Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation

Transcription

1 C. R. Acad. Sc. Pars, Sr. I Probablty Thory Gomtrzaton of Mont-Carlo numrcal analyss of an llptc oprator: strong approxmaton Ana Bla Cruzro a, Paul Mallavn b, Anton Thalmar c a Grupo d Físca-Matmátca UL and Dp. Matmátca IST, Av. Rovsco Pas, Lsboa, Portugal b 10, ru Sant Lous n l Isl, Pars, Franc c Départmnt d mathématqus, Unvrsté d Potrs, téléport, BP 30179, 8696 Futuroscop Chassnul, Franc Rcvd 16 Dcmbr 003; accptd 1 January 004 Prsntd by Paul Mallavn Abstract A on-stp schm s constructd, whch, as th Mlstn schm, has th strong approxmaton proprty of ordr 1; n contrast to th Mlstn schm, our schm dos not nvolv th smulaton of tratd Itô ntgrals of scond ordr. To ct ths artcl: A.B. Cruzro t al., C. R. Acad. Sc. Pars, Sr. I Publshd by Elsvr SAS on bhalf of Académ ds scncs. Résumé Mont-Carlo géométrqu pour un opératur llptqu : approxmaton numérqu fort. On propos un schéma à un pas, qu, comm l schéma d Mlstn, possèd la proprété d approxmaton fort à l ordr 1 ; contrarmnt au schéma d Mlstn, notr schéma n nécsst pas la smulaton d ntégrals térés d Itô du scond dgré. Pour ctr ct artcl : A.B. Cruzro t al., C. R. Acad. Sc. Pars, Sr. I Publshd by Elsvr SAS on bhalf of Académ ds scncs. 1. Introducton Numrcal ntgraton of SDE conssts, an SDE bng fxd, n producng a dscrtzaton schm ladng to a pathws approxmaton of ts soluton. In ths wor w dscuss th rlatd problm of how to stablsh a Mont-Carlo smulaton to a gvn ral strctly llptc scond ordr oprator L, whch lads to good numrcal approxmatons of th fundamntal soluton to th corrspondng hat oprator. Of cours many SDE can b assocatd to L, ach choc corrspondng to a paramtrzaton by th Wnr spac of th Strooc Varadhan soluton of th martngal problm assocatd to L. For applcatons to fnanc all ths paramtrzatons ar quvalnt. W shall prov that thr xsts an optmal paramtrzaton for whch th on-stp Mlstn schm dos not nvolv th computaton of tratd stochastc ntgrals of scond ordr. For our E-mal addrsss: abcruz@math.st.utl.pt A.B. Cruzro, sl@ccr.ussu.fr P. Mallavn, thalmar@math.unv-potrs.fr A. Thalmar X/$ s front mattr 004 Publshd by Elsvr SAS on bhalf of Académ ds scncs. do: /.crma

2 48 A.B. Cruzro t al. / C. R. Acad. Sc. Pars, Sr. I sarch of an optmal paramtrzaton w hav to dscrb th possbl paramtrzatons of th dffuson assocatd to L; t s suffcnt to solv ths problm for th Eucldan Laplacan on R d ; n ths cas t s quvalnt to rplac th standard Brownan moton W on R d by an orthogonal transform W wth an Itô dffrntal of th typ d d W = ΩW t ] dw t, 1 =1 whr t Ω W t s an adaptd procss tang valus n th group Od of orthogonal matrcs. W dnot by OW th famly of all such orthogonal transforms whch s somorphc to POd, th path spac on Od. Pontws multplcaton dfns a group structur on POd OW. Gvn on R d thdataofd + 1 smooth vctor flds A 0,A 1,...,A d, w consdr th Itô SDE dξ W t = A 0 ξw t d dt + A ξw t dw t, ξ W 0 = ξ 0. =1 Throughout ths Not w assum llptcty, that s, for any ξ R d th vctors A 1 ξ,..., A d ξ consttut a bass of R d ; th componnts of a vctor fld U n ths bass ar dnotd U,A ξ whch gvs th dcomposton Uξ= d =1 U,A ξ A ξ. Bychang of paramtrzaton w man th substtuton of W by W n ; w thn gt an Itô procss n W. Ths chang of paramtrzaton dos not chang th nfntsmal gnrator assocatd to whch has th form L = 1,α,β Aα Aβ D αd β + α Aα 0 D α whr D α = / ξ α. Th group OW oprats on th st of llptc SDE on R d and th orbts of ths acton ar classfd by th corrspondng llptc oprators L.. Dfnton of th schm S Dnot by t ε := ε ntgr part of t/ε; w dfn our schm by Z W εt Z W εt ε = A 0 ZW εt ε t t ε + A ZW εt ε W t W t ε A A s Z W εt ε { W t W t ε W s t W s t ε εη s },s A ZW εt ε A s,a ],A Z W ε t ε { W t W t ε W s t W s t ε εη} s, 3,s, whr W s standard Brownan moton on R d,andη s th Kroncr symbol dfnd by ηs = 1f = s and zro othrws. Dnot by PR d th path spac on R d, that s th Banach spac of contnuous maps from 0,T] nto R d, ndowd wth th sup norm: p 1 p = sup t 0,T ] p 1 t p t R d. Fxng ξ 0 R d,ltp ξ0 R d b th subspac of paths startng from ξ 0. Gvn Borl masurs ρ 1,ρ on PR d, dnot by Mρ 1,ρ th st of masurabl maps Ψ : PR d PR d such that Ψ ρ 1 = ρ ;thmong transport norm s 8,5] s dfnd as Ψp d M ρ 1,ρ := nf 1/. Ψ Mρ 1,ρ 1dp] ρ Thorm.1. Assum llptcty and assum th vctor flds A along wth thr frst thr drvatvs to b boundd; fx ξ 0 R d and lt ρ L b th masur on P ξ0 R d dfnd by th soluton of th Strooc Varadhan martngal problm 7] for th llptc oprator L; lt ρ S b th masur obtand by th schm S wth ntal valu Z W ε0 = ξ 0.Thn 1 lm sup ε 0 ε d Mρ L,ρ S = c<. 4

3 A.B. Cruzro t al. / C. R. Acad. Sc. Pars, Sr. I Rmar. Th proof of Thorm.1 wll provd an xplct transport functonal Ψ 0 whch puts th statmnt n a constructv sttng; th constant c s ffctv. 3. Th Mlstn schm Th Mlstn schm for SDE cf., for nstanc, 6], formula 0.3 or ], p. 345; s also 3] s basd on th followng stochastc Taylor xpanson of A along th dffuson tractory: A ξ W t = A ξ W t ε + A A ξ W t ε W t W t ε + Oε, whch lads to ξ W ε t ξ W ε t ε = A ξw ε t ε W t W t ε + t t ε A 0 ξw ε t ε + A A ξ W ε t ε t W s W t ε dw s;, th computaton of t t ε W s W t ε dw s gvs th Mlstn schm ξ W ε t ξ W ε t ε = A ξw ε t ε W t W t ε + t t ε A 0 ξw ε t ε A A ξ W ε t ε W t W t ε W t W t ε εη + R,, whr R = < A,A ]ξ W ε t ε t t ε W s W t ε dw s W s W t ε dw s. It s wll nown that th Mlstn schm has th followng strong approxmaton proprty: E sup t 0,1] t ε ξw t ξ W ε t ] = O ε. 5 Th numrcal dffculty rlatd to th Mlstn schm s how to achv a fast smulaton of R. Th purpos of ths wor s to show that by a chang of paramtrzaton ths smulaton can b avodd. 4. Horzontal paramtrzaton Gvn d ndpndnt vctor flds A 1,...,A d on R d, w ta th vctors A 1 ξ,..., A d ξ as bass at th pont ξ; th functons β,s, calld structural functons, ar dfnd s 1] by: β,l ξ = A,A l ],A ξ, A,A l ]ξ = β,l ξa ξ. Th structural functons ar antsymmtrc wth rspct to th two lowr ndcs. Consdr th conncton functons, dfnd from th structural functons by Γ,s = 1 β,s βs, +, βs. 6 Lt Γ b th d d matrx obtand by fxng th ndx n th thr ndcs functons Γ,. Thn, by mans of th antsymmtry of β, n th two lowr ndcs, Γ s an antsymmtrc matrx: Γ,s + Γ, s = β,s βs, + βs, + βs, β,s + β s, = 0. Th matrx Γ oprats on th coordnat vctors of th bass A s ξ va Γ A s = Γ,s A.ThsgvsΓ A s Γ s A =A,A s ]:thth componnt of th l.h.s. s 1 β,s β s, + βs, β s, + βs, β,s = β,s.ltm = R d E d whr E d s th vctor spac of d d matrcs. Dfn on M vctor flds Ã,= 1,...,d, as follows:

4 484 A.B. Cruzro t al. / C. R. Acad. Sc. Pars, Sr. I à ξ, = l A lξ, N ξ,, N ] s r ξ, = l l r Γ l,l s ξ, ξ R d, E d. 7 l Dnotng for a vctor Z on M by Z H ts procton on R d,whav: Proposton 4.1. Th vctor flds à satsfy th rlaton Ã, à s ] H = 0. Proof. Th horzontal componnt à à s ] H s gvn by à à s ] H = à ÃH s + N ] α β β α q s A q q α,β = l A l s l A l l l β Γ l,l α l l α,β q = l A l s l A l l l s Γ l,l α A α, l l α, η q α ηs β A q usng th fact that α β q s = ηαη q β s. W fnally gt à à s ] H = l l s A l A l α Γ l,l α A α ]. Thrfor th horzontal componnt of th commutator s Ã, à s ] H = l l s A l,a l ] l l s Γ α Γl α,l Aα α, whch vanshs snc Γ l A l Γ l A l =A l,a l ]. Dnot by T th transposd of th matrx and lt à 0 ξ, = A 0 ξ, 1 J, J := d =1 N T N. Consdr th followng Itô SDE on th vctor spac M: dm W = à m W dw + à 0 m W dt, m W 0 = ξ 0, Id. 8 Proposton 4.. Dnot m W t = ξ W t, W t, thn W t s an orthogonal matrx for t 0, and f ξ W t t 0 Lf ξ W s ds s a local martngal, for any f C R d. 9 Proof. W comput th stochastc dffrntal of T : d T ] l l = d l l = m,,p u + l p m u Γ l p,u + v l à 0 l + l à 0 l + l p m v Γ p,v l dw m m p p Γ p,p l m q q Γ l m,p,q,p,q q,q dt, whr th last trm of th drft coms from th Itô contracton dl dl = N TN ] l l l dt = Jl dt. Th frst two trms of th drft ar computd by usng th dfnton of à 0 : l à 0 l + l à 0 l ]= 1 T J l l + T J l l ]. Wrt T = Id+σ, thn th drft tas th form σ J + Jσ/. W comput th coffcnt of dw m :

5 ,p u l p m u Γ l p,u + v A.B. Cruzro t al. / C. R. Acad. Sc. Pars, Sr. I l p m v Γ p,v l = p p m u T ] u l Γ l p,u + v T ] v l Γp,v l. Usng th antsymmtry Γp,l l = Γ p,l l w obtan dσl l = dw m m p σl u Γ p,u l + σl v Γ p,v l 1 σj + Jσ]l l dt. 10 m p u v Eq. 10, togthr wth Eq. 8, gvs an SDE wth local Lpschtz coffcnts for th trpl ξ,,σ; by unqunss of th soluton, as σ0 = 0, w dduc σt= 0forallt 0. In trms of th nw R d -valud Brownan moton W dfnd by d W t := l W t] l dw l,whav d ξ W = A ξ W t d W t + A 0 ξ W t dt Rconstructon of th schm S W want to prov that our schm S s ssntally th procton of th Mlstn schm ξ W ε, W ε for th soluton m W = ξ W, W of th SDE 8. In ordr to wrt th frst componnt ξ W ε w hav to comput th horzontal part of à Ã, whch has bn don n th proof to Proposton 4.1: w gt ξ W ε t ξ W ε t ε = A 0 ξ W εt ε t t ε +,l { W εt ε ] l W εt ε ] l, whr W = W t W t ε.by5 E W t W ε t ] cε, sup t 0,1] Consdr th nw procss ξ W dfnd by W εt ε ] l A l ξ W ε t ε W Al A l E ξ W t ξ W t ε = A 0 ξ W t ε t t ε +,l { W t ε ] l + 1, W t ε ] l sup t 0,1] Al A l Γl,l A ξ W ε t ε } W W εη, ξ W t ξ W ε t ] cε. 1 W t ε ] l A l ξ W t ε W ξ Γl,l A W ε t ε } W W εη. Lmma 5.1. Th procss ξ W has th sam law as th procss Z W ε dfnd n 3. Proof. By Proposton 4., Ŵ l t Ŵ l t ε := W t ε ] l W t W t ε ar th ncrmnts of an R d -valud Brownan moton Ŵ ;w gt ξ W t ξ W t ε = A 0 ξ W t ε t t ε + A A s,s A ξ W t ε Ŵ t Ŵ t ε Γ,s A ξ W t ε Ŵ t Ŵ t ε Ŵ s t Ŵ s t ε εη s.

6 486 A.B. Cruzro t al. / C. R. Acad. Sc. Pars, Sr. I ByEq.6,whav Γ,s A =A,A s ]+ A,A s ],A A + A,A ],A s A, whr th frst trm s antsymmtrc n,s and dos not contrbut; th rmanng sum s symmtrc n,s. Thus w gt,,s Γ,s A Ŵ Ŵ s =,,s A A s,a ],A Ŵ Ŵ s whch provs Lmma 5.1. Lmma 5.. W hav Esup t 0,1] ξ W t ξ W ε t R d ] cε. Proof. Th followng mthod of ntroducng a paramtr and dffrntatng wth rspct to, s contnuously usd n 4]. For 0, 1],lt := W + 1 W ε and dfn th procss ξw by ξw t ξ W t ε = A 0 ξ W t ε t t ε + W t ε ] l A l ξ W t ε W,l { W t ε ] l W t ε ] l Al A l ξ Γl,l A W ε t ε } W W t t ε η., Lt u W := ξ/ ;thnξ W t ξ W ε t = 1 0 u W t d. Dnot by A, A l A l, Γl,l A th matrcs obtand by dffrntatng A, Al A l,γl,l A wth rspct to ξ, and consdr th dlayd matrx SDE dj t t0 = A 0 ξ W t ε dt + W t ε ] l l A ξ W t ε dw { Al A l,l wth ntal condton J t0 t 0 = Id. Thn u W T = J T t 0 T 0 W t ε ] l, W t ε ] l W t ε ] l Γ A ξ W ε t ε } W dw t + W dw t η dt] J t t0 Jt t 1 W 0 t ε W εt ε ] l A l ξ W t ε dw t +,l, Al A l { W t ε W εt ε ] l ξ Γl,l A W ε t ε } W dw t + W dw t η dt] whch along wth 1 provs Lmma 5.. Rfrncs 1] A.-B. Cruzro, P. Mallavn, Rnormalzd dffrntal gomtry on path spac: structural quaton, curvatur, J. Funct. Anal ] P.E. Klodn, E. Platn, Numrcal Soluton of Stochastc Dffrntal Equatons, Sprngr-Vrlag, Brln, ] P. Mallavn, Paramétrx tractorll pour un opératur hypollptqu t rpèr mobl stochastqu, C. R. Acad. Sc. Pars, Sér. A-B A41 A44. 4] P. Mallavn, A. Thalmar, Numrcal rror for SDE: asymptotc xpanson and hyprdstrbutons, C.R. Math. Acad. Sc. Pars, Sér. I ] F. Malru, Convrgnc to qulbrum for granular mda quatons and thr Eulr schms, Ann. Appl. Probab ] G.N. Mlstn, Numrcal Intgraton of Stochastc Dffrntal Equatons, n: Math. Appl., vol. 313, Kluwr Acadmc, Dordrcht, Translatd and rvsd from th 1988 Russan orgnal. 7] D.W. Strooc, S.R.S. Varadhan, Multdmnsonal Dffuson Procsss, n: Grundlhrn Math. Wss., vol. 33, Sprngr-Vrlag, Brln, ] C. Vllan, Topcs n Optmal Transportaton, n: Grad. Stud. Math., vol. 58, Amrcal Mathmatcal Socty, Provdnc, RI, 003.