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1 Edinburgh Research Explorer Localizaion errors in solving sochasic parial differenial equaions in he whole space Ciaion for published version: Gerencsér, M & Gyongy, I 217, 'Localizaion errors in solving sochasic parial differenial equaions in he whole space' Mahemaics of Compuaion, vol. 86, no. 37, pp DOI: 1.19/mcom/321 Digial Objec Idenifier (DOI): 1.19/mcom/321 Lin: Lin o publicaion record in Edinburgh Research Explorer Documen Version: Peer reviewed version Published In: Mahemaics of Compuaion Publisher Righs Saemen: Firs published in Mahemaics of Compuaion in 216, published by he American Mahemaical Sociey General righs Copyrigh for he publicaions made accessible via he Edinburgh Research Explorer is reained by he auhor(s) and / or oher copyrigh owners and i is a condiion of accessing hese publicaions ha users recognise and abide by he legal requiremens associaed wih hese righs. Tae down policy The Universiy of Edinburgh has made every reasonable effor o ensure ha Edinburgh Research Explorer conen complies wih UK legislaion. If you believe ha he public display of his file breaches copyrigh please conac openaccess@ed.ac.u providing deails, and we will remove access o he wor immediaely and invesigae your claim. Download dae: 17. Mar. 219

2 MATHEMATICS OF COMPUTATION Volume, Number, Pages S (XX)- LOCALIZATION ERRORS IN SOLVING STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN THE WHOLE SPACE MÁTÉ GERENCSÉR AND ISTVÁN GYÖNGY Absrac. Cauchy problems wih SPDEs on he whole space are localized o Cauchy problems on a ball of radius R. This localizaion reduces various inds of spaial approximaion schemes o finie dimensional problems. The error is shown o be exponenially small. As an applicaion, a numerical scheme is presened which combines he localizaion and he space and ime discreizaion, and hus is fully implemenable. 1. Inroducion Parabolic, possibly degenerae, linear sochasic parial differenial equaions (SPDEs) are considered. In applicaions such equaions are ofen given on he whole space, which maes he direc implemenaion of discreizaion mehods problemaic. Finie elemen mehods, see e.g., [1], [11], [13], [24], [25] and heir references, mosly rea problems on bounded domains and ofen under srong resricions on he differenial operaors, denoed by L and M below. For finie difference schemes convergence resuls are available on he whole space, see e.g. [26] for he non-degenerae and [2], [7] for he degenerae case, bu he schemes hemselves are infinie sysems of equaions. A naural way o overcome his difficuly is o cu off he equaion ouside of a large ball of radius R. A similar approach o obain error esimaes for a runcaed erminal condiion in deerminisic PDEs of opimal sopping problems is used in [23]. The main resuls of he presen paper are Theorems 2.6 and 3.6. Theorem 2.6 compares soluions of wo SPDEs whose daa agrees on a ball of radius R and esablishes an error esimae of order e δr2 in he supremum norm. The proof relies on ransforming fully degenerae SPDEs o zero order equaions via he mehod of characerisics, whose analysis goes bac o [19] and [18], see also he recen wor [21] and he references herein. Once such a resul is used o esimae he difference beween he original equaion and is runcaion, one can approximae he runcaed equaion wih nown numerical schemes. Our choice is he finie difference mehod for he spaial and he implici Euler mehod for he emporal approximaion. The analysis of he former is invoed from [7], while for he laer one requires an error esimae for he ime discreizaion of he finie difference scheme, which is a his poin a finie dimensional SDE. Of course his esimae needs o be independen of he spaial mesh size, and his is esablished in Theorem 3.1. To he auhors bes 21 Mahemaics Subjec Classificaion. Primary 6H15, 6H35, 65M6. Key words and phrases. Cauchy problem, degenerae sochasic parabolic PDEs, localizaion error, finie difference mehod. 1 c XXXX American Mahemaical Sociey

3 2 M. GERENCSÉR AND I. GYÖNGY nowledge such an analysis of a full discreizaion is also a new resul for degenerae SPDEs, and in fac even in he deerminisic case i improves he resuls of [2] in ha we are no resriced o finie difference schemes which are monoone. An error esimae for he approximaions obained by localized and fully discreized SPDEs is given in Theorem 3.5, which is a special case of Theorem 3.6, where he accuracy of he approximaions is improved by Richardson exrapolaion in he spaial discreizaion. We menion ha an alernaive mehod for localizaion is o inroduce arificial Dirichle boundary condiions on a large ball, his approach is used for deerminisic PDEs in, for example, in [2], [12], and o some exen, in [23]. The order of error in hese wors is e δr, and so he mehod of he presen paper yields an improvemen even in he deerminisic case. In he presen conex he mehod of arificial boundary condiion would presen addiional issues. For insance, if he original SPDE is degenerae, hen afer inroducing he boundary condiions he resuling equaion may no even have a soluion. Even if we do assume nondegeneracy, in he generaliy considered here - which is jusified and moivaed by he filering equaion in signal-observaion models - here is no approximaion heory of SPDEs on bounded domains wih Dirichle boundary condiion, and herefore such a localizaion mehod does no help in finding an efficien numerical scheme. One indicaion of he problem is ha regardless of he smoohness of he daa and of he boundary, soluions of Dirichle problems for SPDEs in general do no have coninuous second derivaives, see [17]. Le us inroduce some noaion used hroughou he paper. All random elemens will be given on a fixed probabiliy space (Ω, F, P ), equipped wih a filraion (F ) of σ-fields F F. We suppose ha his probabiliy space carries a sequence of independen Wiener processes (w ) =1, adaped o he filraion (F ), such ha w ws is independen of F s for each and any s. I is assumed ha F conains all P -null subses of Ω, so ha (Ω, F, P ) is a complee probabiliy space and he σ-fields F are complee. By P we denoe he predicable σ-field of subses of Ω [, ) generaed by (F ). We use he shorhand noaion E α X = [E(X)] α. For p [2, ) and ϑ R we denoe by L p,ϑ (R d, H) he space of measurable mappings f from R d ino a separable Hilber space H, such ha f Lp,ϑ = ( R d (1 + x 2 ) ϑ/2 f(x) p H dx) 1/p <. We do no include he symbol H in he noaion of he norm in L p,ϑ (R d, H). Which H is involved will be clear from he conex. We do he same in oher similar siuaions. In his paper H will be l 2 or R. The space of funcions from L p,ϑ (R d, H), whose generalized derivaives up o order m are also in L p,ϑ (R d, H), is denoed by Wp,ϑ m (Rd, H). By definiion Wp,ϑ (Rd, H) = L p,ϑ (R d, H). The norm f W m p,ϑ of f in Wp,ϑ m (Rd, H) is defined by (1.1) f p W = p,ϑ m D α f p L p,ϑ, α m where D α := D α1 1...Dα d d for muli-indices α := (α 1,..., α d ) {, 1,...} d of lengh α := α 1 + α α d, and D i f is he generalized derivaive of f wih respec o x i for i = 1, 2..., d. We also use he noaion D ij = D i D j and Df = (D 1 f,..., D d f).

4 LOCALIZATION ERRORS 3 When we al abou derivaives up o order m of a funcion for some nonnegaive ineger m, hen we always include he zeroh-order derivaive, i.e. he funcion iself. Unless oherwise indicaed a some places, he summaion convenion wih respec o repeaed ineger valued indices is used hroughou he paper. The consans in our esimaes, usually denoed by N, may change from line o line in he calculaions, bu heir dependencies will always be clear from he saemens. Consider he sochasic equaion 2. Formulaion and main resuls (2.1) du (x) = (L u (x) + f (x)) d + (M u (x) + g (x)) dw =1 on (, x) [, T ] R d =: H T, wih iniial condiion (2.2) u (x) = ψ(x), x R d. Here f and g = (g ) =1 are funcions on Ω H T wih values in R and l 2, respecively, and L and M are second order and firs order differenial operaors of he form L = a ij (x)d i D j + b i (x)d i + c (x), M = σ i (x)d i + µ (x), = 1, 2,..., where he coefficiens a ij, b i, c, σ i and µ are real-valued funcions on Ω H T for i, j = 1, 2,...d, and inegers 1. For an ineger m, p [2, ), and ϑ R he following assumpions ensure he exisence and uniqueness of a W m p,ϑ -valued soluion (u ( )) [,T ]. Assumpion 2.1. For P d dx-almos all (ω,, x) Ω [, T ] R d for all z R d, where α ij (x)z i z j α ij = 2a ij σ i σ j. This is a sandard assumpion in he heory of sochasic PDEs. Below we assume some smoohness on α in x, and on all he coefficiens and free erms of problem (2.1)-(2.2). We will also require ha he nonnegaive symmeric square roo ρ := α possesses bounded second order derivaives in x. Concerning his assumpion we remar ha i is well-nown from [3] ha ρ is Lipschiz coninuous in x if α is bounded and has bounded second order derivaives, bu i is also nown ha he second order derivaives of ρ may no exis in he classical sense, even if α is smooh wih bounded derivaives of arbirary order. Assumpion 2.2. (a) The derivaives in x R d of a ij up o order max(m, 2) are P B(R d )-measurable funcions, bounded by K for all i, j {1, 2,..., d}. (b) The derivaives in x R d of b i and c up o order m are P B(R d )-measurable funcions, bounded by K for all i {1, 2,..., d}. The funcions σ i = (σ i ) =1 and µ = (µ ) =1 are l 2-valued and heir derivaives in x up o order m+1 are P B(R d )- measurable l 2 -valued funcions, bounded by K. (c) The derivaives in x R d of ρ = α up o order m + 1 are P B(R d )- measurable funcions, bounded by K.

5 4 M. GERENCSÉR AND I. GYÖNGY Assumpion 2.3. The iniial value, ψ is an F -measurable random variable wih values in Wp,ϑ m. The free daa, f and g = (g ) =1 are predicable processes wih values in Wp,ϑ m m+1 and Wp,ϑ (l 2), respecively, such ha almos surely (2.3) K p m,p,ϑ (T ) := ψ p W m p,ϑ + T ( f p W + g p,ϑ m p ) W d <. m+1 p,ϑ Definiion 2.1. A Wp,ϑ 1 -valued funcion u, defined on [, T ] Ω, is called a soluion of (2.1)-(2.2) on [, T ] if u is predicable on [, T ] Ω, T u p d < (a.s.), Wp,ϑ 1 and for each ϕ C (R d ) for almos all ω Ω (2.4) (u, ϕ) =(ψ, ϕ) + + { (a ij s D i u s, D j ϕ) + ( b i sd i u s + c s u s + f s, ϕ)} ds (σ ir s D i u s + µ r su s + g r s, ϕ) dw r s for all [, T ], where b i = b i D j a ij, and (, ) denoes he inner produc in he Hilber space of square inegrable real-valued funcions on R d. The following heorem follows from Theorem 2.1 in [6]. Theorem 2.2. Le Assumpions 2.1, 2.2 (a)-(b), and 2.3 wih m hold. Then here exiss a mos one soluion on [, T ]. If Assumpions 2.1, 2.2(a), and 2.3 hold wih m 1, hen here exiss a unique soluion u = (u ) [,T ] on [, T ]. Moreover, u is a Wp,ϑ m -valued wealy coninuous process, i is a srongly coninuous process wih values in W m 1 p,ϑ, and for every q > and n {, 1,..., m} (2.5) E sup u q W NEK q p,ϑ n n,p,ϑ (T ), [,T ] where N is a consan depending only on K, T, d, m, p, ϑ, and q. Remar 2.3. Theorem 2.1 in [6] covers he ϑ = case. We can reduce he case of general ϑ o his by rewriing he equaion for u as an equaion for ũ(x) = u(x)(1+ x 2 ) ϑ/2 (see e.g. [5]). I is easily seen ha he coefficiens of he resuling equaion sill saisfy he condiions of he heorem. Definiion 2.4. A P B(R d )-measurable random field u on H T is called a classical soluion of (2.1)-(2.2), if along wih is derivaives in x up o order 2 i is coninuous in (, x) H T, i saisfies (2.1)-(2.2) almos surely for all (, x) H T, and here exiss a finie random variable ξ and a consan s such ha almos surely D α u (x) ξ(1 + x ) s for all (, x) H T and for α 2. Corollary 2.5. Le assumpions of Theorem 2.2 hold wih m > 2 + d/p. Then here exiss a unique classical soluion u o (2.1)-(2.2). Le us refer o he problem (2.1)-(2.2) as Eq(D), where D sands for he daa D = (ψ, a, b, c, σ, µ, f, g) wih a = (a ij ), b = (b i ), σ = (σ i ), g = (g ) and µ = (µ ). We are ineresed in he error when insead of Eq(D) we solve Eq( D) wih D = ( ψ, ā, b, c, σ, µ, f, ḡ).

6 LOCALIZATION ERRORS 5 Assumpion 2.4. Almos surely (2.6) D = D on [, T ] {x R d : x R}. The main example o eep in mind is when each componen of D is a runcaion of he corresponding componen of D (see Secion 3 below). Le B R = {x R d : x R} for R >. Define K p m,p,ϑ (T ) as Kp m,p,ϑ (T ) wih ψ, f and ḡ in place of ψ, f and g, respecively. The main resul reads as follows. Theorem 2.6. Le ν (, 1) and le Assumpions 2.1, 2.2 (b)-(c) and 2.3 hold wih m > 2 + d/p and ϑ R for D and D. Le also Assumpion 2.4 hold. Then Eq(D) and Eq( D) have a unique classical soluion u and ū, respecively, and for q >, r > 1 (2.7) E sup sup u (x) ū (x) q Ne δr 2E 1/r (K qr qr m,p,ϑ (T ) + K m,p,ϑ (T )), x B νr [,T ] where N and δ are posiive consans, depending on K, d, T, q, r, ϑ, p, and ν. Firs we collec some auxiliary resuls. The following lemma is a version of Kolmogorov s coninuiy crierion, see Theorem 3.4 of [4]. Lemma 2.7. Le x(θ) be a sochasic process paramerized by and coninuous in θ D R p, where D is a direc produc of lower dimensional closed balls. Then for all < α < 1, q 1, and s > p/α, [ ( E x(θ) x(θ ) qs ) ] 1/s E sup x(θ) q N(1 + D ) θ sup(e x(θ) qs ) 1/s + sup θ θ θ where N = N(q, s, α, p), and D is he volume of D. θ θ qsα Lemma 2.8. Le (α ) [,T ] and (β ) [,T ] be F -adaped processes wih values in R d and l 2 (R d ), respecively, in magniude bounded by a consan K. Then for he process (2.8) X = α s ds + β s dw s, [, T ] here exis consans ε = ε(k, T ) > and a N = N(K, T ) such ha E sup e ε X 2 N. T Proof. A somewha more general lemma is proved in [23]. For convenience of he reader we give he proof here. By Iô s formula Y := e X 2 e µ = 1 + e Xs 2 e µs µs { β s 2 + 2α s X s +2 β s X s 2 µ X s 2 } ds + m for any µ R, where (m ) [,T ] is a local maringale saring from. By simple inequaliies 2αX + 2 βx 2 α 2 + X β 2 X 2 K 2 + (2K 2 + 1) X 2.

7 6 M. GERENCSÉR AND I. GYÖNGY Hence for µ = (2K 2 + 1) and for a sopping ime τ T we have EY τn 1 + 2K 2 EY s τn ds, for τ n = τ ρ n, where (ρ n ) n=1 is a localizing sequence of sopping imes for m. Hence, by Gronwall s lemma, EY τn e 2K2T. where N is independen of n. Leing here n, by Faou s lemma we ge Ee Xτ 2 e µt Ee Xτ 2 e µτ e K2 T for sopping imes τ T. Hence applying Lemma 3.2 from [8] for r (, 1) we obain E sup e r Xτ 2 e µt 2 r 1 r erk2t. T To formulae our nex lemma we consider he sochasic differenial equaion (2.9) dx s = α s (X s ) ds + β s (X s ) dw s, where α and β = (β ) are P B(R d )-measurable funcion on Ω [, T ] R d, wih values in R d and l 2 (R d ) such ha hey are bounded in magniude by K and saisfy he Lipschiz condiion in x R d wih a Lipschiz consan M, uniformly in he oher argumens. Then equaion (2.9) wih iniial condiion X = x has a unique soluion X,x = (Xs,x ) s [,T ] for any [, T ] and x R d. Remar 2.9. I is well nown from [19] ha he soluion of (2.8) can be chosen o be coninuous in, x, s. In he following, by Xs,x we always undersand such a coninuous modificaion. Lemma 2.1. Se ˆX,x = X,x x. There exiss a consan δ = δ(d, K, M, T ) > such ha for any R,,x ˆX (2.1) E sup sup e s 2δ N(1 + R d+1/2 ), s T x R and for any R and r,x (2.11) P ( sup sup ˆX s > r) Ne δr2 (1 + R d+1/2 ), s T x R where N = N(d, K, M, T ). Proof. I is easy o see ha (2.1) implies (2.11), so we need only prove he former. For a fixed δ, o be chosen laer, le us use he noaions f(y) = e y 2δ and γ = 2(d + 2) + 1. By Lemma 2.7, we have E sup s T (2.12) + N(1 + R d ) sup s T s T,x sup f( ˆX s ) N(1 + R d ) sup sup (Ef γ,x ( ˆX s )) 1/γ x R s T x R sup x R x R (,x E f( ˆX s ) f( ˆX ),x 1/γ s ) γ. ( 2 + s s 2 + x x 2 ) γ/4

8 LOCALIZATION ERRORS 7 The firs erm above, by Lemma 2.8, provided δ ε/γ, can be esimaed by NR d. As for he second one, f( 1,x ˆX s ) f( ˆX,x s ) = f(ϑ ˆX,x s + (1 ϑ) ˆX,x s )( ˆX,x s ˆX,x ) dϑ. Noice ha f(y) N(δ)f 2 (y), herefore by Jensen s inequaliy and Lemma 2.8 again, provided δ ε/(8γ), we obain E f( ˆX,x s ) f( ˆX,x ) γ NE 1/2 s ˆX,x s ˆX,x s 2γ. Now he he righ-hand side can be esimaed by sandard momen bounds for SDEs, see e.g. Corollary in [15], from which we obain (,x E f( ˆX s ) f( ˆX ),x 1/(2γ) s ) 2γ N(1 + R 1/2 ). ( 2 + s s 2 + x x 2 ) γ/2 Proof of Theorem 2.6. Throughou he proof we will use he consan λ = λ(d, q), which sands for a power of R, and, lie N and δ, may change from line o line. Clearly i suffices o prove Theorem 2.6 wih e δr2 R λ in place of e δr2 in he righ-hand side of inequaliy (2.7). We also assume firs ha q > 1 and ϑ =. The main idea of he proof is based on sochasic represenaion of soluions o linear sochasic PDEs of parabolic ype, see [18], [19], and [21]. This represenaion can be viewed as he generalizaion of he well-nown Feynman-Kac formula and is derived as follows. Firs, we consider an equaion which differs from he original only by an addiional sochasic erm driven by an independen Wiener process. The new equaion is fully degenerae and aing condiional expecaion wih respec o he original filraion of is soluion gives bac u. On he oher hand, he mehod of characerisics allows us o ransform he fully degenerae equaion o a much simpler one. This provides a formula for he represenaion of u, and, more imporanly for our purposes, allows us o compare u and ū on he level of characerisics. Recall ha ρ = (ρ ir (x)) d i,r=1 is he symmeric nonnegaive square roo of α = (2a ij σ i σ j ) d i,j=1 and ρ is he symmeric nonnegaive square roo of ᾱ = (2āij σ i σ j ) d i,j=1. Then due o Assumpion 2.4, ρ = ρ almos surely for all [, T ] and for x R. Le (ŵ r ),r=1...d be a d-dimensional Wiener process, also independen of he σ-algebra F generaed by F for. Consider he problem (2.13) (2.14) dv (x) =(L v (x) + f (x)) d + (M v (x) + g (x)) dw v (x) =ψ(x), + N r v (x) dŵ r where N r = ρ ri D i. Then by Corollary 2.5, (2.13)-(2.14) has a unique classical soluion v, and for each [, T ] and x R d almos surely (2.15) u (x) = E(v (x) F ). Togeher wih (2.13) le us consider he sochasic differenial equaion (2.16) dy = β (Y ) d σ (Y ) dw ρ r (Y ) dŵ r, T, Y = y, s

9 8 M. GERENCSÉR AND I. GYÖNGY where β (y) = b (y) + σ i (y)d i σ (y) + ρ ri (y)d i ρ r (y) + σ (y)µ (y), [, T ], y R d, and σ, ρ r sand for he column vecors (σ 1,..., σ d ), (ρ 1r,..., ρ dr ), respecively. By he Iô-Wenzell formula from [16], for U (y) := v (Y (y)) we have (o ease he noaion we omi he parameer y in Y (y)) dv (Y ) = (L v (Y ) + f (Y )) d + (M v (Y ) + g (Y )) dw + N r v (Y ) dŵ r +(β i D i v (Y ) + a ij D ij v (Y )) d σ i D i v (Y ) dw N r v(y ) dŵ r (2.17) σ i D i (M v (Y ) + g (Y )) d N r N r v (Y ) d. Due o cancellaions on he righ-hand side of (2.17) we obain where du (y) ={γ (Y (y))u (y) + φ (Y (y))} d + {µ (Y (y))u (y) + g (Y (y))} dw, U (y) = ψ(y), γ (x) := c (x) σ i (x)d i µ (x), φ (x) = f (x) σ i (x)d i g. Noice ha in he special case when f =, g =, c =, µ = and ψ(x) = x i for i {1,..., d}, we ge ṽ(y i (y)) = y i for i = 1,..., d, where ṽ i is he soluion of (2.13)- (2.14) wih f = c =, g = µ =, σ = and ψ(x) = x i. Hence for each [, T ] he mapping y Y (y) R d has an inverse, Y 1, for almos every ω, and he mapping x ṽ (x) = (ṽ(x)) i d, defined by he coninuous random field (ṽi ) (,x) HT gives a coninuous modificaion of Y 1. Also, we can wrie v (x) = U (Y 1 ). As we shall see, due o he daa being he same on a large ball, he characerisics Y and Ȳ agree on an even of large probabiliy. This fac and he above represenaion will yield he esimae (2.7). Se Ū(y) = v (Ȳ(y)), where v (x) and Ȳ (y) are defined as v (x) and Y (y) in (2.13)-(2.14) and (2.16), respecively, wih D and ρ in place of D and ρ. Inroduce he noaions B R = [, T ] B R and A R = B R Q d+1. Since u and ū are coninuous in boh variables, (2.18) sup u (x) ū (x) = sup u (x) ū (x) (,x) B νr (,x) A νr Le ν = (1 + ν)/2 and define he even [ Then H := sup Y 1 (x) > ν R (,x) B νr ] [ sup Y (x) > R (,x) B ν R H c = [ Y 1 (x) B ν R, (, x) B νr ] [Y (x) B R, (, x) B ν R], ]. and hus on H c Y (x) = Ȳ(x) for (, x) B ν R, and consequenly, Y 1 (x) = Ȳ 1 (x) for (, x) B νr, v (x) = v (x) for (, x) B νr.

10 LOCALIZATION ERRORS 9 Therefore, by (2.15) and (2.18), and by Doob s, Hölder s, and he condiional Jensen inequaliies, E sup u (x) ū (x) q E (,x) B νr sup [,T ] Q E(1 H sup v τ (x) v τ (x) F ) q (τ,x) A νr (2.19) q (P (H))1/r E 1/r ( sup v τ (x) v τ (x) qr q 1 (τ,x) H T (2.2) 2q 1 q (P (H))1/r V T q 1 wih V T := E 1/r sup v τ (x) qr + E 1/r sup v τ (x) qr, (τ,x) H T (τ,x) H T for r > 1, r = r/(r 1), provided q > 1. By Theorem 2.2 (2.21) V T NE 1/r (K qr qr m,p, (T ) + K m,p, (T )). We can esimae P (H) as follows. Clearly, P (H) P ( sup Y 1 (x) > ν R) + P ( sup Y (x) > R) =: J 1 + J 2. (,x) B νr (,x) B ν R For Ŷ(x) = Y (x) x by (2.11) we have Also, we have J 2 P ( sup Ŷ(x) > (1 ν )R) NR d+1/2 e δ(1 ν) (,x) B ν R J 1 ) 2 R 2. P ( (, x) [, T ] (B 2 l+1 ν R \ B 2l ν R) : Y (x) νr) l= P ( sup Ŷ(x) (2 l ν ν)r). (,x) B 2 l+1 ν R l= Using (2.11) again gives J 1 N We can conclude ha e δ(2l ν ν) 2 R 2 (2 l+1 ν R) d+1 Ne δr2 l= (2.22) P (H) Ne δr2, where N and δ are posiive consans, depending only on d, K and T. Combining his wih (2.2) and (2.21) we can finish he proof of he heorem under he addiional condiions. For general ϑ one applies he same argumens as in Remar 2.3. Finally (2.7) for he case q (, 1] follows easily from sandard argumens using Lemma 3.2 from [8].

11 1 M. GERENCSÉR AND I. GYÖNGY 3. An applicaion - finie differences In his secion we apply Theorem 2.6 o presen a numerical scheme approximaing he iniial value problem (2.1)-(2.2). We mae use of he resuls of [7] on he rae and acceleraion of finie difference approximaions in he spaial variable, which, ogeher wih a ime discreizaion and a runcaion - whose error can be esimaed using Theorem yields a fully implemenable scheme. We shall carry ou he seps of approximaion in he following order: spaial discreizaion by finie differences, localizaion of he finie difference scheme, and discreizaion in ime via implici Euler s mehod. This of course requires an analysis of he Euler scheme, o presen an error esimae for i, which does no depend on he spaial mesh size and he localizaion. Furhermore, in our full discreizaion scheme we shall incorporae Richardson s exrapolaion, which will allow us o improve he accuracy of he scheme in he spaial mesh size h. Firs we inroduce he finie difference approximaion in he spaial variable for (2.1)-(2.2). To his end, le Λ 1 R d be a finie se, conaining he zero vecor, saisfying he following naural condiion: Λ := Λ 1 \ {} is no empy, and if a subse Λ Λ is linearly dependen, hen i is linearly dependen over he raionals. Le h >, and define he grid n G h = {h λ i : λ i Λ 1 Λ 1, n = 1, 2,...}. Due o he assumpion on Λ 1, G h has only finiely many poins in every ball around he origin in R d. Define for λ Λ Λ, he finie difference operaors δλϕ(x) h = 1 (ϕ(x + hλ) 2ϕ(x) + ϕ(x hλ)), 2h and le δ h sand for he ideniy operaor. To approximae he Cauchy problem (2.1) -(2.2), for h > we consider he equaion (3.1) du (x) = (L h u (x) + f (x)) d + (M h, u (x) + g (x))dw on [, T ] G h, wih iniial condiion =1 (3.2) u (x) = ψ(x), where L h and M h, are difference operaors of he form L h (x) = (x)δλδ h κ, h M h, (x) = (x)δλ, h = 1, 2,..., λ,κ Λ 1 a λκ λ Λ 1 b λ, wih some real-valued P B(R d )-measurable a λκ and b λ, on Ω [, T ], such ha (3.3) a λκ (x) K and b λ, (x) 2 K 2 for all λ, κ Λ 1, [, T ], x R d and ω Ω, where K is a consan. Remar 3.1. Here ψ, f and g are he same as in (2.1)-(2.2) and we will assume ha hey saisfy Assumpion 2.3 wih m > d/2, p = 2 and ϑ =. Thus by Sobolev s embedding of W2 m ino C b, he space of bounded coninuous funcions, for all ω we can find a coninuous funcion of x which is equal o ψ almos everywhere, and for each and ω we have coninuous funcions of x which coincide wih f and g for almos every x R d. Here and in he following we always ae such coninuous

12 LOCALIZATION ERRORS 11 modificaions if hey exis, hus we always assume ha ψ, f, and g are coninuous in x for all (for g = (g ) =1 his means, as usual, coninuiy as a funcion wih values in l 2 ). In paricular, erms lie f (x) in (3.1) mae sense. We noe ha for m > d/2 one can use Sobolev s heorem on embedding W2 m o C b o show also ha if Assumpion 2.3 holds wih m > d/2, p = 2 θ =, hen x G h ψ(x) 2 h d + T ( x G h f (x) 2 h d + T g (x) 2 h d) d x G h N ψ(x) 2 m + N f 2 m + g 2 m d < (a.s.), wih a consan N = N(Λ, d), where m := m,2.. (See Lemma 4.2 in [1].) Clearly, for ϕ C (R d ) and λ δ h λϕ(x) λ ϕ(x) := λ i D i ϕ(x) as h. Thus, in order o approximae L and M by L h and M h,, respecively, we need he following compaibiliy condiion. Assumpion 3.1. For every i, j = 1,..., d, = 1,... and for P d dx-almos all (ω,, x) Ω [, T ] R d a ij = a λκ λ i κ j, b i = (a λ + a λ )λ i, c = a, λ,κ Λ λ Λ σ i = λ Λ b λ, λ i, µ = b,. For each x G h equaion (3.1) is a sochasic differenial equaion (SDE), i.e., in general, (3.1)-(3.2) is an infinie sysem of SDEs. To replace his wih a finie sysem we mae he coefficiens, ogeher wih he free and iniial daa, vanish ouside of a large ball by muliplying hem wih a cuoff funcion ζ R, which saisfies he following condiion. Assumpion 3.2. For an ineger m and a real number R > he funcion ζ R is a coninuous funcion wih compac suppor on R d, such ha ζ(x) = 1 for x R and he derivaives of ζ R up o order m + 1 are coninuous funcions, bounded by a consan C. In his way we replace (3.1)-(3.2) wih he sysem of SDEs (3.4) du (x) = (L h,r wih iniial condiion u (x) + f R (x)) d + (M h,r, u (x) + g R, (x)) dw, [, T ], (3.5) u (x) = ψ R (x), for x G h supp ζ R, where supp ζ R is he suppor of ζ R, (3.6) (ψ R, f R, g R, ) := (ζ R ψ, ζ R f, ζ R g ) and L h,r := a λκ,r δλδ h κ, h M h,r, := b λ,r, δλ, h = 1, 2,..., λ,κ Λ 1 λ Λ 1

13 12 M. GERENCSÉR AND I. GYÖNGY wih (3.7) a λκ,r := ζ 2 Ra λκ for λ, κ Λ, (3.8) (a κ,r, a λ,r, b λ,r, ) := (ζ R a κ, ζ R a λ, ζ R b λ, ) for λ, κ Λ 1, 1. A his poin our approximaion is a finie dimensional (affine) linear SDE, whose coefficiens are bounded by K owing o (3.3), and furhermore, by virue of Remar 3.1, for each x, f R (x) and g R (x) are square inegrable in ime under Assumpion 3.2 and 3.5 below wih m > d/2. Hence (3.4)-(3.5) has a unique soluion {u h,r (x) : x G h supp ζ R } [,T], by virue of a well-nown heorem of Iô on finie dimensional SDEs wih Lipschiz coninuous coefficiens. The approximaion of such equaions are well sudied, various ime-discreizaion mehods can be used, each of hem wih heir own advanages and disadvanages. Here we chose he implici Euler mehod, formulaed as follows. We ae a mesh-size τ = T/n for an ineger n 1, and approximae (3.1)-(3.2) by he equaions (3.9) v (x) = ψ R (x), (3.1) + =1 v i (x) = v i 1 (x) + (L h,r τi v i (x) + fτ(i 1) R (x))τ (M h,r, τ(i 1) v i 1(x) + g R, τ(i 1) (x))ξ i, i = 1, 2,..., n, for x G h supp η R, where ξ i = w iτ w (i 1)τ. Remar 3.2. In many applicaions, including he Zaai equaion for nonlinear filering, he driving noise is finie dimensional. If his is no he case, one needs anoher level of approximaion, a which he infinie sum in (3.1) is replaced by is firs m erms. We shall no discuss his here. Remar 3.3. As menioned before, Euler approximaions for SDEs are very well sudied. Therefore, while i is far from immediae ha he error is of he desired order, independenly of h and R, he implemenaion of he scheme goes as usual, see e.g. [14] and is references. To prove o solvabiliy of he fully discreized equaion (3.9)-(3.1) and esimae is error from he rue soluion of (2.1)-(2.2) on he space-ime grid we pose he following assumpions. As for he following we confine ourselves o he L 2 -scale, wihou weighs, we use he shorhand noaion m = W m 2,, =. Assumpion 3.3. For all (ω,, x) Ω [, T ] R d λ,κ Λ (2a λκ b λ, b κ, )z λ z κ for all z = (z λ ) λ Λ, z λ R. In he following assumpions m and l are nonnegaive inegers, as before, and will be more specified in he heorems below.

14 LOCALIZATION ERRORS 13 Assumpion 3.4. The derivaives in x of a λκ up o order max(m, 2) are P B(R d )- measurable funcions, bounded by K for all λ, κ Λ 1. The derivaives in x of b λ = (b λr ) r=1 up o order m + 1 are P B(R d )-measurable l 2 -valued funcions, bounded by K, for all λ Λ 1. Assumpion 3.5. The iniial value, ψ is an F -measurable random variable wih values in W2 m. The free daa, f and g = (g ) =1 are predicable processes wih values in W2 m and W2 m+1 (l 2 ), respecively, such ha almos surely (3.11) K 2 m := ψ 2 m + T ( f 2 m + g 2 ) m+1 d <. Assumpion 3.6. There exiss a consan H such ha E f f s 2 l + E g s g 2 l+1 H s, for all s, [, T ], and D α (a λκ (x) a λκ (x)) 2 H s, s D β (b λ, E f 2 l+1 + E g 2 l+2 H (x) b λ, s (x)) 2 H s, for all ω Ω, x R d, s, [, T ] and muli-indices α and β wih α l and β l + 1. Remar 3.4. If Assumpions 3.1 and 3.3 hold hen (2a ij µ ir µ jr )z i z j = (2 a λκ λ i κ j b λ, λ i b κ, κ j )z i z j λ,κ Λ λ Λ κ Λ = λ,κ Λ (2a λκ b λ, b κ, )(λ i z i )(κ j z j ) for all z = (z 1,..., z d ) R d, i.e., Assumpion 2.1 also holds. Clearly, Assumpions 3.1 and 3.4 imply Assumpion 2.2 (a)-(b). Noice ha Assumpion 3.5 is he same as Assumpion 2.3 wih p = 2 and ϑ =. Thus if Assumpions 2.2 (c), and 3.1 hrough 3.5 hold wih m > 2 + d/2, hen by virue of Corollary 2.5 equaion (2.1) wih iniial condiion (2.2) has a unique classical soluion u = {u (x) : [, T ], x R d }. Now we are in he posiion o formulae he firs main heorem of his secion, wih he noaion G R h = G h B R. Theorem 3.5. Le l > d/2 be an ineger. Le Assumpions 3.1 hrough 3.4 hold wih m 4+l, and le Assumpions 2.2 (c) and Assumpion 3.6 hold wih m 2+l and wih l+1, respecively. Then if τ is sufficienly small, hen for any h >, R > 1 he sysem of equaions (3.9)-(3.1) has a unique soluion (v R,h,τ i ) n i=. Moreover, for any ν (, 1), q > 1 we have E max i=,...,n max x G νr h u τi (x) v h,r,τ i (x) 2 (3.12) N 1 e δr2 E 1/q K 2q l+2 + N 2(h 4 + τ)(1 + EK 2 m), wih consans N 1 and δ > depending only on K, d, T, C q, ν and Λ, and a consan N 2 = N 2 (K, T, d, C, H, Λ ).

15 14 M. GERENCSÉR AND I. GYÖNGY As menioned above we wan o have approximaions wih higher order accuracy in h by exrapolaing from v h,r,τ. Le us recall he mehod of Richardson s exrapolaion. This echnique, firs inroduced in [22], allows one o accelerae he rae of convergence by appropriaely mixing approximaions wih differen mesh sizes, given ha a power expansion of he error in erms of he mesh sizes is available. We shall use his, based on resuls of [7], o obain higher order approximaions wih respec o he spaial mesh size h. To formulae he exrapolaion, le r, V be he (r + 1) (r + 1) Vandermonde marix V ij = (4 (i 1)(j 1) ), (3.13) (c, c 1,..., c r ) := (1,,..., )V 1, and define (3.14) v h,r,τ := r c i v h/2i,r,τ, i= where v h/2i,r,τ denoes he soluion of (3.9)-(3.1) wih h/2 i in place of h. As we shall see, even by mixing only wo approximaions wih differen mesh sizes, ha is, seing r = 1, he exrapolaion increases he order of accuracy in h from 2 o 4. The second main resul of his secion is he following. Theorem 3.6. In addiions o he assumpions of Theorem 3.5 le Assumpions 3.2, 3.4 and 3.5 hold wih m 4r l. Then for he exrapolaion v h,r,τ we have E max i=,...,n max x G νr h u τi (x) v h,r,τ i (x) 2 (3.15) N 1 e δr2 E 1/q K 2q 2+l + N 2(h 2(2r+2) + τ)(1 + EK 2 m) for any ν (, 1) and q > 1, wih consans N 1 and δ >, depending only on K, d, T, C, ν, q and Λ, and a consan N 2 = N 2 (K, T, d, C, H, r, Λ ). These heorems will be proved by using Theorem 2.6, some resuls from [7], summarized below in Theorem 3.9, and he error esimae for he ime-discreizaion, esablished in Theorem 3.1 below. Example 3.7. Consider he equaion du (x) = sin 2 (x)d 2 u (x)d + sin(x)du (x)dw for (, x) [, 1] R, where (w ) [,1] is a 1-dimensional Wiener process, wih he iniial condiion u (x) = (1 + x 2 ) 1. The choice of localizing funcion ζ R is quie arbirary, for he sae of concreeness we ae ζ R (x) := f(x R) f(x 2 R), where f(x) := 2 π arcan ex/(1 x2 ) for x < 1, and f(x) := 1 [1, ) (x) for x 1, while noing ha in pracice a simple mollified indicaor of [ R, R] may be more favourable. Noice ha ζ R (x) = 1 for x R and supp ζ R = [ 3 R, 3 + R]. For an inegers j 1 and n 1 we se h = R/(1j) and τ = 1/n. To use he exrapolaion wih r = 1 in (3.14), we need o solve wo discree equaions wih spaial mesh sizes h = h, h/2, and mix hem according o (3.14), where one can

16 LOCALIZATION ERRORS 15 chec ha he coefficiens are c = 1/3, c 1 = 4/3. Following he seps oulined above, he discree equaion we arrive a is u R, h,τ i ( h) = a( h)(δ hδ hu R, h i )( h)τ + b( h)(δ hu R, h i 1 )( h)(w τi w τ(i 1) ) for i = 1, 2,..., n and =, ±1,..., ± (3 + R)/ h, wih he iniial values u R, h,τ ( h) = (1 + 2 h2 ) 1 ζ R ( h), where a(x) = ζ 2 R (x) sin2 (x), b(x) = ζ R (x) sin(x), and δ hφ(x) = (2 h) 1 [φ(x + h) 2φ(x) + φ(x h)]. For each h one can solve he above equaion recursively in i. Taing he resuling soluions u R,h,τ and u R,h/2,τ, and seing v R,h,τ = (4/3)u R,h/2,τ (1/3)u R,h,τ, we can conclude ha he error E max i, u iτ (h) v R,h,τ i (h) 2, where i runs over, 1,..., n and runs over ± 1,..., ±.9R/h, is of order e δr2 + h 8 + τ. Before summarising some resuls from [7] on finie difference operaors L h, M h and sochasic finie difference schemes in spaial variables we need o mae an imporan remar. Remar 3.8. The concep of a soluion of (3.1)-(3.2), as a process wih values in l 2,h, he space of funcions φ : G h R wih finie norm φ 2 l 2,h = x G h φ(x) 2, is sraighforward. One can, however, also consider (3.1)-(3.2) on he whole space, ha is, for (, x) H T. Namely, when Assumpions 3.4 and 3.5 hold, hen we can loo for an F -adaped L 2 -valued soluion (u h ) [,T ] such ha almos surely for every [, T ] (3.16) u h = ψ + (L h s u h s + f s ) ds + =1 (M h, s u h s + g s ) dw s in he Hilber space L 2, where he firs inegral is undersood as Bochner inegral of L 2 -valued funcions, he sochasic inegrals are undersood as Iô inegrals of L 2 -valued processes, and he convergence of heir infinie sum is undersood in probabiliy, uniformly in [, T ]. Thus, by a well-nown heorem on SDEs in Hilber spaces wih Lipschiz coninuous coefficiens, equaion (3.16) has a unique L 2 -valued coninuous F -adaped soluion u h = (u h ) [,T ]. We refer o such a soluion as an L 2 -valued soluion o (3.1)-(3.2). We can view equaion (3.16) also as an SDE in he Hilber space W2 m, and by he same heorem on exisence and uniqueness of soluion o SDEs in Hilber spaces, we ge a unique W2 m -valued coninuous F -adaped soluion o i. Consequenly, if Assumpions 3.4 and 3.5 hold wih m, hen u = (u h ) [,T ], he L 2 -valued soluion o (3.1)-(3.2) is a W2 m -valued coninuous F -adaped process. Also, i is sraighforward o see ha hese wo conceps of soluions are compaible in he sense ha if m > d/2, hen he resricion of he L 2 -valued soluion o G h solves (3.1)-(3.2) as an l 2,h -valued process. Noe ha (3.4)-(3.5) is a special case of he class of equaions of he form (3.1)-(3.2), and so he above discussion applies o i as well. The analogous conceps will be used for soluions of (3.9)-(3.1). Namely, a sequence of L 2 -valued random variables (v h,r i ) n i= is called an L 2-valued soluion o (3.9)-(3.1) if v h,r i is F iτ -measurable for i =, 1, 2,..., n, and he equaliies hold

17 16 M. GERENCSÉR AND I. GYÖNGY for almos all x R d, for almos all ω Ω. If (v h,r i ) n i= is an L 2-valued soluion o (3.9)-(3.1) such ha v h,r i W2 m (a.s.) for m > d/2 for each i, hen i is easy o see ha he resricion of he coninuous version of v h,r i o G h supp ζ R for each i gives a soluion {v i (x) : x G h supp ζ R, i =, 1,..., n} o (3.9)-(3.1). Theorem 3.9. Le Assumpions 3.3, 3.4 and 3.5 hold wih an ineger m. Then (a) For any φ Wp m and γ m 2(D γ φ, D γ L h φ) + D γ M h, φ 2 N φ 2 m for all ω Ω and [, T ] wih a consan N = N(K, m, d, Λ ); (b) There is a unique L 2 -valued soluion u h of (3.1)-(3.2). I is a W m 2 -valued process wih probabiliy one, and for any q > E sup u h q m NEKm q T wih a consan N = N(K, m, d, Λ, q, T ); (c) If for an ineger r Assumpions 3.1 hrough 3.5 wih m > 4r d/2 hold, hen for any q > E sup max u (x) ū h (x) q Nh q(2r+2) EKm, q T x G h where u is he classical soluion o (2.1)-(2.2), ū h = r i= uh/2i, and N is a consan depending only on K, d, T, q, Λ and m. As discussed above, under Assumpions 3.4 and 3.5 we have a unique L 2 -valued soluion u h,r = (u h,r ) [,T ] o (3.4)-(3.5), and i is a coninuous W2 m -valued process. Theorem 3.1. (i) Le Assumpions 3.2 hrough 3.5 hold wih m. Then for sufficienly small τ here exiss for all h and R > a unique L 2 -valued soluion v h,r,τ o (3.9)-(3.1) such ha v h,r,τ i W2 m for every i =, 1,..., n and ω Ω. (ii) If Assumpion 3.6 holds wih some ineger l and Assumpions 3.1 hrough 3.5 hold wih m = l + 3, hen (3.17) max i n E uh,r τi v h,r,τ i 2 l Nτ(1 + EKm), 2 wih a consan N = N(K, C, H, d, T, l, Λ ). (iii) Le l be an ineger. If Assumpion 3.6 holds wih l + 1 in place of l, and Assumpions 3.1 hrough 3.5 hold wih m = l + 4, hen (3.18) E max i n uh,r τi v h,r,τ i 2 l Nτ(1 + EKm) 2 wih a consan N = N(K, C, H, d, T, l, Λ ). Proof. To prove solvabiliy of he sysem of equaions (3.9)-(3.1) we rewrie (3.1) in he form (3.19) (I τl h,r τi )v i = v i 1 + τf τ(i 1) + =1 (M h,r, τ(i 1) v i 1 + g R, τ(i 1) )ξ i, i = 1, 2,..., n, where I denoes he ideniy operaor. We are going o show by inducion on j n ha for sufficienly small τ for each j here is a sequence of W m 2 -valued random

18 LOCALIZATION ERRORS 17 variables (v i ) j i=, such ha vi is F iτ -measurable, v = ψ R and (3.19) holds for 1 i j. For j = here is nohing o prove. Le j 1 and assume ha our saemen holds for j 1. Consider he equaion (3.2) Dv = X, where D := I τl h,r τj, X := v j 1 + τf τ(j 1) + =1 (M h,r, τ(j 1) v i 1 + g R, τ(j 1) )ξ j. In he following we ae κ o be eiher or m. I easy o see ha D is a bounded linear operaor from W2 κ ino W2 κ, for each ω Ω and τ. Le us define he norm κ in W2 κ by ϕ 2 κ = (I ) κ/2 φ 2 = γ: γ κ C γ D γ φ 2, := d Di 2, where C γ is a posiive ineger for each muli-index γ, γ κ. Thus κ is a Hilber norm which is equivalen o o κ. We denoe he corresponding inner produc in W2 κ by (, ) κ. By virue of Theorem 3.9 (a) for all ω Ω and τ we have (v, Dv) κ = v κ τ(l h,r τj v, v) κ v 2 κ τn v 2 κ, for all v W κ 2 where he dependence of N is as in he heorem, in paricular, i is independen of h, R. Consequenly, for τ < 1/N we have (v, Dv) κ δ v 2 κ for all v W κ 2, ω Ω, where δ = 1 τn >. Hence by he Lax-Milgram lemma for every ω Ω here is a unique v = v κ W2 κ such ha (Dv κ, ϕ) κ = (X, ϕ) κ for all ϕ C (R d ). Since (Y, ϕ) κ = (Y, (I ) κ ϕ) for all Y W κ 2 and ϕ C (R d ), we have (Dv κ, (I ) κ ϕ) = (X, (I ) κ ϕ) for all ϕ C (R d ). Hence, aing ino accoun ha {(I ) m ϕ) : ϕ C (R d )} is dense in W 2 = L 2, we ge ha v m solves (3.2) in L 2 as well, so by uniqueness, v m = v. This means (3.2) has a unique soluion v L 2 for every ω Ω, and v W m 2 for every ω Ω. Since X and Dφ are W m 2 -valued F jτ -measurable random variables for every ϕ W m 2, he unique soluion v W m 2 o (3.2) is also F jτ -measurable. This finishes he inducion, and he proof of saemen (i) of he heorem. For pars (ii) and (iii) EK 2 m < may and will be assumed. Hence by Theorem 3.9 (b), (3.21) E sup u h,r 2 ϱ NEKϱ 2 < for ϱ =, 1, 2,..., m. [,T ] As Assumpion 3.6 also holds, we have (3.22) E ψ 2 l+κ + max E f i 2 l+κ + max E g i n i n i 2 l+κ < wih κ = in par (ii) and κ = 1 in par (iii), and hence, wih he same κ, (3.23) E v h,r,τ i 2 l+κ < for every i =, 1,..., n.

19 18 M. GERENCSÉR AND I. GYÖNGY To sar he proof of (ii), le us fix a muli-index γ wih lengh γ =: ϱ l. From equaions (3.9)-(3.1) and (3.1)-(3.2), we ge ha he error e i := u h,r τi v h,r,τ i is a W2 m -valued F τi -measurable random variable, i =,..., n, and D γ e i is he L 2 - valued soluion of he equaion D γ e i =D γ e i 1 + D γ L h,r τi e i τ + τi τ(i 1) τi + D γ M h,r, τ(i 1) e i 1ξ i + for i = 1,..., n, wih zero iniial condiion, where D γ F s ds τ(i 1) D γ G s dw s F := L h,r u h,r L h,r κ 2() uh,r κ + f R 2() fκ R, κ 1() 2() := κ n 2 () := ( n + 1)/n, G := M h, u h,r M h,r, κ 1() uh,r κ + 1() gr, gκ, κ 1() 1() := κ n 1 () := n /n. To ease noaion we se L i := L h,r τi, M i := M h,r, τi, F i := τi τ(i 1) D γ F s ds, G i := τi τ(i 1) D γ G s dw s, and by using he simple ideniy b 2 a 2 = 2(b, b a) b a 2 wih b := D γ e i and a := D γ e i 1, we ge D γ e i 2 D γ e i 1 2 (3.24) wih = 2(D γ e i, D γ L i e i τ + F i ) + 2(D γ e i, D γ M i 1e i 1 ξ i + G i ) D γ e i D γ e i 1 2 = 2(D γ e i, D γ L i e i τ + F i ) + 2(D γ e i 1, D γ M i 1e i 1 ξ i + G i ) + 2(D γ e i D γ e i 1, D γ M i 1e i 1 ξ i + G i ) D γ e i D γ e i 1 2 = 2(D γ e i, D γ L i ie i τ + F i ) + 2(D γ e i 1, D γ M i 1e i 1 ξ i + G i ) + D γ M i 1e i 1 ξ i + G i 2 D γ L i e i τ + F i 2 I (1) i + I (2) i + I (3) i + I (4) i + I (5) i + I (6) i I (1) i :=2τ(D γ e i, D γ L i e i ) I (2) i :=2(D γ e i, F i ) I (3) i :=2(D γ e i 1, D γ M i 1e i 1 ξ i + G i ) I (4) i := D γ M i 1e i 1 ξ i 2 I (5) i :=2(D γ M i 1e i 1 ξ i, G i ) I (6) i := G i 2. By he Young and Jensen inequaliies, and basic properies of sochasic Iô inegrals we have (3.25) I (2) i τ D γ e i 2 + τ 1 F i 2 τ D γ e i 2 + τi τ(i 1) F s 2 ϱ ds, EI (3) i =,

20 LOCALIZATION ERRORS 19 (3.26) EI (4) i = τe D γ M i 1e i 1 2, (3.27) EI (6) i τi = E τ(i 1) τi D γ G s 2 ds E τ(i 1) G s 2 ϱ ds. By Iô s ideniy for sochasic inegrals τi EI (5) i = 2E D γ M i 1e i 1 (x)d γ G s(x) dx ds. τ(i 1) R d Here by inegraion by pars we drop one derivaive from D γ M i 1 e i 1 on he erm D γ G s, and hen by he Cauchy-Schwarz-Bunyaovsy and Young inequaliies we ge EI (5) i τi τne e i 1 2 ϱ + NE τ(i 1) Using (3.26), by Theorem 3.9 (a) we have (3.28) EI (1) i + EI (4) i NτE e i 2 ϱ. G s 2 ϱ+1 ds. Therefore, by aing expecaions and summing up (3.24) over i from 1 o j n, and over γ for γ l, we ge j T (3.29) E e j 2 l N τ E e i 2 l + N E ( F s 2 l + G s 2 l+1) ds for j = 1,..., n, where N = N (K, C, Λ, l, d) is a consan. Noice ha due o (3.23) and (3.21) E e i 2 l < i = 1, 2,..., n, and due o (3.21) and Assumpions 3.4 and 3.6 we have E T F s 2 l + G s 2 l+1 ds <. Hence he righ-hand side of inequaliy (3.29) is finie. Thus when τ < 1/N, from (3.29) by discree Gronwall s lemma i follows ha (3.3) E e j 2 l N (1 N τ) j E T for j = 1,..., n. Now we are going o show ha (3.31) E F 2 l + ( F 2 l + E G 2 l+1 Nτ(EK 2 l+3 + 1) G 2 l+1) d for all [, T ] wih a consan N = N(K, C, c, d, T, l, Λ ). To esimae E F 2 ϱ, firs noice ha due o Assumpion 3.6, (3.32) E f R f R κ 1() 2 l Nτ, and due o Assumpions 3.2, 3.4 and 3.6 wih L h,r u h,r L h,r κ 2() uh,r κ 2() 2 l 2A 1 () + 2A 2 () A 1 () := (L h,r L h,r κ 2() )uh,r κ 2() 2 l Nτ u h,r κ 2() 2 l+2,

21 2 M. GERENCSÉR AND I. GYÖNGY A 2 () := L h,r (u h,r u h,r κ 2() ) 2 l N u h,r By virue of Theorem (b) and Assumpion 3.2, κ 2() uh,r 2 l+2. (3.33) EA 1 () Nτ sup E u h,r 2 l+2 NτEKl+2. 2 [,T ] To esimae A 2 we show ha (3.34) E u h,r u h,r s 2 l+2 N s (1 + EK 2 l+3) for all s, [, T ]. To his end we mollify u h,r in he spaial variable. We ae a nonnegaive φ C (R d ) suppored on he uni ball in R d such ha i has uni inegral, and for funcions ϕ, which are locally inegrable in x R d, we define ϕ (ε) by ϕ (ε) (x) := ε d ϕ(y)φ((x y)/ε) dy, x R d, for ε >. R We will mae use of he following nown and easily verifiable properies of ϕ (ε). For inegers r and ε >, (3.35) ϕ (ε) ϕ r ε ϕ r+1 for ϕ W2 r+1, and (3.36) ϕ (ε) r ϕ r, (D i ϕ) (ε) r = D i ϕ (ε) r N i ε ϕ r, ϕ W2 r for i = 1, 2,..., d, where N i depends only on he sup and L 1 norms of D i φ. Thus by (3.35) (3.37) u h,r + d u h,r s 2 l+2 = u h,r u h,r s 2 l+1 + u h,r D i u h,r u h,r u h,r s 2 l+1 + d (D i u h,r ) (ε) 2 l+1 + u h,r s 2 l+1 + d d D i (u h,r d D i (u h,r + Nε 2 ( u h,r l+2 + u h,r s l+2 ). (D i u h,r s D i (u h,r Since u h,r saisfies (3.1)-(3.2), for s T we have (3.38) wih u h,r B 1 := B 2 := u h,r s 2 l+1 2B 1 + 2B 2 and using (3.36) we have d (3.39) D i (u h,r s s (L h,r r (M h,r, r u h,r r u h,r r + fr R ) dr + g R, r u h,r s ) (ε) 2 l+1 u h,r s ) 2 l+1 (D i u h,r s ) (ε) 2 l+1 u h,r s ) (ε) 2 l+1 2 l+1, ) dwr 2 l+1, u h,r s ) (ε) 2 l+1 N ε 2 B 1 + B 3,

22 LOCALIZATION ERRORS 21 wih B 3 := I is easy o see ha d s EB 1 ( s) (D i M h,r, r (3.4) N( s) 2 (sup T s u h,r r + D i g R, r ) dwr E L h,r s u h,r s + fs R 2 l+1 ds E u 2 l+3 + sup E f 2 l+1). T Using Iô s ideniy, for B 2 we obain EB 2 =E Mr h,r, u h,r r + gr 2 l+1 dr (3.41) s N( s)(sup T and in he same way, for B 3 we have (3.42) EB 3 N( s)(sup T From (3.37) hrough (3.41) we ge E u h,r for every ε >, where Choosing here ε := s 1/2 gives E u h,r E u h,r 2 l+1. 2 l+2 + sup E g 2 l+1), T 2 l+3 + sup E g 2 l+2). T u h,r s 2 l+2 N(ε 2 + ( s) 2 ε 2 + s )J J := sup{e u h,r 2 l+3 + E g 2 l+2 + E f 2 l+1}. T E u h,r u h,r s 2 l+2 3N( s)j. Hence we obain (3.34), since by Theorem 3.9 and Assumpion 3.6 we have From (3.34) we ge he esimae J NEK 2 l+3 + H. EA 1 Nτ(EK 2 l+3 + 1), which ogeher wih (3.33) and (3.32) shows ha E F 2 l is esimaed by he righhand side of he inequaliy (3.31). Similarly, by maing use of (3.34), we can ge he same esimae for E G 2 l+1, which finishes he proof of (3.31). From (3.3) and (3.31) we have max E e j 2 ϱ τn(1 N τ) n (EK ϱ+3 + 1) j n for τ < N 1, wih a consan N = N(K, C, H, d, l, T, Λ ). Hence noicing ha lim (1 N τ) n = e NT, n we obain (3.17), which complees he proof of (ii). To prove (iii) noice firs ha by using (ii) wih l + 1 in place of l we have n (3.43) E e i 2 l+1 N(1 + EKm) 2

23 22 M. GERENCSÉR AND I. GYÖNGY wih m = l + 4. Fixing a muli-index γ as in (ii), wih γ = ϱ l, we revisi (3.24) and observe ha I (5) i I (4) i + I (6) i and by Theorem 3.9 I (1) i Nτ e i 2 l. Thus summing up (3.24) over i = 1,..., j, and recalling also (3.25) and (3.26), we obain n T E max 1 j n Dγ e i 2 NτE e i 2 l + N E F s 2 l ds +N T Hence, noicing ha E G s 2 l ds + τ n E D γ M i 1e i E max D γ M i 1e i 1 2 N e i 2 l+1, =1 by using (3.43) and (3.31), we ge (3.44) E max 1 i n Dγ e i 2 Nτ(EKm 2 + 1) + E max 1 j n Clearly, j where (Z ) [,T ] is defined by jτ I (3) i = Z dw, j = 1, 2,..., n, j 1 j n j I (3) i. I (3) i. Z = 2(D γ e i 1, D γ M i 1e i 1 + D γ G ) for ( i 1, i ], i = 1, 2,..., n. I is easy o see ha T Z 2 d 4 max 1 i n Dγ e i 2 ( n T ) D γ M i 1e i 1 2 τ + G s 2 l ds ( 1 36 max n 1 i n Dγ e i e i 1 2 l+1τ + 2 T G s 2 l ds). Hence by he Davis inequaliy we have ( ) 1/2 T S n E sup Zs dws 3E Zs 2 ds [,T ] 1 2 E max 1 i n Dγ e i τ n T E e i 1 2 l E G s 2 l ds <. Using his, and esimaes (3.31) and (3.43), we obain (iii) from (3.44). By virue of Remar 3.8, by using Sobolev s heorem on embedding W2 m for m > d/2, we ge he following corollary. ino C b

24 LOCALIZATION ERRORS 23 Corollary (i) Le Assumpions 3.2 hrough 3.5 hold wih m > d/2. Then for sufficienly small τ here exiss for all h and R > a unique soluion v h,r,τ = {v h,r,τ iτ (x) : x G h supp ζ R } o (3.9)-(3.1). (ii) If Assumpion 3.6 holds wih some ineger l > d/2 and Assumpions 3.1 hrough 3.5 hold wih m > d/2 + 3, hen (3.45) max E max u h,r τi i n x G h v h,r,τ i 2 Nτ(1 + EKm), 2 wih a consan N = N(K, C, H, d, T, l, Λ ). (iii) If Assumpion 3.6 holds wih l > d/2 + 1 and Assumpions 3.1 hrough 3.5 hold wih m > d/2 + 4, hen (3.46) E max max u h,r τi v h,r,τ i 2 Nτ(1 + EKm) 2 i n x G h wih a consan N = N(K, C, H, d, T, l, Λ ). Proof of Theorems 3.5 and 3.6. The solvabiliy of (3.9)-(3.1) has already been discussed above, so we need only prove he esimaes (3.12) and (3.15). A naural way o separae he errors of he differen ypes of approximaions would be o wrie u τi (x) v h,r,τ i (x) u τi (x) u h τi(x) + u h τi(x) u h,r τi (x) + u h,r τi (x) v h,r,τ i (x). However, in such a decomposiion we canno direcly esimae he middle erm on he righ-hand side, ha is, he localizaion error of he finie difference equaion. Therefore we inroduce u,r, he classical soluion of (2.1)-(2.2) wih daa D := D R = (ζ R ψ, ζ 2 Ra, ζ R b, ζ R c, ζ R σ, ζ R µ, ζ R f, ζ R g). Clearly he pair D, D saisfies Assumpion 2.4. Also, as he finie difference coefficiens a, b are compaible wih he daa D in he sense ha Assumpion 3.1 is saisfied, i follows ha he coefficiens a R, b R, as defined in (3.7)-(3.8), are compaible wih he daa D in he same sense. Therefore in he decomposiion u τi (x) v h,r,τ i (x) u τi (x) u,r (x) + u,r(x) uh,r τi τi τi (x) + u h,r τi (x) v h,r,τ i (x) he firs erm can be reaed by Theorem 2.6, he second by Theorem 3.9 (c), and he hird by Corollary 3.11 (iii). Adding up he resuling errors we ge he esimae (3.12), and he dependence of he consans also follows from he invoed heorems. Similarly, for (3.15) we wrie u τi (x) v h,r,τ i (x) u τi (x) u,r τi (x) + u,r τi (x) ūh,r τi (x) r + c j u h/2j,r τi (x) v h/2j,r,τ i (x), j= where ū h,r = r j= c ju h/2j,r, and follow he same seps as above. Remar As i can be easily seen from he las sep of he proof, Assumpion 3.6 can be weaened o α-hölder coninuiy for any fixed α >, a he cos of lowering he rae from 1/2 o α (1/2). To decrease he spaial regulariy condiions, in paricular, he erm d/2, one would need he generalizaion of he resuls of [7], and subsequenly, of Theorem 3.1, o arbirary Sobolev spaces W m p. Parial resuls in his direcion can be found in [5].

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