdkσ(x)−σ(y)k≤K|x−y|;(L)4A.POPIER•Boundednesscondition:∀x∈Rd|b(x)|+kσ(x)k≤K;(B)•Uniformellipticity:thereexistsaconstantα0suchthat∀x∈Rdσσ∗(x)≥αId.(E)Intherestofthispaper(L),(B)and(E)aresupposedtobesatisfied.Undertheseassumptions,fromaresultofYuVeretennikov[24]and[25],equation(4)hasauniquestrongsolutionXx.Foreachx∈D,wedefinethestoppingtimeτ=τx=inf{t≥0,Xxt/∈D}.(5)Ourstoppingtimesatisfiesthefollowingtwoproperties.SinceDisboundedandsincetheconditions(B)and(E)holdeverypointx∈∂Disregular.(C1)Inparticular,ifx∈∂D,τx=0a.s.(see[3],Corollary3.2).Thisassump-tion(C1)isimportanttodefineasingularsolution(seeDefinition2below).Moreover,since(L),(B)and(E)hold,we