1. Introduction. We consider a d-dimensional stochastic differential equation (SDE) defined by

Transcription

1 SIAM J. CONROL OPIM. Vol. 43, No. 5, pp c 5 Sociey for Indusrial and Applied Mahemaics SENSIIVIY ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES, AND APPLICAION O SOCHASIC OPIMAL CONROL EMMANUEL GOBE AND RÉMI MUNOS Absrac. We consider a mulidimensional diffusion process X α whose dynamics depends on a parameer α. Our firs purpose is o wrie as an expecaion he sensiiviy αjα for he expeced cos Jα =EfX α, in order o evaluae i using Mone Carlo simulaions. his issue arises, for example, from sochasic conrol problems where he conroller is parameerized, which reduces he conrol problem o a parameric opimizaion one or from model misspecificaions in finance. Previous evaluaions of αjα using simulaions were limied o smooh cos funcions f or o diffusion coefficiens no depending on α see Yang and Kushner, SIAM J. Conrol Opim., , pp In his paper, we cover he general case, deriving hree new approaches o evaluae αjα, which we call he Malliavin calculus approach, he adjoin approach, and he maringale approach. o accomplish his, we leverage Iô calculus, Malliavin calculus, and maringale argumens. In he second par of his work, we provide discreizaion procedures o simulae he relevan random variables; hen we analyze heir respecive errors. his analysis proves ha he discreizaion error is essenially linear wih respec o he ime sep. his resul, which was already known in some specific siuaions, appears o be rue in his much wider conex. Finally, we provide numerical experimens in random mechanics and finance and compare he differen mehods in erms of variance, complexiy, compuaional ime, and ime discreizaion error. Key words. sensiiviy analysis, parameerized conrol, Malliavin calculus, weak approximaion AMS subjec classificaions. 9C31, 93E, 6H3 DOI /S Inroducion. We consider a d-dimensional sochasic differenial equaion SDE defined by 1.1 X = x + bs, X s,α ds + q j=1 σ j s, X s,α dw j s, where α is a parameer aking values in A R m and W is a sandard Brownian moion in R q on a filered probabiliy space Ω, F, F, P, wih he usual assumpions on he filraion F. We firs aim a evaluaing he sensiiviy w.r.. α of he expeced cos 1. Jα =E fx, for a given erminal cos f and for a fixed ime. he sensiiviy of more general funcionals including insananeous coss like E g, X d+fx = Eg, X d+ EfX will follow by discreizing he inegral and applying he sensiiviy esimaor for each ime. his evaluaion is a ypical issue raised in various applicaions. A firs example is he analysis of he impac on he expeced cos Jα of a misspecificaion of a sochasic model defined by a SDE wih coefficiens b, x and σ j, x 1 j q. he issue Received by he ediors December 3, ; acceped for publicaion in revised form July 5, 4; published elecronically March 11, 5. hp:// Cenre de Mahémaiques Appliquées, Ecole Polyechnique, 9118 Palaiseau Cedex, France emmanuel.gobe@polyechnique.fr, remi.munos@polyechnique.fr. 1676

2 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1677 may be formulaed by seing b, x, α = b, x + m i=1 α iφ i, x and analogously for σ j, x, α 1 j q, hen compuing he sensiiviies a he poin α =. In finance, misspecificaions in opion pricing procedures usually concern he diffusion coefficiens σ j, x 1 j q he volailiy of he asses. here are also some connecions wih he so-called model risk problem see Cvianić and Karazas [CK99]. Sochasic conrol is anoher field requiring sensiiviy analysis. For insance, if he conrolled SDE is defined by dx = b, X,u d + q j=1 σ j, X,u dw j, he problem is o find he maximal value of EfX among he admissible policies u. In low dimensions say 1 or, numerical mehods based on he dynamic programming principle can be successfully implemened see Kushner and Dupuis [KD1] for some references, bu hey become inefficien in higher dimensions. Alernaively, one can use policy search algorihms see [BB1] and references herein. I consiss in seeking a good policy in a feedback form using a parameric represenaion, ha is, u = u, X,α: in ha case, one pus b, x, α = b, x, u, x, α and σ j, x, α = σ j, x, u, x, α. he policy funcion u, x, α can be parameerized hrough a linear approximaion a linear combinaion of basis funcions or hrough a nonlinear one e.g., wih neural neworks, see Rumelhar and McClelland [RM86] or Haykin [Hay94] for general references. hen, one migh use a sandard parameric opimizaion procedure such as he sochasic gradien mehod or oher sochasic approximaion algorihms see Polyak [Pol87]; Benvenise, Meivier, and Prioure [BMP9]; Kushner and Yin [KY97], which require sensiiviy esimaions of Jα w.r.. α, such as α Jα. his gradien is he quaniy we will focus on in his paper. Since he seing is a priori mulidimensional, we propose a Mone Carlo approach for he numerical compuaions. he evaluaion of Jα is sandard and has been widely sudied. For an inroducion o numerical approximaions of SDEs, we refer he reader o Kloeden and Plaen [KP95], for insance. o our knowledge, here are hree differen approaches o compue α Jα in our conex: 1. he resampling mehod see Glasserman and Yao [GY9], L Ecuyer and Perron [LP94] for insance, which consiss in compuing differen values of Jα for some close values of he parameer α and hen forming some appropriae differences o approximae he derivaives. However, no only is i cosly when he dimension of he parameer α is large, bu i also provides biased esimaors.. he pahwise mehod proposed in our conex by Yang and Kushner [YK91], which consiss in puing he gradien inside he expecaion, involving f and α X. hen, α Jα is expressed as an expecaion see Proposiion 1.1 below and Mone Carlo mehods can be used. One limiaion of his mehod is ha he cos funcion f has o be smooh. 3. he so-called likelihood mehod or score mehod inroduced by Glynn [Gly86, Gly87], Reiman and Weiss [RW86]; see also Broadie and Glasserman [BG96] for applicaions o he compuaion of Greeks in finance, in which he gradien is rewrien as EfX H for some random variable H. here is no uniqueness in his represenaion, since we can add o H any random variables orhogonal o X. Unlike he pahwise mehod, his mehod is no limied o smooh cos funcions. Usually, H is equal o α logpα, X, where pα,. is he densiy w.r.. he Lebesgue measure of he law of X. his has some srong limiaions in our conex since his quaniy is generally unknown. However, Yang and Kushner [YK91] provide explici weighs H, under he resricions ha α concerns only b and no σ j and ha he diffusion coefficien is ellipic, using he Girsanov heorem see Proposiion.6.

3 1678 EMMANUEL GOBE AND RÉMI MUNOS A firs purpose of his work is o handle more general siuaions where boh coefficiens defining he SDE 1.1 depend on α. o address his issue, we provide hree new approaches o express he sensiiviy of Jα wih respec o α. 1. he firs one is an exension of he likelihood approach mehod o he case of diffusion coefficiens depending on α. I uses a direc inegraion-by-pars formula of he Malliavin calculus. his idea has been used recenly in a financial conex in he paper by Fournié e al. [FLL + 99] o compue opion prices sensiiviies. hese echniques have also been used efficienly by he firs auhor o derive asympoic properies of saisical procedures when we esimae parameers defining a SDE see [Gob1b, Gob]. Acually, our rue conribuion concerns essenially a siuaion where ellipiciy is replaced by a weaker bu sandard nondegeneracy condiion, which addresses random mechanics problems or porfolio opimizaion problems in finance.. he second approach is raher differen from previous mehods. Indeed, we iniially focus on he adjoin poin of view see Bensoussan [Ben88] or Peng [Pen9] o finally derive new formulae, involving again some inegraion-bypars formula, bu wrien in a simple way using only Iô s calculus. In sochasic conrol problems, adjoin processes are relaed o backward SDEs see Yong and Zhou [YZ99], e.g., and heir simulaion is an exremely difficul and cosly ask. Here, we circumven his difficuly since we only need o express hem as explici condiional expecaions, which is feasible. 3. he hird approach follows from maringale argumens applied o he expeced cos and leads o an original represenaion, which appears o be surprisingly simple. o compare hese new mehods wih he previous ones, we will measure in secion 5, on he one hand, he variance of he random variables involved in he resuling formulae for α Jα, and on he oher hand, he compuaional ime. Surprisingly, he hree mehods ha we propose behave similarly in erms of variance, bu he mos efficien in erms of compuaional ime is cerainly he maringale approach see ables 5.1, 5., 5.3, 5.4, and 5.5. Anoher elemen of comparison is he influence of he ime sep h, which is used o approximaely simulae he random variables. he analysis of hese discreizaion errors is he second significan par of his work. he relevan random variables are essenially wrien as he produc of he cos funcion fx by a random variable H, and simulaions are based on Euler schemes. Alhough H has a complex form, we firs propose an approximaion algorihm and hen we analyze he induced error w.r.. he ime sep h. his par of he paper is original: previous resuls in he lieraure concern he approximaion of EfX see Bally and alay [B96a] or more generally of some smooh funcionals of X see Kohasu-Higa and Peersson [KHP], [KHP]. Here, regarding he echniques, we improve esimaes given in [KHP] since we do no need o add a small perurbaion o he processes. Our mulidimensional framework also raises exra difficulies compared o [KHP], and we develop specific localizaion echniques ha are ineresing for hemselves. Ouline of he paper. In he following, we make some assumpions and define he noaions which will be used hroughou he paper. We also recall he pahwise approach in Proposiion 1.1. In secion, afer giving some sandard facs on he Malliavin calculus, we develop our hree approaches o compuing he sensiiviy of Jα w.r.. α: hese are he so-called Malliavin calculus approach Proposiions.5 and.8, he adjoin approach heorem.11, and he maringale approach heorem.1. In secion 3, we provide simulaion procedures o compue α Jα by

4 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1679 he usual Mone Carlo approach using he mehods developed before and analyze he influence of he ime sep h used for Euler-ype schemes. A significan par of he paper covers hese analyses which have never been developed before in he lieraure. he approximaion resuls are saed in heorems 3.1, 3., 3.4, and 3.5, while heir proofs are posponed o secion 4. Finally, numerical experimens in secion 5 illusrae he developed mehods: we compare he compuaional ime, he complexiy, he variance, and he ime discreizaion error of he esimaors on many examples borrowed from finance and conrol. Assumpions. In our applicaions, he parameer is a priori mulidimensional, bu since in he following we will look a sensiiviies w.r.. α coordinaewise, i is no a resricion o assume ha he parameer space A is a subse of R m = 1. he process defined in 1.1 depends on he parameer α, bu we deliberaely omi his dependence in he noaion. Furhermore, he iniial condiion X = x is fixed hroughou he paper. We noe σ j, he jh column vecor of σ. o sudy he sensiiviy of J defined in 1. w.r.. α, we may assume ha coefficiens are smooh enough: in wha follows, k is an ineger greaer han. Assumpion R k. he funcions b and σ are of class C 1 w.r.. he variables, x, α, and for some η>, he following Hölder coninuiy condiion holds: g, x, α g, x, α sup,x,α,α [, ] R d A A α α η < for g = α b and g = α σ. Furhermore, for any α A, he funcions b,,α, σ,,α, α b,,α, and α σ,,α are of class C k/,k w.r.., x; he funcions α b and α σ are uniformly bounded in, x, α, and he derivaives of b, σ, α b, and α σ w.r.., x are uniformly bounded as well. Noe ha b and σ may be unbounded. We do no asser ha he assumpion above is he weakes possible, bu i is sufficien for our purpose. A several places, he diffusion coefficien will be required o be uniformly ellipic, in he following sense. Assumpion E. σ is a squared marix q = d such ha he marix σσ saisfies a uniform ellipiciy condiion:, x [,] R d, [σσ ], x, α µ min I d for a real number µ min >. Noaion. Sensiiviy esimaors. o clarify he connecion beween our mehods and he esimaors H which are derived, we will wrie H P ah. for he pahwise approach Proposiion 1.1, H Mall.Ell. resp., H Mall.Gen. for he Malliavin calculus approach in he ellipic case resp., in he general case Proposiions.5 and.8, H b,adj. and H σ,adj. for he adjoin approach heorem.11, and H Mar. for he maringale approach heorem.1. he subscrip refers o he ime in he expeced cos 1.. heir approximaions using some discreizaion procedure wih N ime seps will be denoed H P ah.,n, H Mall.Ell.,N, and so on. Differeniaion. As usual, derivaives w.r.. α will be simply denoed wih a do, for insance, α J = J. If no ambiguiy is possible, we will omi o wrie explicily he parameer α in b, σ j. We adop he following usual convenion on he gradiens: if ψ : R p R p1 is a differeniable funcion, is gradien x ψx = x1 ψx,..., xp ψx akes values in R p1 R p.a many places, x ψx will simply be denoed ψ x.

5 168 EMMANUEL GOBE AND RÉMI MUNOS Linear algebra. he rh column of a marix A will be denoed A r or A r, if A is a ime dependen marix, and he rh elemen of a vecor a will be denoed a r or a r, if a is a ime dependen vecor. A sands for he ranspose of A. For a marix A, he marix obained by keeping only he las r rows resp., he las r columns will be denoed Π R r A resp., Π C r A. For i {1,...,d}, we se e i = 1, where 1 is he ih coordinae. Consans. We will keep he same noaion K for all finie, nonnegaive, and nondecreasing funcions: hey do no depend on x, he funcion f, or furher discreizaion seps h, bu hey may depend on he coefficiens b and σ. he generic noaion Kx, sands for any funcion bounded by K 1 + x Q for Q. When a funcion gs, x, α is evaluaed a x = Xs α, we may someimes use he shor noaion g s if no ambiguiy is possible. For insance, 1.1 may be wrien as X = x + b sds + q j=1 σ j,sdws j. Oher processes relaed o X. o he diffusion X under R, we may associae is flow, i.e., he Jacobian marix Y := x X, he inverse of is flow Z = Y 1, and he pahwise derivaive of X w.r.. α, which we denoe Ẋ see Kunia [Kun84]. hese processes solve Y =I d + Z =I d Ẋ = b s Y s ds + Z s b s q j=1 σ j,s Y s dw j s, q σ j,s ds j=1 ḃs + b s Ẋs ds + q j=1 q j=1 Z s σ j,s dw j s, σj,s + σ j,s Ẋs dw j s. Acually, since he process Ẋ saisfies a linear equaion, i can also simply be wrien using Y and Z apply heorem 56 from p. 71 of Proer [Pro9]: q q 1.6 Ẋ = Y Z s ḃ s σ j,s σ j,s ds + σ j,s dws j. j=1 If f is coninuously differeniable wih an appropriae growh condiion in order o apply he Lebesgue differeniaion heorem, one immediaely obains he following resul see also Yang and Kushner [YK91], which we call he pahwise approach. Proposiion 1.1. Assume R. One has Jα =E H P ah. wih H P ah. = f X Ẋ. Hence, he gradien can sill be wrien as an expecaion, which is crucial for a Mone Carlo evaluaion. One purpose of he paper is o exend his resul o he case of nondiffereniable funcions, by essenially wriing Jα =E fx H for some random variable H. In wha follows, we will make wo ypes of assumpion on f. Assumpion H. f is a bounded measurable funcion. Acually, he above boundedness propery of f is no imporan, since in wha follows, we essenially use he fac ha he random variable fx belongs o any L p. However, his assumpion simplifies he analysis. j=1

6 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1681 Assumpion H. f is a bounded measurable funcion and saisfies he following coninuiy esimae for p > 1: fx fx L p d < +. his L p -smoohness assumpion of fx fx is obviously saisfied for uniformly Hölder funcions wih exponen β, bu also for some nonsmooh funcions, such as he indicaor funcion of a domain. Proposiion 1.. Le D be a domain of R d : suppose ha eiher i has a compac and smooh boundary say, of class C ; see [G77], or i is a convex polyhedron D = I i=1 D i, where D i 1 i I are half-spaces. Assume E, R, and bounded coefficiens b and σ. hen, he funcion f = 1 D saisfies he assumpion H for any p > 1. Proof. Since fx fx p L E 1 DX p 1 D X PX D, X / D +PX / D, X D, we only need o prove ha PX D, X / D K β wih β>. Now, recall he sandard exponenial inequaliy P X u x δ K exp c δ u wih c> available for u ],] and δ see, e.g., Lemma 4.1 in [Gob]. Combining his wih he Markov propery, i follows ha PX D, X / D K E1 X / D exp c d X,D c. hen, a direc esimaion of he above expecaion using in paricular a Gaussian upper bound for he densiy of he law of X see Friedman [Fri64] yields easily he required esimae wih β = 1 see Lemma.8 in [Gob1a] for deails.. Sensiiviy formulae. In his secion, we presen hree differen approaches o evaluae Jα. Before his, we inroduce he Malliavin calculus maerial necessary o our compuaions..1. Some basic resuls on he Malliavin calculus. he reader may refer o Nualar [Nua95] secion. for he case of diffusion processes for a deailed exposiion of his secion. Pu H = L [,], R q : we will consider elemens of H wrien as a row vecor. For h. H, denoe by W h he Wiener sochasic inegral h dw. Le S denoe he class of random variables of he form F = fw h 1,...,Wh N, where f is a C -funcion wih derivaives having a polynomial growh, h 1,...,h N H N and N 1. For F S, we define DF =D F := D 1 F,...,D q F [, ], is derivaive, as he H-valued random variable given by D F = N i=1 x i fw h 1,..., W h N h i. he operaor D is closable as an operaor from L p Ω o L p Ω, H, for any p 1. Is domain is denoed by D 1,p w.r.. he norm F 1,p = [E F p + E DF p H ]1/p. We can define he ieraion of he operaor D in such a way ha for a smooh random variable F, he derivaive D k F is a random variable wih values on H k. As in he case k = 1, he operaor D k is closable from S L p Ω ino L p Ω; H k, p 1. If we define he norm F k,p =[E F p + k j=1 E Dj F p H ] 1/p, j we denoe is domain by D k,p. Finally, se D k, = p 1 D k,p and D = k,p 1 D k,p. One has he following chain rule propery. Proposiion.1. Fix p 1. For f Cb 1Rd, R and F =F 1,...,F d a random vecor whose componens belong o D 1,p, ff D 1,p and for, one has

7 168 EMMANUEL GOBE AND RÉMI MUNOS D ff = f F D F, wih he noaion D F 1 D F =. D F d R d R q. We now inroduce δ, he Skorohod inegral, defined as he adjoin operaor of D. Definiion.. δ is a linear operaor on L [,] Ω, R q wih values in L Ω such ha 1. he domain of δ denoed by Domδ is he se of processes u L [,] Ω, R q such ha E D F u d cu F L for any F D 1,.. if u belongs o Domδ, hen δu is he elemen of L Ω characerized by he inegraion-by-pars formula F D 1,, E Fδu.1 = E D F u d. In he following proposiion, we ouline a few properies of he Skorohod inegral. Proposiion he space of weakly differeniable H-valued variables D 1, H belongs o Domδ.. If u is an adaped process belonging o L [,] Ω, R q, hen he Skorohod inegral and he Iô inegral coincide: δu = u dw. 3. If F belongs o D 1,, hen for any u Domδ such ha EF u d < +, one has. δf u =Fδu D F u d, whenever he righ-hand side above belongs o L Ω. Concerning he soluion of SDEs, i is well known ha under R k k for any, he random variable X resp., Y, Z, and Ẋ belongs o D k, resp., D k 1,. Furhermore, one has he following esimaes: E sup D r1,...,r k U p K,x for U = X wih 1 k k or U = Y,Z, Ẋ wih 1 k k 1. Besides, D s X is given by.3 D s X = Y Z s σs, X s 1 s. Finally, we recall some sandard resuls relaed o he inegraion-by-pars formulae. he Malliavin covariance marix of a smooh random variable F is defined by.4 γ F = D F [D F ] d. Proposiion.4. Le γ be a muli-index, F be a random variable in D k1, such ha deγ F is almos surely posiive wih 1/deγ F p 1 L p and G belongs o D k,. hen for any smooh funcion g wih polynomial growh, provided ha k 1 and k are large enough depending on γ, here exiss a random variable H γ F, G in any L p such ha E[ γ gf G] =E[gF H γ F, G].

8 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1683 Moreover, for any arbirary even A we have H γ F, G1 A L p C [γ F ] 1 1 A p3 L q 3 F p1 k 1,q 1 G k,q for some consans C, p 1,p 3,q 1,q,q 3 depending on p and γ. Proof. See Proposiions 3..1 and 3.. in Nualar [Nua98, pp ] when A = Ω. For any oher even A, see Proposiion.4 from Bally and alay [B96a]. he consrucion of H γ F, G is based on he equaliy.1 and involves ieraed Skorohod inegrals. We do no really need o make i explici a his sage... Firs approach: Direc Malliavin calculus compuaions. Here, he guiding idea is o sar from Proposiion 1.1 and apply resuls like Proposiion.4 o ge Jα =EfX H. Neverheless, here are several ways o do his, depending on wheher he diffusion coefficien is ellipic see also [FLL + 99] in ha siuaion or no...1. Ellipic case. Consider firs ha he assumpion E is fulfilled. Proposiion.5. Assume R, E, and H. One has Jα =E H Mall.Ell., where H Mall.Ell. = 1 fx δ [σ 1 Y Z Ẋ ] belongs o p 1 L p. Proof. We can consider ha f is smooh, he general case being obained using an L -approximaion of f wih some smooh and compacly suppored funcions. As a consequence of.3 and Assumpion E, D X is inverible for any [,]: hus, for such, using he chain rule Proposiion.1, one ges ha f X =D fx σ 1 Y Z. Inegraing in ime over [,] and using Proposiion 1.1, one ges ha Jα = 1 d ED fx σ 1 Y Z Ẋ. An applicaion of he relaion.1 complees he proof of Proposiion.5 he L p -esimaes follow from Proposiion.4. When he parameer eners he drif coefficien only, he laws of X for wo differen values of α are equivalen owing o he Girsanov heorem. Exploiing his possible change of measure direcly, a simplified expression for Jα can be found: his is he likelihood raio mehod or score mehod from Kushner and Yang [YK91]. Proposiion.6. Assume R, E, and H. Suppose ha he parameer of ineres α is no in he diffusion coefficien. hen, one has Jα =E fx [σ 1 ḃ ] dw. Proof. Insead of using he Girsanov heorem, we leverage he paricular form of Ẋ given in 1.6 o prove his. Indeed, f X Ẋ = f X Y Z ḃd = d D fx [σ 1 ḃ ], and he resul follows using General nondegenerae case. here are many siuaions where he ellipiciy Assumpion E is oo sringen and canno be fulfilled. o illusrae his, le us rewrie he SDE in he following way, spliing is srucure ino wo pars:.5 ds dx = = dv bs, X,α b V, X,α d + σs, X,α σ V, X,α dw.

9 1684 EMMANUEL GOBE AND RÉMI MUNOS Here, S is d r-dimensional, V r-dimensional, and he dimension of W is arbirary. he cos funcion of ineres may involve only he value of V : Jα =EfV. Noe ha considering r = d reduces o he previous siuaion. We now give wo examples ha moivae he saemen of Proposiion.7 below. a In random mechanics see Krée and Soize [KS86], he pair posiion/velociy dx v d dx = = canno saisfy an ellipiciy condiion, bu dv weaker assumpions such as hypoellipiciy are more realisic. b For porfolio opimizaion in finance for a recen review, see, e.g., Runggaldier [Run], r usually equals 1. S describes he dynamic of he risky asses, while V is he wealh process, corresponding o he value of a self-financed porfolio invesed in a nonrisky asse wih insananeous reurn r, S and in he asses S w.r.. he sraegy ξ = {ξ i, X : 1 i d 1} : dv = ξ, X ds +V ξ, X S r, S d see e.g. Karazas and Shreve [KS98]. I is clear ha he resuling diffusion coefficien S for he whole process X = canno saisfy an ellipiciy condiion. V Neverheless, requiring ha he marix σ V σv, x saisfy an ellipiciy ype condiion is no very resricing in ha framework. We se γ for he Malliavin covariance marix of V : γ = D V [D V ] d. his allows o reformulae Assumpion E as he following. Assumpion E. deγ is almos surely posiive and for any p 1, one has 1/deγ L p < +. We now bring ogeher sandard resuls relaed o Assumpion E. Proposiion.7. Assumpion E is fulfilled in he following siuaions. 1. Hypoellipic case wih r = d under R. he Lie algebra generaed by he vecor fields + A, x := + d i=1 b 1 q j=1 σ j σ j i, x xi, A j, x := d i=1 σ i,j, x xi for 1 j q spans R d+1 a he poin,x : dim span Lie + A,A j, 1 j q,x =d +1.. Parially ellipic case wih r 1 under R. For a real number µ min >, one has x R d, [σ V σ V ], x, α µ min I d. Proof. he saemen 1 is sandard and we refer o Caiaux and Mesnager [CM] for a recen accoun on he subjec. he saemen is also sandard: see, for insance, he argumens in Nualar [Nua98, pp ]. Now, we are in a posiion o give a sensiiviy formula under E. Proposiion.8. Assume R, E, and H. One has Jα =E H Mall.Gen. where H Mall.Gen. = fv δ V γ 1 D V belongs o p 1 L p. Proof. Assumpion E validaes see Nualar [Nua98, Proposiion 3..1] he following compuaions, adaped from he ones used for Proposiion.5. he chain

10 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1685 rule propery yields f V = D fv [D V ] γ 1 d, and hus Ef V V = E D fv [D V ] γ 1 V d. Proposiion.8 now follows from.1. Proposiion.8 is also valid under E in he case r = d, bu he formula in Proposiion.5 is acually a bi simpler o implemen..3. A second approach based on he adjoin poin of view Anoher represenaion of he sensiiviy of Jα. If we se u, x = E fx X = x, omiing o indicae he dependence w.r.. α, we have ha Jα defined in 1. equals u,x. Under smoohness assumpions on b and σ and he nondegeneracy hypohesis on he infiniesimal generaor of X, i is well known see Caiaux and Mesnager [CM] ha u is he smooh soluion of he parial differenial equaion PDE u, x+ d b i, x xi u, x+ 1 i=1 u,x=fx. d [σσ ] i,j, x x i,x j u, x = for<, i,j=1 Our purpose is o give anoher expression for Jα of Proposiion 1.1. he idea is simple: i consiss in formally differeniaing he PDE above w.r.. α and in reinerpreing he derivaive as an expecaion. his is now saed and jusified rigorously. Lemma.9. Assume R 3, E, and H. One has d Jα = E ḃ i, xi u, X + 1 d [σσ ] i,j, x i,x j u, X d. i=1 i,j=1 Proof. his is a sandard fac ha under R 3 and E, u is wice differeniable w.r.. x see he argumens of Lemma.1 below, where he proof is skeched. he echnical difficuly in he following compuaions comes from he possible explosion of derivaives of u for close o, when f is nonsmooh. For his reason, we firs prove useful uniform esimaes: for any muli-index γ wih γ, any smooh random variable G D, and any parameers α and α, one has.6 sup [, [ E[G γ xu, X α ] K,x f G γ γ,p. Indeed, for /, firs apply Proposiion.4: hen, use u, x f combined wih some specific esimaes for H γ X α,g L p K,x G γ γ,p available under he ellipiciy condiion E see heorem 1. and Corollary 3.7 in Kusuoka and Sroock [KS84], or secion 4.1. in [Gob] for a brief review. For /, noe ha using he Markov propery, one has xu, γ x = xe γ u +,X,x + = 1 γ γ E x γ u +,X,x + G γ + wih G γ + D + γ γ, and Xs,y s sanding for he process saring from y a ime. Again applying he inegraion-by-pars formula wih he ellipic esimaes gives xu, γ x K,x f and.6 follows [ + γ ]

11 1686 EMMANUEL GOBE AND RÉMI MUNOS since + 4. Now, for ɛ R, he difference Jα + ɛ Jα equals E fx α+ɛ fx α = E u,x α+ɛ u,x α+ɛ d = E u, X α+ɛ + b i, X α+ɛ,α+ ɛ xi u, X α+ɛ i=1 + 1 d [σσ ] i,j, X α+ɛ,α+ ɛ x i,x j u, X α+ɛ d i,j=1 d = E b i, X α+ɛ,α+ ɛ b i, X α+ɛ,α xi u, X α+ɛ + 1 i=1 d [σσ ] i,j, X α+ɛ,α+ ɛ [σσ ] i,j, X α+ɛ,α x i,x j u, X α+ɛ d, i,j=1 where a he las equaliy we used he PDE solved by u o remove he erm u.now, divide by ɛ and ake is limi o : he resul follows owing o he uniform esimaes.6. Noe ha he formulaion of Lemma.9 is srongly relaed o a form of he sochasic maximum principle he Ponryagin principle for opimal conrol problems: he processes [ xi u, X ] i < and [ x i,x j u, X ] i,j < are he so-called adjoin processes see Bensoussan [Ben88] for convex conrol domains, or more generally Peng [Pen9] and solve backward SDEs. Usually in hese problems, he funcion f is smooh. Here, since he law of X has a smooh densiy w.r.. he Lebesgue measure, we can remove he regulariy condiion on f. Noe also ha Lemma.9 remains valid under a hypoellipiciy hypohesis condiion 1 in Proposiion.7. However, he derivaion of racable formulae below relies srongly on he ellipiciy propery..3.. ransformaion using Iô Malliavin inegraion-by-pars formulae. he aim of his secion is o ransform he expression for Jα in erms of explici quaniies. o remove he nonexplici erms xi u and x i,x j u, we may use some inegraion-by-pars formulae, bu here, o keep more racable expressions, we are going o derive Bismu-ype formulae, i.e., involving only Iô inegrals insead of Skorohod inegrals see Bismu [Bis84]; Elworhy, Le Jan, and Li [EJL99]; and references herein, using a maringale argumen see also halmaier [ha97] or, more recenly, Picard [Pic]. In he cied references, his approach has been used o compue esimaes of he gradien of u. Here, we exend i o suppor higher derivaives. he basic ool is given by he following lemma. Lemma.1. Assume R, E, and H and define M = u, X Y for <. hen M =M < is an R 1 R d -valued maringale. Proof. Firs, we jusify ha u is coninuously differeniable w.r. x under R and E. If f is smooh, his is clear even wihou E, bu.1 below also shows ha under E, u can be expressed wihou he derivaive of f. his easily leads o our asserion see he proof of Proposiion 3. in [FLL + 99]. Now, he Markov propery ensures ha u, X,x < is a maringale for any x R d. Hence, is derivaive w.r.. x i.e., M < is also a maringale see Arnaudon and halmaier [A98].

12 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1687 We now sae a heorem which, if combined wih Lemma.9, leads o an alernaive represenaion for Jα. heorem.11. Assume R 3 and E. Under H, one has.7 E d i=1 ḃ i, xi u, X d = EH b,adj., where H b,adj. Z = fx d ḃ Under H, one has [σ 1 s Y s ] dw s belongs o p 1 Lp..8 E d i,j=1 [σσ ] i,j, x i,x j u, X d = EH σ,adj., where H σ,adj. = d [ ei Z d i,j=1 + [σσ e j ] i,j, [fx fx ] [σ 1 s Y s ] dw s ] + ei { x [ Z [ Z + + [σ 1 s Y s ] dw s ] [σ 1 s Y s ] dw s ] Z e j } belongs o p<p L p. Proof. Equaliy.7. Firs, Clark and Ocone s formula [Nua95, p. 4] gives uτ,x τ =u, X + τ ED s[uτ,x τ ] F s dw s for τ<. Using.3 and he maringale propery of Lemma.1, we ge ED s [uτ,x τ ] F s = Eu τ,x τ Y τ Z s σ s F s =u s, X s σ s. Hence, i gives an explici form o he predicable represenaion heorem:.9 τ τ uτ,x τ =u, X + u s, X s σ s dw s he case τ = is obained by passing o he limi. Noe ha his represenaion holds under R. Since u, X Y < is a maringale, we obain ha u 1, X Y = E u s, X s Y s ds F [ ][ 1 ] = E u s, X s σ s dw s [σs 1 Y s ] dw s F.1 = E = E fx u, X fx [ [ [σ 1 s Y s ] dw s ] F [σ 1 s Y s ] dw s ] F,

13 1688 EMMANUEL GOBE AND RÉMI MUNOS where for he hird equaliy we used.9 wih τ = and u,x =fx. Now he proof of.7 is sraighforward. Equaliy.8. Noe ha a sligh modificaion of he preceding argumens namely, inegraing over [, + /] insead of [, ] and applying.9 wih τ = + / e i ] [σs 1 Y s ] dw s F. Differenia- leads o xi u, X =E u +,X + [Z ing w.r.. x on boh sides and using.1 yields + xi u, X Y = E u + +E u [ = E [fx fx ],X +,X + e i +[fx fx ] x Y + e i + [ e i Z {[ x ] [σs 1 Y s ] e i dw s + {[ Z Z [ Z [σ 1 s Y s ] dw s ] F [σ 1 s Y s ] dw s ]} F + [σ 1 s Y s ] dw s ]} F [σ 1 s Y s ] dw s ] noe ha he fx erms have no conribuion in he expecaion. Rearranging his las expression leads o.8. he L p -esimaes can be jusified using he generalized Minkowski inequaliy and sandard esimaes from he sochasic calculus:.11 H b,adj. L p H σ,adj. L p K,x f ḃ, X Z fx fx L p [σs 1 Y s ] dw s d K,x L p d f d, for p<p <p. Remark.1. he fx erms in H σ,adj. seem o be crucial o ensure is L p inegrabiliy: numerical experimens in secion 5 illusrae his fac..4. A hird approach using maringales. We emphasize he dependence on α of he expeced cos by denoing uα,, x =EfX α Xα = x: hence, Jα = uα,,x. From he esimaes proved in Lemma.9, his is a differeniable funcion w.r.. α and one has uα,, x K,x f and u α,, x K,x f. Furhermore, using heorem.11 and he L p -esimaes.11 under H, one ges uα,, x K,x [ f + fx,x fx,x s L p s for p <p. Consequenly, if we pu gr =E uα, r, X r, we easily obain gr K,x[ f r + fx fx s L p r s ds] and hus, lim r gr =. For any r s, one has Euα, r, X r = Euα, s, X s = 1 r Euα, s, X s ds r ] ds

14 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1689 using he Markov propery; hence, by differeniaion w.r.. α, one ges.1 E uα, r, X r = 1 r = 1 r r r ds E uα, s, X s +u α, s, X s Ẋs u α, r, X r Ẋr ds E uα, s, X s +u α, s, X s [Ẋs Y s Z r Ẋ r ], where we used a he las equaliy he maringale propery of M = u α,, X Y beween = s and = r see Lemma.1. Now, pu hr = 1 r u r ds E α, s, X s [Ẋs Y s Z r Ẋ r ] : one has derived he following inegral equaion:.13 g = 1 gs ds + h. Before solving i, we express hr using only f: for his, we use he predicable represenaion.9, which immediaely gives.14 hr = 1 r E fx fx r r [σ 1 s Ẋs Y s Z r Ẋ r ] dw s. Noe again ha he erm wih fx r has no conribuion and is pu only o jusify ha hr K,x fx fx r L p use he Burkholder Davis Gundy inequaliies and sraighforward upper bounds for Ẋs Y s Z r Ẋ r L q K,x s r, from which we deduce ha he inegral h d is convergen because of H. o solve 1 he inegral equaion above, noe ha [ gs ds] = h, and hus by inegraion, we have 1 gs ds = C hr r dr. he consan C equals since boh inegrals in he previous equaliy converge o when goes o use lim g = and H. Plug his new equaliy ino.13, use.14, and ake = wih Ẋ = o ge he following represenaion for Jα: his is he main resul of his secion. heorem.1. Assume R, E, and H. hen, one has Jα =EH Mar. wih.15 H Mar. = fx + [σs 1 Ẋ s ] dw s dr [fx fx r ] r r [σ 1 s Ẋs Y s Z r Ẋ r ] dw s. Furhermore, he random variable H Mar. belongs o p<p L p. his mehod is called he maringale approach because i is based on he equaliy.1, which is a consequence of he maringale propery of [ uα, s, X s +u α, s, X s Ẋs] s<. Proof. Wha remains o be proved is he L p esimae of H Mar. : his can be easily obained by combining Minkowski s inequaliy, Hölder s inequaliy, Assumpion H, and sandard sochasic calculus inequaliies as before.

15 169 EMMANUEL GOBE AND RÉMI MUNOS Remark.. When he parameer is no involved in he diffusion coefficien, i is easy o see ha he improved esimae Ẋs Y s Z r Ẋ r Lq K,xs r is available: hus, his allows us o remove fx r erms in he expression of H Mar. wihou changing he finieness of he L p -norm of he new H Mar.. In oher words, only Assumpion H is needed. Besides, when α is only in he drif coefficien and hese fx r erms are suppressed, his represenaion coincides wih ha of heorem.11. Indeed, le us wrie P r = r [σ 1 s Ẋs Y s Z r Ẋ r ] dw s = r [σ 1 s Ẋ s ] dw s [Z r Ẋ r ] r [σ 1 s Y s ] dw s := P 1,r P,r, where P 1,r = [σ 1 s Ẋ s ] dw s [Z r Ẋ r ] [σ 1 s Y s ] dw s and P,r = r [σ 1 s Ẋ s ] dw s [Z r Ẋ r ] r [σ 1 s Y s ] dw s. From he fac ha Z r Ẋ r = r Z sḃsds see 1.6, one ges dp,r = [Z r ḃ r ] r [σ 1 Y ] dw dr, hence P,r is of bounded variaion. P 1,r is also of bounded variaion, since Z r Ẋ r is. hus, one obains dp r = ḃr Zr r [σ 1 Y ] dw dr: furhermore, since P =, one has P r L p K,x r 3/. Using an inegraion-by-pars formula in.15 finally complees our asserion: H Mar. = fx 1 P + P r r dr = fx dp r r = H b,adj.. Consequenly, his maringale approach does no provide any new elemens when he parameer is no in he diffusion coefficien. On he conrary, if σ depends on α, he represenaion wih he adjoin poin of view is differen from he maringale one see numerical experimens. However, we mus admi ha his maringale approach remains somewha myserious o us. 3. Mone Carlo simulaion and analysis of he discreizaion error. In his secion, we discuss he numerical implemenaion of he formulae derived in his paper o compue he sensiiviy of Jα w.r.. α. hese formulae are wrien as expecaions of some funcionals of he process X and relaed ones: a sandard way o proceed consiss in drawing independen simulaions, approximaing he funcional using Euler schemes, and averaging independen samples of he resuling funcional o ge an esimaion of he expecaion see secion 5. Here, we focus on he impac of he ime sep h = /N N is he number of discreizaion imes in he regular mesh of he inerval [,] in he simulaion of he funcional: i is well known ha for he evaluaion of EfX, he discreizaion error using an Euler scheme is of order h see Bally and alay [B96a] for measurable funcions f, or Kohasu-Higa and Peersson [KHP] if f is a disribuion and for more general discreizaion schemes. We recall ha he error on he processes called he srong error is much easier o analyze han he one on he expecaions he weak error: he firs one is essenially of order h see [KP95] bu his is no relevan for he curren issues. Besides, he quaniy of ineres here has a more complex srucure ha is essenially EfX H, where H is one of he random variables resuling from our compuaions. In general, H involves Iô or Skorohod inegrals: our firs purpose is o give some approximaion procedure o simulae hese weighs using only he incremens of he Brownian moion compued along he regular mesh wih ime sep h. Our second purpose is o analyze he error induced by his discreizaion procedure: generally speaking, he weak error is sill a mos linear w.r.. h, as for EfX. he proofs are quie inricae and we pospone hem o secion 4. For he sake of clariy, we assume R, ha is, b and σ of class C, bu approximaion resuls only depend on a finie number of coefficiens derivaives.

16 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1691 Approximaion procedure. We consider a regular mesh of he inerval [,], wih N discreizaion imes i = ih, where h = /N is he ime sep. Denoe φ = sup{ i : i }. he processes we need o simulae are essenially X, Y,Z,Ẋ, and we approximae hem using a sandard Euler scheme as follows: 3.1 X N = x + 3. Y N =I d Z N =I d bφs,x N φs ds + q j=1 b φs,x N φs Y N φs ds + q Z N φs b σ j φs,x N φs dw j s, j=1 q q σ j φs,xφs N ds j=1 σ jφs,x N φs Y N φs dw j s, j=1 Z N φs σ jφs,x N φs dw j s, 3.4 Ẋ N = N ḃφs,xφs +b φs,xφs φs N ẊN ds q + σ j φs,xφs N +σ jφs,xφs N ẊN φs j=1 dw j s. Noe ha only he incremens W j i+1 W j i ;1 j q i N 1 of he Brownian moion are needed o ge values of X N, Z N, Y N, Ẋ N a imes i i N Pahwise approach. heorem 3.1. Assume R. hen, one has Jα E f X N ẊN C, x, fh, under eiher one of he wo following assumpions on f and X: A1 f is of class C 4 b : one may pu C, x, f =K,x 1 α 4 α f in ha case. A f is coninuously differeniable wih a bounded gradien and he nondegeneracy condiion E holds: in ha case, C, x, f may be se o K,x f 1/deγ q L p for some posiive numbers p and q. Noe ha in he case A1, only hree addiional derivaives of he funcion f are required o ge he order 1 w.r.. h: his is a sligh improvemen compared o resuls in alay and ubaro [L9], where four derivaives are needed. 3.. Malliavin calculus approach Ellipic case. One needs o define he approximaion for he random variable H Mall.Ell. := δ [σ 1,X Y Z Ẋ ] involved in Proposiion.5. Basic

17 169 EMMANUEL GOBE AND RÉMI MUNOS algebra using he equaliy. gives H Mall.Ell. = = d δ [σ 1,X Y ] i [Z Ẋ ] i i=1 d [Z Ẋ ] i [σ 1 s, X s Y s ] i dw s i=1 d i=1 D s [Z Ẋ ] i [σ 1 s, X s Y s ] i ds. he new quaniies involved are D s Z j,k, and D s Ẋ k,. We now indicae how o simulae hem. he R d -valued process X Ẋ forms a new sochasic differenial equaion see 1.5: we denoe he flow of his exended sysem by Ŷ and is inverse by Ẑ. As we did for Y and Z, we can define heir Euler scheme as in 3. and 3.3, which we denoe Ŷ N and ẐN. he Malliavin derivaive of his sysem follows from.3. Hence, one has 3.5 D s Ẋ =Π R ˆ d Y Ẑ s.. σ j s, X s. σ j s, X s +σ j s, X s Ẋs.., and we naurally approximae i by 3.6 [D s Ẋ ] N =Π R d Ŷ N ẐN s. σ j s, Xs N. σ j s, X N s +σ j s, XN s ẊN s... he same approach can be developed for he ch column of he ranspose of Z, since X Z c forms a new SDE see 1.4: he associaed flow and is inverse, respecively denoed Ŷ c and Ẑc, enable us o derive a simple expression for D s [Z c ] analogously o 3.5 and 3.6. As a consequence, one ges 3.7 D s [Z Ẋ ] i =1 s A βj,i, B βj,i,s, where A βj,i, and B βj,i,s are given by some appropriae coordinaes of he processes Ŷ,Ŷ c 1 c d on one hand; and Ẑs, Ẑc s 1 c d, σ j s, X s, σ j s, X s, σ j s, X s, Ẋ s, Z s on he oher hand; in order o keep hings clear, we do no develop heir expression furher we refer o a echnical repor [GM3] for full deails. Finally, we approximae H Mall.Ell. by H Mall.Ell.,N = d [Z N i=1 d i=1 j Ẋ N ] i [σ 1 φs,xφs N Y φs N ] i dw s A N βj,i, BN βj,i,φs [σ 1 φs,xφs N Y φs N ] i ds, j which can be simulaed using only he Brownian incremens as before. We now sae ha he approximaion above converges a order 1 w.r.. he ime sep. heorem 3.. Assume R, E, and H. For some q, one has Jα E fx N H Mall.Ell.,N K,x f q h.

18 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1693 Remark 3.1. Insead of basing he compuaions of Malliavin derivaives for differen adaped processes U, D i U j i j N, on he equaliy.3, an alernaive approach would be o derive equaions solved by D i U i and hen approximae hem wih a discreizaion scheme for each i. However, his approach would require essenially ON operaions, insead of ON in our case General nondegenerae case. Denoe by d1,d he d 1 d marix wih for each elemen. Simple algebra yields ha V γ 1 D sv is equal o where V γ 1 ΠR r Y Z s σs, X s = 1,d r V F i = = Y d r,d r d r,r r,d r γ 1 d r,d r d r,r r,d r γ 1 d F i [Z s σs, X s ] i, i=1 d r,1 V i = j Y Z s σs, X s U κi,j, γ 1 βi,j,γi,j, wih he random variables U κi,j, i,j being expressed as a produc of coordinaes of Y and V. As before, we do no develop heir expression o keep he formulae easy o manipulae, and we refer o [GM3] for more deails. Hence, he random variable of ineres in Proposiion.8, i.e., H Mall.Gen.,is d d δ V γ 1 D V = F i [Z s σs, X s ] i dw s D s F i [Z s σs, X s ] i ds. i=1 By he chain rule, he Malliavin derivaive of F i is relaed o ha of U κi,j, i.e., coordinaes of Y and V and ha of γ 1 βi,j,γi,j: he laer can be expressed in erms of γ 1 and D s γ see Lemma.1.6 in Nualar [Nua95, p. 89] and we obain H Mall.Gen. + i,j,k,l i,j = i,j U κi,j, γ 1 βi,j,γi,j i=1 [Z s σs, X s ] i dw s γ 1 βi,j,γi,j D s U κi,j, [Z s σs, X s ] i ds U κi,j, γ 1 βi,j,kγ 1 l,γi,j D s γ k,l, [Z s σs, X s ] i ds. Analogously o he ellipic case, he inegrals above may be discreized. Furhermore, he random variables U κi,j, may be approximaed by Uκi,j, N, defined by he same produc of coordinaes of Y N and V N as he one defining U κi,j,. Is weak derivaive can be compued as in 3.7: indeed, wih he same argumens, one may prove ha 3.11 D s U κi,j, = 1 s Û κi,j,k, Ǔ βi,j,k,s, k where Ûκi,j,k, i,j,k resp., Ǔβi,j,k,s i,j,k are appropriae real values resp., vecors a ime resp., a ime s of some exended sysems of SDEs. hen, he

19 1694 EMMANUEL GOBE AND RÉMI MUNOS naural approximaion is 3.1 [D s U κi,j, ] N = 1 s Ûκi,j,k, N Ǔ βi,j,k,s N. Acually, wha differs from he ellipic case are he Malliavin covariance marix γ and is weak derivaive. Even if γ = ΠR r Y Z s σs, X s [Π R r Y Z s σs, X s ] ds is almos surely inverible wih an inverse in any L p, a naive approximaion may no saisfy hese inveribiliy properies: for his reason, we add a small perurbaion in is discreizaion as follows: 3.13 γ N = k Π R r Y N Z N φs σφs,xn φs [ΠR r Y N Z N φs σφs,xn φs ] ds + N I d. his allows us o sae he following resul. Lemma 3.3. Assume R and E. hen, for any p 1, one has for some posiive numbers p 1 and q 1 : 1/deγ N L p K,x 1/deγ q1 L p 1 wih a consan K,x independen of N. Proof. I is easy o check ha γ N γ L p K,x h use Lemma 4.3 below. Moreover, he eigenvalues of γ N are all greaer han h; hence deγn hr, and one deduces Edeγ N p =E deγ N p 1 deγ N 1 deγ + E deγ N p 1 deγ N > 1 deγ h rp deγ deγ N P 1 + p Edeγ p deγ h rp q deγ deγ N q L p 1 deγ q L p + p Edeγ p where p 1 and p are conjugae numbers. ake q =rp o ge he resul. o deal wih he weak derivaive of γ, one needs o rewrie γ k,l, = i A ɛk,l,i, B ηk,l,i,udu, where A ɛk,l,i, resp., B ηk,l,i,u are producs of coordinaes of Y resp., Z u and σu, X u. As for 3.7, he Malliavin derivaive of A ɛk,l,i, and B ηk,l,i,u can be expressed as D s A ɛk,l,i, = 1 s C ɛk,l,i,j, D ɛk,l,i,j,s, j D s B ηk,l,i,u = 1 s u Hence, for s, one has D s γ k,l, = C ɛk,l,i,j, i,j A ɛk,l,i, F ηk,l,i,j,s i,j j E ηk,l,i,j,uf ηk,l,i,j,s. B ηk,l,i,udu s D ɛk,l,i,j,s E ηk,l,i,j,udu,

20 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1695 which can be approximaed by 3.15 [D s γ k,l, ] N = i,j C N ɛk,l,i,j, + i,j A N ɛk,l,i, F N ηk,l,i,j,s B N ηk,l,i,φu du D N ɛk,l,i,j,s s E N ηk,l,i,j,φu du. We now urn o he global approximaion of he weigh H Mall.Gen. : 3.16 H Mall.Gen.,N = i,j Uκi,j, N [γn 1 ] βi,j,γi,j [Zφs N σφs,xn φs ] i dw s [γ N 1 ] βi,j,γi,j [D φs U κi,j, ] N [Zφs N σφs,xn φs ] i ds i,j + Uκi,j, N [γn 1 ] βi,j,k [γ N 1 ] l,γi,j i,j,k,l [D φs γ k,l, ] N [Z N φs σφs,xn φs ] i ds. We are now in a posiion o sae he following approximaion resul. heorem 3.4. Assume R, E, and H. For some posiive numbers p and q, one has: Jα EfV N H Mall.Gen.,N K,x f 1/deγ q L ph. In he hypoellipic case case 1 in Proposiion.7, noe ha he weak approximaion resul above holds rue under a nondegeneracy condiion saed only a he iniial poin,x, which is a significan improvemen compared o [B96a] or more recenly in [Z4], where he condiion was saed in he whole space Adjoin approach. o approximae H b,adj. we propose he following naural esimaes: 3.19 N 1 H b,adj.,n = fx N h ḃ k,x N Z N k k k 3. H σ,adj.,n N 1 = h k= k= d i,j=1 [ e j k [ ei k [ { x + ei k [σσ ] i,j k,x N k [fx N fx N k ] Z N k φ + k + φ k Z N k k + φ k Z N k k and H σ,adj. from heorem.11, k [σ 1 φs,x N φs Y N φs ] dw s, [σ 1 φs,x N φs Y N φs ] dw s ] [σ 1 φs,x N φs Y N φs ] dw s ] } [σ 1 φs,xφs N Y φs N ] dw s Z Nk e j. ]

21 1696 EMMANUEL GOBE AND RÉMI MUNOS Derivaives x Yφs N and xz N k are obained by a direc differeniaion in 3. and 3.3: we do no make he equaions explici; hey coincide wih hose of he Euler procedure applied o x Y and x Z defined in 1.3 and 1.4. hese approximaions also induce a discreizaion error in he compuaion of Jα of order 1 w.r.. h. heorem 3.5. Assume R, E, and H. For some p, one has Jα E H b,adj.,n he proof is posponed o secion H σ,adj.,n f K,x p h Maringale approach. he naural approximaion of H Mar. heorem.1 may be given by defined in H Mar.,N = fxn [σ 1 φs,xφs N ẊN φs ] dw s + dr [fxn fxn φr ] φr [σ 1 φs,xφs N ẊN φs Y φs N ZN φr φrẋn φr ] dw s. Unforunaely, we have no been able o analyze he approximaion error Jα EH Mar.,N under he fairly general assumpion H. Indeed, an immediae issue o handle would be o quanify he qualiy of he approximaion of dr E [fx fx r] r r [σ 1 s Ẋs Y s Z r Ẋ r ] dw s by is Riemann sum, which seems o be far from obvious under H. 4. Proof of he resuls on he discreizaion error analysis. his secion is devoed o he proof of secion 3 s heorems analyzing he discreizaion error. he rick o prove hese esimaes for EfX usually relies on he Markov propery: one decomposes he error using he PDE solved by he funcion, x EfX x see Bally and alay [B96a], bu his makes no sense in our siuaion. Anoher way o proceed consiss in cleverly using he dualiy relaionship.1 wih some sochasic expansion o ge he righ order see Kohasu-Higa [KH1] or [KHP]. During he revision of his work, Kohasu-Higa brough o our aenion anoher paper [KHP] where he approximaion of some smooh funcionals of SDEs is successfully analyzed in his way. Here, we also adop his approach. However, he funcionals of ineres are much more complex. Moreover, exra echnicaliies compared o [KHP] are required, because of he necessiy for our Malliavin calculus compuaions o inroduce a localizaion facor ψ N,ɛ. o clarify he argumens, we firs sae a quie general resul, whose saemen enables us o reduce he proof of our heorems o check ha a sochasic expansion holds rue A more general resul. By convenion, we se dw s = ds. Firs, we need o define some paricular forms of sochasic expansions.

22 ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES 1697 Definiion 4.1. he real random variable U which in general depends on N saisfies propery P if i can be wrien as q U = c U, i,j c U,1 i,j c U, i,j sdw s i dw j i,j= + q i,j,k= c U, i,j,k φ c U,1 i,j,k [ c U, i,j,k s s c U,3 i,j,k udw u i φs for some adaped processes {c U,i1 i,j,cu,i i,j,k : i, j, k q, i 1, i 3} possibly depending on N and if, for each [,], hey belong o D wih Sobolev norms saisfying sup N, [, ] c U,i 1 i,j k,p + c U,i i,j,k k,p < for any k,p 1. heorem 4.. Assume R and ha H N H saisfies propery P. hen, 1. if f is of class Cb 3, one has EfV H EfV N H N K,x α f h; α 3 dw j s. under E and H, one has EfV H EfV N H N K,x f 1/deγ q L p h. In he saemen above, V corresponds o some coordinaes of a SDE X as i is defined in.5, bu we can also simply consider V = X. heorem 4. is proved a he end of his secion, and for a while, we focus on is applicaions o derive he announced resuls abou he discreizaion errors. Remember ha he approximaion of he weighs H is essenially based on an Euler scheme applied o a sysem of SDEs. For his reason, he verificaion of propery P is ighly conneced o he decomposiion of he error, beween a Brownian SDE and is Euler approximaion, in erms of a sochasic expansion. his is he purpose of he following sandard lemma for more general driven semimaringales, see Jacod and Proer [JP98]. Lemma 4.3. Consider a general d -dimensional SDE X defined by C coefficiens wih bounded derivaives, and X N is Euler approximaion: X = x + X N = x + bs, Xs ds + q j=1 bφs, XN φs ds + σ j s, X s dw j s, q j=1 σ j φs, X N φs dw j s. hen, for each, each componen of X X N saisfies P. Namely, for 1 k d, one has q X k, X k, N = c X, i,j,k c X,1 s i,j,k s c X, i,j,k udw u i dw j s i,j= for some adaped processes {c X,i 1 i,j,k : i, j q, 1 k d, i 1 } saisfying sup N, [, ] c X,i 1 i,j,k k,p < for any k,p 1. φs ] dw k

23 1698 EMMANUEL GOBE AND RÉMI MUNOS Proof. One has X X N = b s X s X s N ds+ q j=1 σ j s X s X s N dws j + [ bs, X s N bφs, X φs N ] ds + q j=1 [ σ js, X s N σ j φs, X φs N ] dw s j wih a s = 1 xas, X s N + λ X s X s N dλ for a = b or a = σ j. Now, consider he unique soluion of he linear equaion E =I d + b se s ds + q j=1 σ j se s dws j. From heorem 56 p. 71 in Proer [Pro9], one deduces ha { X X N 1 = E E s [ bs, X s N bφs, X φs N ] + } q σ js[ σ j s, X s N σ j φs, X φs N ] ds j=1 q E E 1 s [ σ j s, X s N σ j φs, X φs N ] dw s j ; j=1 hen, by applying Iô s formula beween φs and s, we can easily complee he proof of Lemma Proof of heorem 3.4 general nondegenerae case. Owing o heorem 4., we only have o prove ha H Mall.Gen. H Mall.Gen.,N saisfies propery P. hus, i is enough o separaely look a each facor in H Mall.Gen. and H Mall.Gen.,N,by proving ha heir difference is of he form c U, i,j cu,1 i,j φ cu, i,j sdw s i dw j or c U, i,j,k cu,1 i,j,k [ cu, i,j,k s s φs cu,3 i,j,k udw ] u i dw j s dw k, while he oher facors jus belong o D wih uniformly bounded Sobolev norms. a he fac ha he difference U κi,j, Uκi,j, N involved in 3.8, 3.1, 3.16, and 3.18 saisfies P can be derived from an applicaion of Lemma 4.3 by noicing ha U κi,j, is he produc of coordinaes of Y and V. b Using he expressions of γ and γ N, one ges γ k,l, γk,l, N = δ k,lh+e 3,1,k,l + E 3,,k,l wih E 3,1,k,l = E 3,,k,l = [ Π R r Y Z s σs, X s [Π R r Y Z s σs, X s ] ] k,l ds [ Π R r Y N Z N s σs, X N s [Π R r Y N Z N s σs, X N s ] ] k,l ds, [ Π R r Y N Z N s σs, X N s [Π R r Y N Z N s σs, X N s ] ] k,l ds [ Π R r Y N Z N φs σφs,xn φs [ΠR r Y N Z N φs σφs,xn φs ] ] k,l ds. Using Lemma 4.3 and he relaion as, X s as, Xs N =a sx s Xs N wih a s = 1 xas, Xs N + λx s Xs N dλ available for smooh funcions a, i is sraighforward o see ha E 3,1,k,l can be wrien as a sum of erms saisfying P. he same conclusion holds for E 3,,k,l if we apply Iô s formula beween φs and s. Finally, as 1/deγ and 1/deγ N belong o any Lp p 1 according o Lemma 3.3, i follows ha he difference [γ 1 ] k,l [γn 1 ] k,l involved in 3.8, 3.9, 3.1, 3.16, 3.17, and 3.18 saisfies P.