Weak convergence of error processes in discretizat

arXiv:0711.1439v2[math.PR]12Feb2008WeakonvergeneoferrorproessesindisretizationsofstohastiintegralsandBesovspaesStefanGeissandAnniToivolaDepartmentofMathematisandStatistisP.O.Box35(MaD)FIN-40014UniversityofJyvskylFinlandgeissmaths.jyu.andatoivolamaths.jyu.February12,2008AbstratWeonsiderweakonvergeneoftheresalederrorproessesarisingfromRiemanndisretizationsofertainstohastiintegralsandrelatetheLp-integrabilityoftheweaklimittothefrationalsmoothnessintheMalliavinsenseofthestohastiintegral.Keywords:Besovspaes,stohastiintegrals,approximation,weakonver-geneMathematisSubjetlassiation:60H05,41A25,46E50,60G44IntrodutionQuantitativeapproximationproblemsforstohastiintegralsappearnaturallyinStohastiFinane.Considerastohastiintegralg(X1)=Eg(X1)+Z10∂G∂x(u,Xu)dXu,wherethediusionX=(Xt)t∈[0,1]modelsaprieproess,g(X1)∈L2isapay-oofaEuropeantypeoption,andGsolvesaorrespondingparabolibakwardPDEwithg(x)=G(1,x).WelookattheRiemannapproximationn−1Xi=0∂G∂x(ti,Xti)Xti+1−Xti
1alongadeterministitime-netτ=(ti)ni=1andtheerrorproessC(τ)=(Ct(τ))t∈[0,1]givenbyCt(τ):=Zt0∂G∂x(u,Xu)dXu−n−1Xi=0∂G∂x(ti,Xti)Xti+1∧t−Xti∧t
.TheproessC(τ)desribesthehedgingerrorwhihourswhenaontinuouslyadjustedportfolioisreplaedbyaportfoliowhihisadjustedonlyatthetime-knotst0,...,tn−1.Givenasequeneoftime-netsτn=(tni)ni=0,oneisinterestedintherateofonvergeneofC(τn)towardszeroasn→∞.Thereare(atleast)twoprinipalwaystomeasurethesizeofC(τn).Firstly,oneanusestrongriteria,likeLp-norms,whereonetypiallylooksforestimatesoftheformkC1(τn)kLp≤cn−θ(1)forsomeθ0.Seondly,oneaninvestigatetheweakonvergeneofthere-saledproesses√nC(τn).Apriori,therearenogeneralpriniplestodedueaertainstrongonvergenefromaweaklimitortogotheotherwayround.Coneptsofweakonvergeneareofpartiularinterestinappliationsbeausetheyalreadyprovidetheneededinformationinmanyasesandpromisepoten-tiallybetterapproximationratesthanobtainedunderstrongriteria.Resultsaboutweakonvergeneinourontextareobtainedforexamplein[18℄(seealso[3℄)and[11,13,19℄.Forthegeneraltheorythereaderisreferredto[15℄(seealso[17℄).GobetandTemamhaveshownin[11,Theorems1and3℄thatfortheBinaryoption(i.e.g(x)=χ[K,∞)(x)forK0andXbeingthegeometriBrownianmotion)inaseofequidistanttime-netsτnthesalingfatorfortheweakon-vergeneanbetakenton12whereastheL2-ratein(1)isθ=14.Intuitively,onewouldexpetthatthesalingexponent12andθoinide.Indeed,forpay-ofuntionsghavingsomefrationalsmoothnessintheMalliavinsense(liketheBinaryoptionwithsmoothnessβ∈0,12
;seeExample1.2)theL2-rateθ=12analwaysbeahievedbyusingappropriatenon-equidistanttime-nets,see[4,10℄1.Fromthis,twoquestionsbeomenatural:Isthereaonnetionbetweenfrationalsmoothnessandweakonvergene?And,donon-equidistanttime-netshaveapositiveeetontheweakonvergene?Theaimofthispaperistoanswerbothquestionstothepositive.Toseetheonnetionsbetweenfrationalsmoothnessofgandweakonvergeneof√nC(τn)onehastohangethepointofview:insteadoflookingforapproxi-mationrates(whihwouldorrespondtothesalingfatorforC(τn))onehastolookattheLp-integrabilityoftheweaklimit.Thishasrelevaneforappliationswheregoodtail-estimatesfortheweaklimitaredesirable.Thepaperisorganizedasfollows:•Afterintroduingthenotationweformulateourbasiresult,Theorem2.1,whereweharaterizetheexisteneofasquareintegrableweaklimit1Non-equidistanttime-netshavebeenalsousedinotherpaperslike,forexample,[12,16℄.2of√nC(τn,β)bytheonditionthatgorg(exp(·−12))(dependingonthediusionX)belongstotheBesovspaeBβ2,2(γ).Theparameterβ∈(0,1]isthefrationalsmoothnessintheMalliavinsenseandτn,βaretime-netsadaptedtothesmoothnessβ.Hene,ifgorg(exp(·−12))haveanon-trivialfrationalsmoothnessandifweusetherighttime-nets,thenwealwaysgetasquare-integrableweaklimit.TheoneptoffrationalsmoothnessallowsustoonsideratonethelargelassoffuntionsSβ∈(0,1]Bβ2,2(γ)whihontainsallexamplesusuallystudiedintheliteratureinthisontext.FortheBinaryoptionthismeans:theweaklimitforequidistanttime-netsin[11℄isnotsquare-integrable,butbeomessquareintegrableforthetime-netsτn,βaslongasβ∈(0,12)beauseofExample1.2below.•TheL2-settingofTheorem2.1isextendedinSetion3totheLp-setting,p∈[2,∞).Corollary3.4givesnearlyoptimalonditionsthattheweaklimitisLp-integrable.AsanappliationfortheBinaryoptionweomputeinExample3.7thebestpossibleLp-integrabilityoftheweaklimitprovidedthattheτn,β-netsareused.Inpartiular,thisexampleshowshowtheintegrabilityanbeimprovedtoanyp∈[2,∞)byusingnetsτn,βwithaβsmallenoughor,equivalently,byusingnetswithasuientlyhighonentrationofthetime-knotslosetothenaltime-pointt=1.•TheupperestimatefortheLp-integrabilityoftheweaklimitinExample3.7fortheBinaryoptionhasamoregeneralbakground:inCorollary3.11weassumethatghasaloalsingularityoforderη≥0(measuredintermsofasharpfuntion)anddedueanupperboundfortheLp-integrabilityoftheweaklimit.•InSetion4weproveTheorem2.1:rstly,wederivetheexisteneoftheweaklimiton[0,T]withT∈(0,1).Seondly,asthemainpart,wehavetodealwithasingularityofourapproximationproblemattimet=1beauseoftheblow-upoftheMalliavinderivativeofE(g(X1)|Ft)ast↑1.Thedegreeofthisblow-upisonnetedtothefrationalsmoothnessβofg.Theusedtime-netsτn,βareessentialastheyarehosentoompen