Dimensional reduction in nonlinear filtering: A homogenization approach

The Annals of Applie d Pr obabil ity 2013, V ol. 23, N o. 6, 2290 –2326 DOI: 10.1214 /12-AAP901 c  Institute of Mathematical Statistics , 2 013 DIMENSIONAL REDUCTION IN NO NLINEAR FIL TERING: A HOMOG ENIZA TION APPR OA CH By Peter Imkell er 3 , N. Sri Namachchiv a y a 1 , Nicolas Perko wsk i 2 , 4 and Hoong C. Yeong 1 Humb oldt-Universit¨ at zu Berlin, U niversity of Il linois at Urb ana-Champ aign, Hu mb oldt-Universit¨ at zu Berlin and University of Il linois at Urb ana-Champ aign W e prop ose a homogenized filter for m ultiscale signals, whic h allo ws us to redu ce the dimension of t he system. W e prov e that the nonlinear filter c onv erges to o ur homogenized filter with rate √ ε . This is achiev ed by a suitable asymptotic expansion of the du al of the Zak ai equation, and by probabilistically representing the correction terms with the help of BDSD Es. 1. In tro du ction. Filtering theory is an established field in applied proba- bilit y and decision and con trol systems, whic h is imp ortant in man y p ractical applications from iner tial guidance of aircrafts and sp acecrafts to w eather and climate prediction. It pr o vides a recursive algo rithm f or estimating a signal or state of a r andom dy n amical system based on noisy measuremen ts. More pr ecisely , filtering pr ob lems consist of an unobserv able signal p ro cess X def = { X t : t ≥ 0 } and an observ ation pr o cess Y def = { Y t : t ≥ 0 } that is a func- tion of X corrupted b y noise. The main ob jectiv e of filtering theory is to get the b est estimate of X t based on the inf orm ation Y t def = σ { Y s : 0 ≤ s ≤ t } . This is giv en b y the conditional distribution π t of X t giv en Y t or equiv a- len tly , the conditional exp ectations E [ f ( X t ) |Y t ] f or a r ic h enough class of functions. S ince this estimate minimizes the m ean square err or loss, we call Received December 2011; revised S eptember 2012. 1 Supp orted by the NS F under Grant num b ers CMMI 10-30144 and EFRI 10-24772 and by A FOSR und er Grant num b er F A9550-12-1-039 0. 2 Supp orted by a Ph.D. scholarship of the Berlin Mathematical School. 3 F unded by NSF Grant number CMMI 10-30144. 4 F unded by NSF Grant num b er EFRI 10-24772 and by the Berlin Mathematical School. AMS 2000 subje ct classific ations. 60G35, 35B27, 60H15, 60H35. Key wor ds and phr ases. Nonlinear filtering, dimensional reduction, homogenization, particle filtering, asymptotic expansion, SPDE, BDSDE. This is an electr o nic repr int of the origina l a rticle published by the Institute of Mathematical Statistics in The Annals of Applie d Pr ob ability , 2013, V ol. 23, No. 6, 2 290– 2326 . This r eprint differs from the orig inal in pagination and typo graphic detail. 1 2 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G π t the optimal fi lter. The goal of fi ltering theory is to c haracterize this con- ditional distribu tion effectiv ely . In simplified prob lems where the signal and the observ ation m o dels are linear and Gaussian, the filtering equation is finite-dimensional, and the solution is the w ell-kno wn Kalman–Bucy fi lter. In more realistic problems, nonlinearities in the mo d els lead to more com- plicated equ ations for π t , d efi ned by Zak ai ( 196 9 ) and F ujisaki, Kallianpur and Ku n ita ( 1972 ), whic h d escrib e the ev olution of the conditional distrib u- tion in the space of probabilit y measures; see, for example, Bain and Cr isan ( 2009 ), K allianpur ( 1980 ), Liptser and Shirya ev ( 2001 ). It is impractical to imp lemen t a numerical solution to such infinite d i- mensional sto chasti c evol ution equations of th e general nonlinear filtering problem b y finite difference or finite elemen t appro ximations. Therefore, extended Kalman filter algorithms, w hic h use linear appro ximations to the signal dynamics and observ ation, h a ve b een used extensivel y in seve ral appli- cations. These p ro vide essentia lly a fi rst-order ap p ro xim ation to an infinite dimensional problem and can p erform quite p oorly in problems with strong nonlinearities. P article filters hav e b een w ell established for th e implemen- tation of nonlinear filtering in science and engineering applications. Doucet, de F reitas and Gord on ( 2001 ) and Arulampalam et al. ( 2002 ) p ro vide com- prehensive insight into particle filtering. Ho wev er, due to d imensionalit y issues [see, e.g., Snyder et al . ( 2008 )] and computational complexities that arise in represent ing the signal densit y using a high num b er of p articles, the problem of particle filtering in h igh dimens ions is still not completely resolv ed. As a result of these difficulties, we h a ve established a no v el parti- cle filtering metho d Pa rk, Namac hc h iv a y a and Y eong ( 2011 ) for multiscale signal and observ ation pro cesses that com bines the homogenization with fil- tering tec hniqu es. The theoretical basis for this new capabilit y is p resen ted in this p ap er. The results presented here are set within the cont ext of s lo w-fast d ynam- ical systems, where the r ates of c hange of differen t v ariables differ by orders of magnitud e. Multiple time scales o ccur in m o dels throu gh ou t the science and engineering field. F o r examp le, climate ev olution is go v ern ed b y fast at- mospheric and slo w o ceanic dynamics and state dynamics in electric p o w er systems consists of fast- and slo wly-v aryin g elemen ts. This p ap er addresses the effects of the m ultiscale signal and observ ation pro cesses via the stu dy of the Zak ai equation. W e construct a lo wer d imensional Zak ai equation in a canonical wa y . This problem h as also b een studied in P ark, So wers and Sri Namac h c hiv a y a ( 2010 ) using a differen t appr oac h fr om w hat is pr esen ted here. In mo derate dimensional p roblems, particle fi lters are an attractiv e al- ternativ e to numerical approximati on of the sto c h astic partial differen tial equations (SPDEs) by finite d ifference or finite elemen t metho ds . F or the reduced nonlinear mo del an appropr iate form of particle filter can b e a viable DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 3 and usefu l sc h eme. Hence, L ingala et al. ( 2012 ) p resen ts the numerical solu- tion of the low er dimensional sto c hastic p artial differenti al equ ation d eriv ed here, as it is applied to a c haotic high-dimensional m ultiscale sys tem. In general, this pap er p ro vid es rigorous mathematical results that supp ort the numerical algo rithms based on the idea that sto c hastically a verage d mo dels p r o vide qualita tiv ely useful results wh ic h are p oten tially helpful in dev eloping inexp ensiv e lo wer dimens ional filtering as demonstrated by P ark, Namac hc h iv a y a and Y eo ng ( 2011 ) in the conte xt of homogenized particle filters and by Harlim and K ang ( 2012 ) in the con text of av er aged ensem b le Kalman filters. Th e con vergence of the optimal filter to th e homogenized filter is sh o wn u sing bac kward s to c hastic different ial equations (BSDEs) and asymptotic tec hniques. Let us describ e the main result. W e assu m e the signal is giv en as s olution of th e t w o time scale sto c h astic differen tial equ ation (SDE) dX ε t = b ( X ε t , Z ε t ) dt + σ ( X ε t , Z ε t ) dV t , d Z ε t = 1 ε f ( X ε t , Z ε t ) dt + 1 √ ε g ( X ε t , Z ε t ) dW t . Here X ε is the slo w comp onent, and Z ε is the fast comp onen t. W e assume that for ev ery fixed x , the solution Z x of d Z x t = f ( x , Z x t ) dt + g ( x, Z x t ) dW t is ergo d ic and con v erges r apidly to its uniqu e stationary distribution. In this case it is w ell known that X ε con verge s in distribu tion to a diffu sion X 0 whic h is go verned by an SDE dX 0 t = ¯ b ( X 0 t ) dt + ¯ σ ( X 0 t ) dV t . This X 0 is used to construct an av eraged filter π 0 . W e denote the optimal filter for the full system by π ε . Define the x -marginal of π ε as π ε,x , th at is, Z ϕ ( x ) π ε,x t ( dx ) = Z ϕ ( x ) π ε t ( dx, dz ) . Our main resu lt is th en the follo win g: Theorem. Under the assumptions state d in The or em 3.1 , f or e v ery p ≥ 1 and T ≥ 0 ther e exists C > 0 , such that for e v ery ϕ ∈ C 4 b ( E Q [ | π ε,x T ( ϕ ) − π 0 T ( ϕ ) | p ]) 1 /p ≤ √ εC k ϕ k 4 , ∞ . In p articular, ther e exists a metric d on the sp ac e of pr ob ability me asur es, such that d ge ne r ates the top olo gy of we ak c onver genc e, and such that for every T ≥ 0 ther e exi sts C > 0 such that E Q [ d ( π ε,x T , π 0 T )] ≤ √ εC. 4 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G W e b egin in Section 2 by present ing the general formula tion of the multi - scale nonlinear filtering problem. Here w e describ e the m easure-v alued Z ak ai equation and introdu ce the homogenized equations th at w e seek to d eriv e for the reduced dimension unn ormalized filter. Section 3 presen ts the for- mal asymptotic expansion of the m ulti scale Zak ai equation that results in sev eral SPDEs. W e also p resen t the main r esults of this pap er in this sec- tion. Section 4 pro vid es the pr ob ab ilistic represen tation of the SPDEs, that is, w e d escrib e the solutions of the infin ite d imensional S PDEs by fi nite d i- mensional bac kward doub ly s to c hastic differen tial equations (BDSDEs). W e restate some of th e results in this conte xt d ue to Rozo vski ˘ ı ( 1990 ) and Pa r- doux and P eng ( 1994 ) at the end of th is sectio n. W e present some of the preliminary results of Pardoux and V e retennik ov ( 2003 ) on con vergence of the trans ition fun ction of Z x in Section 5 . These estimates are used in the pro of of the main r esults pr esen ted in Section 6 . 2. F o rm u lation of multisca le nonlinear filtering problems. Let (Ω , F , ( F t ) , Q ) b e a filtered p robabilit y space that sup p orts a ( k + l + d )-dimensional standard Bro wn ian motion ( V , W, B ). Let the signal ( X ε , Z ε ) b e a t wo time scale diffu sion pro cess with a fast comp onent Z ε and a slow comp onen t X ε , dX ε t = b ( X ε t , Z ε t ) dt + σ ( X ε t , Z ε t ) dV t , (1) d Z ε t = 1 ε f ( X ε t , Z ε t ) dt + 1 √ ε g ( X ε t , Z ε t ) dW t , where X ε t ∈ R m , Z ε t ∈ R n , W t ∈ R l and V t ∈ R k are indep enden t standard Bro wnian motions, b : R m + n → R m , σ : R m + n → R m × k , f : R m + n → R n , g : R m + n → R n × l . All the f unctions ab o v e are assumed to b e Borel measurable. F or fixed x ∈ R m , d efine d Z x t = f ( x, Z x t ) dt + g ( x, Z x t ) dW t . (2) Assume that for all x ∈ R m , Z x is ergo dic and con v erges rapidly to wards its stationary measure µ ( x, · ). W e will make this precise later. The d -dimensional observ ation Y ε is giv en by Y ε t = Z t 0 h ( X ε s , Z ε s ) ds + B t with Borel-measurable h : R m + n → R d . B is assumed to b e a d -dimensional standard Bro wnian motion that is indep enden t of W and V . Define Y ε t = σ ( Y ε s : 0 ≤ s ≤ t ) ∨ N , where N are th e Q -negligible sets. F or a fin ite measure π on R m + n and for a b ounded measurable fun ction ϕ on R m + n denote π ( ϕ ) = R ϕ ( x, z ) π ( dx, dz ). Then our aim is to calculate the measure-v alued p ro cess ( π ε t , t ≥ 0) determined by π ε t ( ϕ ) = E [ ϕ ( X ε t , Z ε t ) |Y ε t ] . DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 5 Define the Girsano v transform d P ε d Q    F t = D ε t = exp  − Z t 0 h ( X ε s , Z ε s ) ∗ dB s − 1 2 Z t 0 | h ( X ε s , Z ε s ) | 2 ds  . Under P ε , th e observ ation pro cess, Y ε , is a Bro wnian motion and indep en- den t of ( X ε , Z ε ). By the Kallianpur–Str ieb el form ula, E Q [ ϕ ( X ε t , Z ε t ) |Y ε t ] = E P ε [ ϕ ( X ε t , Z ε t )( d Q /d P ε ) | F t |Y ε t ] E P ε [( d Q /d P ε ) | F t |Y ε t ] with d Q d P ε    F t = ˜ D ε t = exp  Z t 0 h ( X ε s , Z ε s ) ∗ d Y ε s − 1 2 Z t 0 | h ( X ε s , Z ε s ) | 2 ds  . So if we define ρ ε t ( ϕ ) = E P ε  ϕ ( X ε t , Z ε t ) exp  Z t 0 h ( X ε s , Z ε s ) ∗ d Y ε s − 1 2 Z t 0 | h ( X ε s , Z ε s ) | 2 ds     Y ε t  , then π ε t ( ϕ ) = ρ ε t ( ϕ ) ρ ε t (1) . Denote by L ε = 1 ε L F + L S the d ifferen tial op erato r asso ciated to ( X ε , Z ε ). That is, L F = n X i =1 f i ( x, z ) ∂ ∂ z i + 1 2 n X i,j =1 ( g g ∗ ) ij ( x, z ) ∂ 2 ∂ z i ∂ z j , L S = m X i =1 b i ( x, z ) ∂ ∂ x i + 1 2 m X i,j =1 ( σ σ ∗ ) ij ( x, z ) ∂ 2 ∂ x i ∂ x j , where · ∗ denotes th e transp ose of a matrix or a v ector. Then the unnorm alized measure-v alued pro cess, ρ ε , satisfies the Zak ai equation dρ ε t ( ϕ ) = ρ ε t ( L ε ϕ ) dt + ρ ε t ( hϕ ) d Y ε t , (3) ρ ε 0 ( ϕ ) = E Q [ ϕ ( X ε 0 , Z ε 0 )] for every ϕ ∈ C 2 b ( R m + n , R ) ; see, for exa mple, Bain and Crisan ( 2009 ). F or k ≥ 0, C k b is the sp ace of k times con tinuously differenti able functions f , suc h that f and all its p artial deriv ativ es up to order k are b oun ded. The theory of stochastic av eraging [see, e.g., Pa panicolaou, Stro o ck and V aradhan ( 1977 )] tells us that un der suitable conditions, X ε con verge s in la w to X 0 as ε → 0 , where X 0 is the solution of an SDE dX 0 t = ¯ b ( X 0 t ) dt + ¯ σ ( X 0 t ) dW t for su itably av eraged ¯ b and ¯ σ . Denote the generator of X 0 b y ¯ L . 6 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G W e w an t to sho w that as long as we are only intereste d in estimating the slow c omp onent , w e can tak e adv an tage of this f act. More pr ecisely , we wa n t to find a homogenized (unnnormalized) filter ρ 0 , s uc h th at for small ε , ρ ε,x whic h is the x -marginal of ρ ε t , is close to ρ 0 . Th e x -marginal of ρ ε t is d efi ned as ρ ε,x t ( ϕ ) = Z R m + n ϕ ( x ) ρ ε t ( dx, dz ) for eve ry measurable b oun d ed ϕ : R m → R , an d ρ 0 is the solution of dρ 0 t ( ϕ ) = ρ 0 t ( ¯ L ϕ ) dt + ρ 0 t ( ¯ hϕ ) d Y ε t , (4) ρ 0 0 ( ϕ ) = E Q [ ϕ ( X 0 0 )] , where ¯ h is a suitably a v eraged v er s ion of h . The measure-v alued pro cesses π 0 and π ε,x are then d efined in terms of ρ 0 and ρ ε,x as π ε w as defined in terms of ρ ε , π 0 t ( ϕ ) = ρ 0 t ( ϕ ) ρ 0 t (1) and π ε,x t ( ϕ ) = ρ ε,x t ( ϕ ) ρ ε,x t ( ϕ ) . Note that the homogenized filter is still driv en by the real observ ation Y ε and not by a “homogenized observ ation,” whic h is practical for imp lemen tation of the homogenized filter in applications since such homogenized observ ation is u sually n ot a v ailable. Ho wev er, should suc h homogenized observ ation b e a v ailable, using it wo uld lead to loss of in formation for estimating the signal compared to us in g the actual observ ation. In this p ap er, we will p r o ve L 1 -con ve rgence of the actual fi lter to th e homogenized filter, that is, w e will show th at for an y T > 0, lim ε → 0 E [ d ( π ε,x T , π 0 T )] = 0 , where d denotes a suitable distance on the space of probability measures that generates the top ology of we ak con ve rgence. Th is con verge nce r esult is sho w n in Pa r k, So w ers and Sri Namac hchiv a ya ( 2010 ) for a t wo-dimensional m u ltiscale s ignal pro cess with no dr ift in the fast comp onen t SDE. Here, we extend the result to an R m + n -dimensional signal pr o cess with d rift and diffu- sion co efficien ts of the fast and s lo w comp onen ts d ep endent on b oth comp o- nen ts. The pro of of P ark, Sow ers and Sr i Namac hc h iv a y a ( 2010 ) is based on represent ing the s lo w comp onen t as a time-c hanged Bro wnian motion under a su itable measure, w hic h cannot b e extended easily to the multidimensional setting w e assume here. Based on ( 3 ) and ( 4 ), the filter conv ergence p r oblem is a problem of ho- mogenizatio n of a S PDE. In P ap an icolaou, Str o o c k and V aradhan ( 1977 ), homogenizatio n of diffusion pro cesses with p eriodic s tructures is done using the martingale problem app roac h. In P apanicolao u and Kohler ( 1975 ) and DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 7 Chapter 2 of Bensoussan, Lions and P apanicolaou ( 1978 ), limit b eha vior of sto c hastic p ro cesses is studied using asymptotic analysis. Bensoussan, Li- ons and P ap an icolaou ( 1978 ) s tudy linear SPDEs with p erio dic co efficient s and also used a probabilistic approac h in Chapter 3. Homogenization in the nonlinear filtering problem fr amew ork has b een stud ied in Bensoussan and Blank enship ( 1986 ) and I c hihara ( 20 04 ) via asymp totic analysis on a dual represent ation of the nonlinear filtering equation. As far as w e are aw are, Ic hih ara ( 2004 ) has used BSDEs for studying homogenization of Zak ai-t yp e SPDEs for the first time. Ou r conv ergence pr o of applies BSDE tec hniqu es by in voking the du al representat ion of the filtering equation and u sing asymp - totic analysis to determin e the limit b eha v ior of the solution of the bac k- w ard equation. P ardoux and V eretenniko v ( 2003 ) giv e precise estimates for the transition fu nction of an ergo dic SDE of the t yp e ( 2 ), and these results are used in our pro of. T o our kno w ledge, suc h metho d of homogenization for SP DEs com b ining BSDE and asymptotic metho ds has n ot b een done b efore. T o ou r knowledge, a r esult presen ted in Chapter 6 of Ku shner ( 199 0 ) is the closest to the r esults presented in this pap er. In T heorem 6.3.1 of Kush- ner ( 1990 ) it is sh own that for a fixed test fun ction, the difference of the unnormalized actual and homogenized filters for m u ltiscale jump-diffusion pro cesses conv erges to zero in distribution. S tandard results then giv e con- v ergence in probabilit y of the fixed time marginals. Kushn er ( 1990 )’s metho d of p r o of is b y av eraging the co efficien ts of the SDEs for the unnorm alized filters and sh owing that the limits of b oth fi lters satisfy the same S DE that p ossesses a un ique solution. W e obtain L p con verge n ce of th e measure v al- ued pro cess, not just for fixed test functions, and w e are able to quanti fy the r ate of con v ergence, wh ich, to the b est of our kno wledge, has n ot b een ac hiev ed b efore in h omogenizatio n of nonlinear filters. In Kleptsin a, Liptser and Serebrovski ( 1997 ), conv ergence of the nonlin- ear filter is sh own in a very general setting, based on conv ergence in tota l v ariation distance of the law of ( X ε , Y ε ). T h is is then applied to tw o exam- ples. Sin ce the d iffu sion m atrix of our slo w comp onent is allo wed to d ep end on the fast comp onent, our r esu lts are not a sp ecial case. I n the examples of Kleptsina, Liptser an d Serebrovski ( 1997 ), X ε con verge s to ¯ X in probabil- it y , which is no longer th e case in our setting. How ev er it migh t b e p ossible to apply the total v ariation tec hniques dev elop ed in Kleptsina, Liptser and Serebro vski ( 1997 ) to obtain con v ergence in our setting. Only the r ate of con verge n ce cannot b e determined with these tec h n iques. F or a giv en b ounded test fun ction ϕ and termin al time T , we follo w P ard oux ( 1979 ) in introdu cing the asso ciated dual pro cess v ε,T ,ϕ t ( x, z ), which is a dynamic v ersion of E P ε [ ϕ ( X ε T ) ˜ D ε T |Y ε T ], v ε,T ,ϕ t ( x, z ) = E P ε t,x,z [ ϕ ( X ε T ) ˜ D ε t,T |Y ε t,T ] , 8 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G where P ε t,x,z is the measur e und er w h ic h X ε and Z ε are go verned by the same dynamics as under P ε , b ut ( X ε , Z ε ) sta ys in ( x, z ) u n til time t , and then it starts to follo w the SDE dynamics. ˜ D ε t,T = ˜ D ε T ( ˜ D ε t ) − 1 ; and Y ε t,T = σ ( Y ε r − Y ε t : t ≤ r ≤ T ) ∨ N (recall that N denotes the Q -negligible sets). F rom th e Mark o v prop erty of ( X ε , Z ε ) it follo w s that for an y t ∈ [0 , T ] : ρ ε t ( v ε,T ,ϕ t ) = ρ ε,x T ( ϕ ). In particular [b ecause at time 0, ρ ε is jus t the s tarting d istribution of ( X ε , Z ε )], ρ ε,x T ( ϕ ) = Z v ε,T ,ϕ 0 ( x, z ) Q ( X ε 0 ,Z ε 0 ) ( dx, dz ) . Similarly in tro duce v 0 ,T ,ϕ t ( x ) = E P ε t,x [ ϕ ( X 0 T ) ˜ D 0 t,T |Y ε t,T ] , where ˜ D 0 t,T = exp  Z T t ¯ h ( X 0 r ) ∗ d Y ε r − 1 2 Z T t | ¯ h ( X 0 r ) | 2 dr  and P ε t,x is the measure und er wh ic h X 0 is go v ern ed by the same dynamics as und er P ε , but sta ys in x until time t . W e can also s ho w that for any t ∈ [0 , T ] : ρ 0 t ( v 0 ,T ,ϕ t ) = ρ 0 T ( ϕ ), so that ρ 0 T ( ϕ ) = Z v 0 ,T ,ϕ 0 ( x ) Q X 0 0 ( dx ) . Note that Q X 0 0 = Q X ε 0 b ecause the homogenized pro cess has the same start- ing distribution as the unhomogenized one. No w fix T and ϕ ∈ C 2 b ( R m , R ) and w rite v ε t = v ε,T ,ϕ t and v 0 t = v 0 ,T ,ϕ t . Our aim is to sh o w that for nice test fu nctions ϕ , and for the dual p ro cesses v ε and v 0 defined abov e, E [ | v ε 0 ( x, z ) − v 0 0 ( x ) | p ] is small (in a w a y that will dep end on x and z ). Then E [ | ρ ε,x T ( ϕ ) − ρ 0 T ( ϕ ) | p ] = E      Z ( v ε 0 ( x, z ) − v 0 0 ( x )) Q ( X ε 0 ,Z ε 0 ) ( dx, dz )     p  ≤ E  Z | v ε 0 ( x, z ) − v 0 0 ( x ) | p Q ( X ε 0 ,Z ε 0 ) ( dx, dz )  = Z E [ | v ε 0 ( x, z ) − v 0 0 ( x ) | p ] Q ( X ε 0 ,Z ε 0 ) ( dx, dz ) will also b e small as long as Q ( X ε 0 ,Z ε 0 ) is well b eha v ed. 3. F ormal expansions of the filtering equations and the main resu lts. Be- fore we con tinue, let us c hange notation: F or large parts of this article w e will only work u nder P ε , and the pro cess Y ε is a Brownian motion un der DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 9 P ε whic h is in d ep end ent of ( X ε , Z ε , X 0 ). Therefore from no w on w e w rite P instead of P ε and B instead of Y ε to facilitate th e reading. The d istribution and n otation f or the Marko v pro cesses ( X ε , Z ε , X 0 ) d o n ot c h ange. The key p oin t is now th at v ε and v 0 solv e bac kward S PDEs − dv ε t ( x, z ) = L ε v ε t ( x, z ) dt + h ( x, z ) ∗ v ε t ( x, z ) d ← B t , (5) v ε T ( x, z ) = ϕ ( x ) and − dv 0 t ( x ) = ¯ L v 0 t ( x, z ) dt + ¯ h ( x ) ∗ v 0 t ( x ) d ← B t , (6) v 0 T ( x ) = ϕ ( x ) . Here and ev erywhere in th is article, d ← B denotes I tˆ o’s bac kward integ ral. W e formally expand v ε as v ε t ( x, z ) = u 0 t ( x, z ) + εu 1 t/ε ( x, z ) + ε 2 u 2 t/ε ( x, z ) . Note th at rigorously this do es not make an y sen se b ecause: • W e w ork with equations w ith terminal conditions. But when we send ε → 0, then t/ε con verges to infi nit y . S o for wh ic h time s hould the termin al condition of, for example, u 1 b e defined? • The terms in this exp an s ion w ill all b e sto c h astic. T hen if u 1 is adapted to F B , the sto c h astic in tegral R T t u 1 s/ε ( x, z ) d ← B s a priori do es not mak e an y sense for ε < 1. Ho we v er if we do s uc h a formal asymptotic expansion, and then call v 0 ( t, x ) = u 0 ( t, x ) , ψ 1 ( t, x, z ) = εu 1 t/ε ( x, z ) , R ( t, x, z ) = ε 2 u 2 t/ε ( x, z ) (of course all terms except v 0 dep end on ε , which we omit in the n otation to facilitat e the reading), then these terms ha ve to solve the follo wing equations: − dv 0 t ( x ) = ¯ L v 0 t ( x, z ) dt + ¯ h ( x ) ∗ v 0 t ( x ) d ← B t , − dψ 1 t ( x, z ) = 1 ε L F ψ 1 t ( x, z ) dt + ( L S − ¯ L ) v 0 t ( x ) dt (7) + ( h ( x, z ) − ¯ h ( x )) ∗ v 0 t ( x ) d ← B t , − dR t ( x, z ) = L ε R t ( x, z ) dt + L S ψ 1 t ( x, z ) dt (8) + h ( x, z ) ∗ ( ψ 1 t ( x, z ) + R t ( x, z )) d ← B t with termin al conditions v 0 ( T , x ) = ϕ ( x ) , ψ 1 ( T , x, z ) = R ( T , x, z ) = 0 . 10 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G Note that the equation for v 0 is exactly the desired equation ( 6 ). By existence and u niqueness of the solutions to these line ar equations, we can app ly sup erp osition to obtain that then ind eed v ε t ( x, z ) = v 0 t ( x ) + ψ 1 t ( x, z ) + R t ( x, z ) . Therefore the pr oblem of sho win g L p -con ve rgence of v ε to v 0 reduces to sho w in g L p -con ve rgence of ψ 1 + R to 0. T o achiev e this, we will giv e p roba- bilistic rep resen tations of ψ 1 and R in terms of bac kw ard doub ly sto chastic differen tial equations. This will allo w u s to app ly the existing estimates f or the transition fu nction of Z x from Pardoux and V eretenniko v ( 2003 ). It will b e con venien t f or us to work with fun ctions that are sm o other in their x -comp onen t than they are in their z -comp onent or vice v ersa. T o do so, introdu ce the fun ction spaces C k ,l ( R m × R n , R d ): F or θ : R m × R n → R d , θ = θ ( x, z ) , write θ ∈ C k ,l ( R m × R n , R d ), if θ is k times con tinuously differen tiable in its x -comp onents and l times con tin u ou s ly differen tiable in its z -comp onents. If θ as well as its partial d eriv ativ es up to ord er ( k , l ) are b ound ed, write θ ∈ C k ,l b ( R m × R n , R d ). In tro duce th e follo wing assumptions: (H stat ) F or the existence of a stationary distribution µ ( x, dz ) for Z x , w e supp ose that there exist M 0 > 0 , α > 0, suc h that for all | z | ≥ M 0 sup x h f ( x, z ) , z i ≤ − C | z | α . F or the uniqu eness of the stationary distribution µ ( x, dz ) of Z x , w e supp ose uniform ellipticit y , that is, that there are 0 < λ ≤ Λ < ∞ , su c h that λI ≤ g g ∗ ( x, y ) ≤ Λ I in the sense of p ositiv e semi-definite matrices ( I is the unit m atrix). (HF k ,l ) T he co efficie n ts of th e fast d iffusion satisfy f ∈ C k ,l b ( R m × R n , R n ) and g ∈ C k ,l b ( R m × R n , R n × k ). (HS k ,l ) The co efficients of the slo w d iffusion satisfy b ∈ C k ,l b ( R m × R n , R m ) and σ ∈ C k ,l b ( R m × R n , R m × k ). (HO k ,l ) The observ ation function h satisfies h ∈ C k ,l b ( R m × R n , R d ). W e will usu ally write p ∞ ( x, dz ) in stead of µ ( x, dz ). Also introdu ce the notation p t ( z , θ ; x ) := Z R n θ ( x, z ′ ) p t ( z , z ′ ; x ) dz ′ := E z [ θ ( Z x t )] , where z denotes the starting p oint of Z x , and z ′ 7→ p t ( z , z ′ ; x ) is the density of Z x t if at time 0 it is started in z . Note that the density exists for all DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 11 t > 0 under the condition (H stat ), b ecause of the uniform ellipticit y of g g ∗ . Similarly p ∞ ( θ ; x ) = Z R n θ ( x, z ) p ∞ ( x, dz ) . Let the different ial op erator ¯ L b e defin ed as ¯ L = m X i =1 ¯ b i ( x ) ∂ ∂ x i + 1 2 m X i,j =1 ¯ a ij ( x, z ) ∂ 2 ∂ x i ∂ x j , where ¯ b ( x ) = p ∞ ( b ; x ) and ¯ a = p ∞ ( σ σ ∗ ; x ). Also define ¯ h ( x ) = p ∞ ( h ; x ). W e introduce the follo wing n otation: A m ultiindex α = ( α 1 , . . . , α m ) ∈ N n 0 is of order | α | = α 1 + · · · + α m . Giv en s uc h a multiindex, defi ne the differen tial op erator D α = ∂ | α | ∂ x α 1 1 · · · x α m m . Finally in tr o duce the follo wing norms for f ∈ C k b ( R m , R n ): k f k k , ∞ = X | α |≤ k k D α f k ∞ , where k · k ∞ is the usu al supremum norm. Our main result is: Theorem 3.1. Assume (H stat ) , (HF 8 , 4 ) , (HS 7 , 4 ) , (HO 8 , 4 ) and that the initial distribution Q ( X ε 0 ,Z ε 0 ) has finite moments of every or der. Then for every p ≥ 1 and T ≥ 0 ther e exists C > 0 , such that for every ϕ ∈ C 4 b ( E Q [ | π ε,x T ( ϕ ) − π 0 T ( ϕ ) | p ]) 1 /p ≤ √ εC k ϕ k 4 , ∞ . In p articular, ther e exists a metric d on the sp ac e of pr ob ability me asur es, such that d ge ne r ates the top olo gy of we ak c onver genc e, and such that for every T ≥ 0 ther e exi sts C > 0 , such that E Q [ d ( π ε,x T , π 0 T )] ≤ √ εC. This result will b e prov en in S ection 6 . In particular we can u se Borel–Can telli to conclude that if ( ε n ) conv er ges quic kly enough to 0, th en π ε n will a.s. con v erge weakly to π 0 . The ideas are rather simp le: W e represent the bac kw ard SPDEs by fin ite- dimensional sto c hastic equatio ns (this will b e BDSDEs). The diffusion op- erators get r eplaced by the asso ciat ed diffusions. W e are able to solv e those 12 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G finite-dimensional equations explicitly , or at least give explicit estimates up to an app lication of Gron w all. This allo ws us to estimate ψ 1 and R in terms of the transition function of the fast diffusion. But P ardoux and V ereten- nik ov ( 2003 ) prov ed ve ry precise estimates for this transition function. These estimates allo w us to obtain the con ve rgence. While the ideas are simp le, th e precise form ulation and the actual pr o ofs are qu ite tec hnical. W e start by describing the pr obabilistic repr esen tation. 4. Probabilistic representa tion of SPDEs. In this section, we derive p rob- abilistic represen tations for SPDEs of the form − dψ ( ω , t, x ) = L ψ ( ω , t, x ) dt + f ( ω , t, x ) dt + ( g ( ω , t, x ) + G ( ω , t, x ) ψ ( ω , t, x )) d ← B t , (9) ψ ( T , x ) = ϕ ( ω , x ) , where ψ : Ω × [0 , T ] × R m → R , f : Ω × [0 , T ] × R m → R , g : Ω × [0 , T ] × R m → R 1 × d , and G : Ω × [0 , T ] × R m → R 1 × d , ϕ : Ω × R m → R are all join tly mea- surable, and ( B t : t ∈ [0 , T ]) is a d -dimensional stand ard Bro w nian m otion under the measure P . Equation ( 9 ) represents the general form of equa- tions ( 7 ) an d ( 8 ) for the corrector ψ 1 t ( x, z ) and error R t ( x, z ), resp ectiv ely . The d ifferen tial op erato r L is giv en by L = m X i =1 b i ( x ) ∂ ∂ x i + 1 2 m X i,j =1 a ij ( x ) ∂ 2 ∂ x i ∂ x j for measurab le b : R m → R m and a : R m → S m × m ( S m × m denotes p ositiv e semidefinite symmetric matrices). W e will r epresen t these equations in terms of BDSDEs as introdu ced by Pardoux and Peng ( 1994 ). Note that for these linear equations it is p ossible to give a F eynman–Kac t yp e represent ation without using BDSDEs. Th is is done, for example, in Rozo vski ˘ ı ( 1990 ) (“The Metho d of Sto chastic Ch aracteristics”) . Ho wev er the BDSDE-representati on has the adv anta ge that it p ermits us to app ly Gronw all’s lemma. This would not b e p ossible with the metho d of s to c hastic charac teristics. A BDSDE is an integ ral equation of the form Y t = ξ + Z T t f ( s, Y s , Z s ) ds + Z T t g ( s, Y s , Z s ) d ← B s − Z T t Z s dW s , where B an d W are ind ep endent Brownian motions. T he solution ( Y t , Z t ) will b e F B t,T ∨ F W t -measurable. Starting from the notion of BDSDEs, w e can defi ne forward-bac kw ard d oubly sto chastic differenti al equ ations. L et σ = a 1 / 2 and X t,x s = x + Z s t b ( X t,x s ) ds + Z s t σ ( X t,x s ) dW s for s ≥ t, X t,x s = x for s ≤ t. DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 13 W e then define the follo w in g BDSDE: − d Y t,x s = f ( s, X t,x s ) ds + ( g ( s, X t,x s ) ds + G ( s, X t,x s ) Y t,x s ) d ← B s − Z t,x s dW s , Y t,x T = ϕ ( X t,x T ) . It tur ns out that Y giv es a finite-dimensional probabilistic repr esen tation for equation ( 9 ), more pr ecisely we h av e Y t,x t = ψ ( t, x ). This is not completely co v ered by Pardoux and P eng ( 1994 ), b ecause we hav e random unboun ded co efficien ts, and b ecause we do not assume the diffusion matrix a to ha ve a smooth square ro ot. On the ot her side, the equation is of a p articularly simple linear t yp e. In the remainder of this section, we giv e the precise statemen t and pro of f or this representati on. Th is can b e skip p ed at first reading. W e will not b e able to get an existence result for classical solutions of the ab o ve SP DE fr om the theory of BDSDEs: Th is is due to th e fact that f or this we w ould need smo othness prop erties of a squ are ro ot of a . But ev en when a is smo oth, in th e degenerate elliptic case it do es n ot need to ha ve a smo oth squ are r o ot; see, for example, Str o ock ( 2008 ), Ch apter 2.3. In stead w e will use the existence result of Rozo vsk i ˘ ı ( 1990 ) an d only repro v e th e uniqueness resu lt of P ard oux and Peng ( 1994 ) in our setting. This will w ork under L ip sc hitz con tin uity of a 1 / 2 . Define for 0 ≤ t ≤ s ≤ T F 0 ,B t,s = σ ( B u − B t : t ≤ u ≤ s ) and F B t,s as the completion of F 0 ,B t,s under P . Int ro duce the sp ace of adapted random fields of p olynomial gro wth: Definition 1. P T ( R m , R n ) is the space of ran d om fields H : Ω × [0 , T ] × R m → R n that are join tly measurable in ( ω , t, x ) , and for fixed ( t, x ), ω 7→ H ( ω , t, x ) is F B t,T -measurable. F urther for fi xed ω outside a n ull set, H has to b e join tly con tinuous in ( t, x ) , and it has to satisfy the f ollo wing inequalit y: F or eve ry p ≥ 1 there is C p > 0, q > 0, such that for all x ∈ R m , E h sup 0 ≤ t ≤ T | H ( t, x ) | p i ≤ C p (1 + | x | q ) . W e make the follo w in g assumptions on the co efficien ts of the SPDE: (S k ) f and g a re k times con tinuously differen tiable and the partial deriv ativ es up to ord er k are all in P T . G is ( k + 1) times con tin uously differen tiable and the partial d eriv ativ es up to order ( k + 1) are all u ni- formly b ounded in ( ω , t, x ). ϕ is k times cont in u ou s ly differen tiable, and all partial deriv ativ es of order 0 to k gro w at most p olynomially . 14 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G W e mak e the follo wing assu mptions on the co efficients of the differential op erator L : (D k ) b ∈ C k b ( R m , R m ), a ∈ C k b ( R m , S m × m ), and a is d egenerate elliptic: F or ev er y ξ ∈ R m and every x ∈ R m , h a ( x ) ξ , ξ i = m X i,j =1 a ij ( x ) ξ i ξ j ≥ 0 . Then we ha ve th e follo win g result: Pr oposition 4.1. Assume (S k ) and (D k ) for some k ≥ 3 . Then e qua- tion ( 9 ) has a unique classic al solution ψ in the sense that for every fixe d ω outside a nul l set, ψ ( ω , · , · ) ∈ C 0 ,k − 1 ([0 , T ] × R d , R ) , ψ and its p artial deriva- tives ar e in P T ( R m , R ) , and ψ solves the inte gr al e quation. If ˜ ψ is any other solution of the inte gr al e quation, then ψ and ˜ ψ ar e indistinguishable. If fur- ther f , g and ϕ as wel l as their derivatives up to or der k ar e uniformly b ounde d in ( ω , t, x ) , then for any p > 0 ther e exist C p , q > 0 (only dep ending on p , the dimensions involve d, the b ounds on a, b and G , and on T), such that for al l | α | ≤ k − 1 and x ∈ R m , E h sup t ≤ T | D α ψ ( t, x ) | p i ≤ C (1 + | x | q ) E h k ϕ k p k , ∞ + sup t ≤ T k f ( t, · ) k p k , ∞ + su p t ≤ T k g ( t, · ) k p k , ∞ i . Pr oof. This is a com bin ation of Theorem 4.3.2 and Corollary 4.3.2 of Rozo vski ˘ ı ( 1990 ) (The claimed b ound is only giv en f or the equation in unw eigh ted Sob olev spaces, in Corollary 4.2.2. But from that we can deduce the result for the w eigh ted Sob olev case). Th e only thing we need to verify is that our p olynomial gro wth assumption on the co efficients is compatible with the Sob olev norm condition there. But if θ ∈ P T ( R m , R n ), then for an y p ≥ 1 there certainly is an r < 0 suc h that θ take s its v alues in th e w eigh ted L p -space with we igh t (1 + | x | 2 ) r / 2 , E  sup 0 ≤ t ≤ T Z | θ ( t, x ) | p (1 + | x | 2 ) r / 2 dx  ≤ E  Z sup 0 ≤ t ≤ T 1 | θ ( t, x ) | p (1 + | x | 2 ) r / 2 dx  = Z E h sup 0 ≤ t ≤ T | θ ( t, x ) | p i (1 + | x | 2 ) r / 2 dx ≤ Z C p (1 + | x | q )(1 + | x | 2 ) r / 2 dx < ∞ for small enough r .  DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 15 No w we com bine this result w ith the theory of BDSDEs: Let ( W t : t ∈ [0 , T ]) b e an n -dimensional s tand ard Brownian motion that is indep enden t of B . F or 0 ≤ t ≤ s , F W t,s is defined analogously to F B t,s . F or 0 ≤ t ≤ T we set F t = F B t,T ∨ F W t . Note that this is n ot a filtration, as it is neither d ecreasing nor increasing in t . Introdu ce th e f ollo wing notation: • H 2 T ( R m ) is the space of measurable R m -v alued p r o cesses Y s.t. , Y t is F t - measurable and E  Z T 0 | Y t | 2 dt  < ∞ . • S 2 T ( R m ) is the sp ace of con tinuous adapted R m -v alued pro cesses Y s.t. Y t ∈ F t and E h sup 0 ≤ t ≤ T | Y t | 2 i < ∞ . A BDSDE is an integ ral equation of the form Y t = ξ + Z T t f ( s, · , Y s , Z s ) ds + Z T t g ( s, · , Y s , Z s ) d ← B s − Z T t Z s dW s , (10) where f : [0 , T ] × Ω × R × R 1 × n → R , g : [0 , T ] × Ω × R × R 1 × n → R 1 × l , and for fixed y ∈ R , z ∈ R 1 × n the pro cesses ( ω , t ) 7→ f ( t, ω , x, z ) and ( ω , t ) 7→ g ( t, ω , x, z ) are ( F B 0 ,T ∨ F W T ) ⊗ B ( R )-measurab le, an d for eve ry t , f ( t, · , x, z ) and g ( t, · , x, z ) are F t -measurable. ( Y , Z ) will b e called solution of ( 10 ) if ( Y , Z ) ∈ S 2 T ( R ) × H 2 T ( R 1 × n ) and if the couple solve s the integral equation. W e will also write the equation in differential form − d Y t = f ( t, Y t , Z t ) dt + g ( t, Y t , Z t ) d ← B t − Z t dW t . Observe that with suitable adaptations, all of the follo w ing results also hold in the m u ltidimensional case, that is, for Y ∈ R m . W e restrict to one- dimensional Y for simp licit y and b ecause ultimately w e are only in terested in that case. P ard oux and Peng ( 1994 ) sho w that un der the follo wing conditions, equa- tion ( 10 ) has a u nique solution: • ξ ∈ L 2 (Ω , F T , P ; R ); • for any ( y , z ) ∈ R × R 1 × n : f ( · , · , y , z ) ∈ H 2 T ( R ) and g ( · , · , y , z ) ∈ H 2 T ( R 1 × k ); 16 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G • f and g satisfy Lipsc hitz conditions and g is a contract ion in z : there exist constan ts L > 0 and 0 < α < 1 s.t. for an y ( ω , t ) and y 1 , y 2 , z 1 , z 2 , | f ( t, ω , y 1 , z 1 ) − f ( t, ω , y 2 , z 2 ) | 2 ≤ L ( | y 1 − y 2 | 2 + | z 1 − z 2 | 2 ) and | g ( t, ω , y 1 , z 1 ) − g ( t, ω , y 2 , z 2 ) | 2 ≤ L | y 1 − y 2 | 2 + α | z 1 − z 2 | 2 . No w w e w ant to asso ciate a diffusion X to the differen tial op erator L . T o do so, assume that (D k ) is satisfied for some k ≥ 2 . Then σ := a 1 / 2 is Lipsc h itz con tin uous b y Lemma 2.3.3 of Stro o ck ( 2008 ). Hence for every ( t, x ) ∈ [0 , T ] × R m , there exists a str ong solution of the SDE X t,x s = x + Z s t b ( X t,x s ) ds + Z s t σ ( X t,x s ) dW s for s ≥ t, X t,x s = x for s ≤ t. Asso ciate the follo wing BDSDE to ( 9 ): − d Y t,x s = f ( s, X t,x s ) ds + ( g ( s, X t,x s ) + G ( s, X t,x s ) Y t,x s ) d ← B s − Z t,x s dW s , (11) Y t,x T = ϕ ( X t,x T ) . Under the assump tions (S k ) and (D k ) f or k ≥ 2 , this equation has a un ique solution. Pr oposition 4.2. Assume (S k ) and (D k ) for some k ≥ 3 . Then the unique classic al solution ψ of the SPDE ( 9 ) i s given by ψ ( t, x ) = Y t,x t , wher e ( Y t,x , Z t,x ) is the uniq u e solution of the BDSDE ( 11 ). W e can giv e exactly the same pro of as in Pardoux and Peng ( 1994 ), Theorem 3.1 , taking adv an tage of the indep end ence of B and W . F or the reader’s con venience , w e includ e it here. Pr oof. Let ψ b e a classical solution of ( 9 ). It suffices to sh o w that ( ψ ( s , X t,x s ) , D ψ ( s, X t,x s ) σ ( X t,x s ) : t ≤ s ≤ T ) solv es the BDSDE ( 11 ). Here D ψ is the gradient of ψ . F or this pu rp ose, consider a partition t = t 0 < t 1 < · · · < t n = T of [ t, T ]. Then ψ ( t, X t,x t ) = ψ ( T , X t,x T ) + n − 1 X i =0 ( ψ ( t i , X t,x t i ) − ψ ( t i +1 , X t,x t i +1 )) = ϕ ( X t,x T ) + n − 1 X i =0 ( ψ ( t i , X t,x t i ) − ψ ( t i +1 , X t,x t i +1 )) DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 17 and ψ ( t i , X t,x t i ) − ψ ( t i +1 , X t,x t i +1 ) = ( ψ ( t i , X t,x t i ) − ψ ( t i , X t,x t i +1 )) + ( ψ ( t i , X t,x t i +1 ) − ψ ( t i +1 , X t,x t i +1 )) = −  Z t i +1 t i L ψ ( t i , X t,x s ) ds + Z t i +1 t i D ψ ( t i , X t,x s ) σ ( X t,x s ) dW s  + Z t i +1 t i ( L ψ ( s, X t,x t i +1 ) + f ( s, X t,x t i +1 )) ds + Z t i +1 t i ( g ( s, X t,x t i +1 ) + G ( X t,x t i +1 ) ψ ( s, X t,x t i +1 )) d ← B s . This is ju stified b ecause X t,x and ψ are indep enden t and b ecause ψ gro w s p olynomially , hence w e can apply Itˆ o’s formula. W e also us ed th e fact that ψ is a classica l solution to ( 9 ). If w e let the mesh size tend to 0, then b y con tinuit y of X t,x and ψ , the r esult follo ws .  5. Preliminary estimates. Th e notation D α x indicates that the differen- tial op erator D α is only acting on the x -v ariables. The follo wing resu lt will h elp u s to justify the BDSDE-represen tations on the deep er lev els. Recall that p t ( z , θ ; x ) = E [ θ ( x, Z x t ) | Z x 0 = z ] . Pr oposition 5.1. Assume (HF k ,l ) . L et θ ∈ C k ,l ( R m × R n , R ) satisfy f or some C, p > 0 X | α |≤ k X | β |≤ l | D α x D β z θ ( x, z ) | ≤ C (1 + | x | p + | z | p ) . Then ( t, x, z ) 7→ p t ( z , θ ; x ) ∈ C 0 ,k ,l ( R + × R m × R n , R ) and ther e exist C 1 , p 1 > 0 , such that for al l ( t, x, z ) ∈ [0 , ∞ ) × R m × R n X | α |≤ k X | β |≤ l | D α x D β z p t ( z , θ ; x ) | ≤ C 1 e C 1 t (1 + | x | p 1 + | z | p 1 ) . If the b ound on the derivatives of θ c an b e chosen u ni f ormly in x , that is, X | α |≤ k X | β |≤ l sup x | D α x D β z θ ( x, z ) | ≤ C (1 + | z | p ) , then the b ound on the derivatives of p t ( z , θ ; x ) is also uniform i n x , X | α |≤ k X | β |≤ l sup x | D α x D β z p t ( z , θ ; x ) | ≤ C 1 e C 1 t (1 + | z | p 1 ) . 18 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G Pr oof. Note th at p t ( z , θ ; x ) = E [ θ ( x, Z x t ) | Z x 0 = z ] = E [ θ ( X t , Z t ) | ( X 0 , Z 0 ) = ( x, z )] is the solution of Kolmogoro v’s bac kward equation asso ciated to ( X, Z ), where X t = X 0 , Z t = Z 0 + Z t 0 f ( X s , Z s ) ds + Z t 0 g ( X s , Z s ) dW s . In this formulation, the firs t result is standard; cf., for example, Stro o c k ( 2008 ), C orollary 2.2.8. The second statemen t can b e p ro ven in the same wa y as Stro o ck ( 2008 ), Corollary 2.2.8.  Some results from P ardoux and V eretenniko v ( 2003 ) are collected in the follo wing pr op osition: Pr oposition 5.2. A ssu me (H stat ) and (HF k , 3 ) . L et θ ∈ C k , 0 ( R m × R n , R ) satisfy for some C, p > 0 , X | α |≤ k sup x | D α x θ ( x, z ) | ≤ C (1 + | z | p ) . Then: (1) x 7→ p ∞ ( θ ; x ) ∈ C k b ( R m , R ) . (2) Assume add itional ly th at θ satisfies the c entering c ondition Z R n θ ( x, z ) p ∞ ( x, dz ) = 0 for al l x , and that θ ∈ C k , 1 ( R m × R n , R ) and X | α |≤ k X | β |≤ 1 sup x | D β z D α x θ ( x, z ) | ≤ C (1 + | z | p ) . Then ( x, z ) 7→ Z ∞ 0 p t ( z , θ ; x ) dt ∈ C k , 1 ( R m × R n , R ) , and for every q > 0 ther e exist C 1 , q 1 > 0 , such that for every z ∈ R n X | α |≤ k X | β |≤ 1 Z ∞ 0 sup x | D β z D α x p t ( z , θ ; x ) | q dt ≤ C 1 (1 + | z | q 1 ) . DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 19 Pr oof. The statemen ts in the p rop osition are tak en from T heorems 1 and 2 and Prop osition 1 of P ard oux and V eretennik o v ( 2003 ): (1) W e get from Theorem 1 of Pa rdoux and V eretennik o v ( 2003 ), that for an y q > 0 there exists C q > 0, suc h that for any ( x, z , z ′ ) ∈ R m × R n × R n , X | α |≤ k sup x | D α x p ∞ ( z ′ ; x ) | ≤ C q 1 + | z ′ | q . So if we c ho ose q large enough and differentia te p ∞ ( θ ; x ) under the inte gral sign, then we obtain the fir st claim. (Of course here we ha ve to use the gro wth constraint on θ and its der iv ativ es.) (2) This follo ws from the b oun ds on the deriv ativ es of p t ( z , θ ; x ) that are giv en in P ardou x and V eretenniko v ( 2003 ), Theorem 2, formulas (14) and (15): F or an y k > 0 there exist C k , m k > 0, s u c h that for an y ( t, x, z ) ∈ [1 , ∞ ) × R m × R n , X | α |≤ k X | β |≤ 1 | D β z D α x p t ( z , θ ; x ) | ≤ C k 1 + | z | m k (1 + t ) k . W e com b ine th is estimate with Prop osition 5.1 , from where w e obtain for ( t, x, z ) ∈ R + × R m × R n X | α |≤ k X | β |≤ l sup x | D α x D β z p t ( z , θ ; x ) | ≤ C 1 e C 1 t (1 + | z | p 1 ) . W e choose k suc h that q k > 1 and use th e first estimate on [1 , ∞ ) and the second estimate on [0 , 1) . The result follo w s.  W e will also need some momen t b ounds for th e diffusions X ε and Z ε . Pr oposition 5.3. Assume (H stat ) and that the c o efficients b and σ and f and g of the fast and slow motion ar e b ounde d and glob al ly Lipschitz c ontinuous. Then for any p ≥ 1 ther e exists C p > 0 , such that sup ( t,ε,x ) ∈ [0 , ∞ ) × [0 , 1] × R m E [ | Z ε t | p | ( X ε 0 , Z ε 0 ) = ( x, z )] ≤ C p (1 + | z | p ) . Also , for every T > 0 and ev ery p ≥ 1 ther e exist C ( p, T ) , q > 0 , such that sup ( t,ε ) ∈ [0 ,T ] × [ 0 , 1] E [ | X ε t | p | ( X ε 0 , Z ε 0 ) = ( x, z )] ≤ C ( p, T )(1 + | x | p ) . Pr oof. The fir st claim can b e pr o ven exactly as in V eretennik ov ( 1997 ): First write ¯ Z ε t := Z ε tε 2 . T hen d ¯ Z ε t = f ( X ε ε 2 t , ¯ Z ε t ) dt + g ( X ε ε 2 t , ¯ Z ε t ) d ¯ W ε t , 20 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G where ¯ W ε t := 1 /εW ε 2 t is a Wie ner pro cess. Next, introd uce the same time c hange as in P ard oux and V eretenniko v [( 2001 ), p age 1063 ], κ ( x, z ) := | g ( x, z ) ∗ z | / | z | , γ ε ( t ) := Z t 0 κ 2 ( X ε ε 2 s , ¯ Z ε s ) ds, τ ε ( t ) := ( γ ε ) − 1 ( t ) . Define ˜ Z ε t := ¯ Z ε τ ε ( t ) . Then d ˜ Z ε t = κ − 2 ( X ε ε 2 t , ˜ Z ε t ) f ( X ε ε 2 t , ˜ Z ε t ) dt + κ − 1 ( X ε ε 2 t , ˜ Z ε t ) g ( X ε ε 2 t , ˜ Z ε t ) d ˜ W ε t with a new standard Bro w nian motion ˜ W ε . Now we are in a p ositio n to just copy the pr o of of Lemma 1 in V eretennik o v ( 1997 ) (whic h we do not d o here) to get the fi rst r esult. The second claim is obvious, b ecause the co efficien ts of X ε are b ounded.  No w we we are able to imp ose conditions on the co efficien ts of the diffu- sions that guarante e smo othness of th e co efficients of ¯ L . Recall that ¯ L was defined as ¯ L = m X i =1 ¯ b i ( x ) ∂ ∂ x i + 1 2 m X i,j =1 ¯ a ij ( x, z ) ∂ 2 ∂ x i ∂ x j , where ¯ b = p ∞ ( b ; x ) and ¯ a = p ∞ ( σ σ ∗ ; x ). Pr oposition 5.4. Assume (HF k , 3 ) , (HS k , 0 ) and (HO k , 0 ) . Then ¯ b ∈ C k b ( R m , R m ) , ¯ a ∈ C k b ( R m , S m × m ) , ¯ h ∈ C k b ( R m , R k ) . Pr oof. All the terms of ¯ b , ¯ a and ¯ h are of the form p ∞ ( θ ; x ). So by Prop osition 5.2 , we on ly need to v erify that the r esp ectiv e θ are in C k , 0 and satisfy th e p olynomial b oun d X | α |≤ k sup x | D α x θ ( x, z ) | ≤ C (1 + | z | p ) for s ome C, p > 0. But we ev en assumed them to b e in C k , 0 b , so the result follo ws.  6. Proof of the main result. W e will find con v ergence rates for the correc- tor and remainder terms that are expressed in term s of v 0 and its deriv ativ es. So no w w e giv e b oun ds on v 0 and its deriv ativ es in terms of the test function ϕ . T his is necessary b ecause we d o not only wan t to sho w con vergence of the filter int egrating fixed test functions, but with r esp ect to a suitable d istance on the space of probabilit y measures. DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 21 Lemma 6.1. L et k ≥ 2 and assume ¯ b, ¯ a, ϕ ∈ C k +1 b and ¯ h ∈ C k +2 b . Then v 0 ∈ C 0 ,k ([0 , T ] × R m , R ) , and for any p ≥ 1 ther e exist C p , q > 0 , indep endent of ϕ , su c h that for al l x ∈ R m , X | α |≤ k E h sup 0 ≤ t ≤ T | D α v 0 t ( x ) | p i ≤ C p (1 + | x | q ) k ϕ k p k , ∞ . In p articular, v 0 and al l its p artial derivatives up to or der (0 , k ) ar e in P T ( R m , R ) . Pr oof. This is a simple application of P rop osition 4.1 , noting that equation ( 6 ) for v 0 is of the typ e ( 9 ) with f = 0, g = 0, and G = ¯ h ∗ .  W e will pro ve L p -con ve rgence of ψ 1 and R separately: Lemma 6.2. L e t k , l ≥ 2 . A ssume (H stat ) , (HF k +1 ,l +1 ) , (HS k +1 ,l +1 ) and (HO k +1 ,l +1 ) . Also assume v 0 ∈ C 0 ,k +1 ([0 , T ] × R m , R ) , and that al l its p artial derivatives in x up to or der k + 1 ar e in P T ( R m , R ) . Final ly assume ¯ a, ¯ b, ¯ h ∈ C k b . Then ψ 1 ∈ C 0 ,k ,l ([0 , T ] × R m × R n , R ) , and ψ 1 as wel l as its p artial derivatives up to or der (0 , k , l ) ar e in P T ( R m × R n , R ) . F or any p ≥ 1 ther e exist C p , q > 0 , indep endent of ϕ , such that for any ( x, z ) ∈ R m + n and any ε ∈ (0 , 1) X | α |≤ k − 1 sup 0 ≤ t ≤ T E [ | D α x ψ 1 t ( x, z ) | p ] ≤ ε p/ 2 C p (1 + | z | q ) X 0 ≤| α |≤ k +1 E h sup 0 ≤ t ≤ T | D α x v 0 t ( x ) | p i . Pr oof. ψ 1 t ( x, z ) solv es the BSPDE − dψ 1 t ( x, z ) =  1 ε L F ψ 1 t ( x, z ) + ( L S − ¯ L ) v 0 t ( x )  dt + [ h ( x, z ) − ¯ h ( x )] ∗ v 0 t ( x ) d ← B t , (12) ψ 1 T ( x, z ) = 0 . Existence of the solutio n ψ 1 and its deriv ativ es as w ell as the p olynomial gro wth all follo w from Prop osition 4.1 . W rite Z ε,x, ( t,z ) for th e solution of the SDE d Z ε,x, ( t,z ) s = 1 ε f ( x, Z ε,x, ( t,z ) s ) ds + 1 √ ε g ( x, Z ε,x, ( t,z ) s ) dW t , s ≥ t, Z ε,x, ( t,z ) s = z , s ≤ t. 22 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G W e consider ( x, Z ε,x, ( t,z ) ) as a j oin t d iffusion, just as in the pr o of of Pr op osi- tion 5.1 ( x h as generator 0). By Pr op osition 4.2 , the solution of ( 12 ) is giv en b y θ ( t,x,z )(1) t , the uniqu e solution to the BDSDE − dθ ( t,x,z )(1) s = ( L S ( · , Z ε,x, ( t,z ) s ) − ¯ L ) v 0 s ( x ) ds + ( h ( x, Z ε,x, ( t,z ) s ) − ¯ h ( x )) ∗ v 0 s ( x ) d ← B s + γ t,x,z s dW s , θ ( t,x,z )(1) T = 0 . W e will d rop sup erscripts ( t, x, z ) for θ ( t,x,z )(1) s and write θ 1 s instead. Similarly , w e wr ite Z ε,x s instead of Z ε,x, ( t,z ) s . ψ 1 t ( x, z ) is F B t,T -measurable, hence so is θ 1 t . W e can then write θ 1 t = E [ θ 1 t |F B t,T ], where E [ θ 1 t |F B t,T ] = E  Z T t ( L S − ¯ L ) v 0 s ( x ) ds |F B t,T  + E  Z T t [ h ( x, Z ε,x s ) − ¯ h ( x )] ∗ v 0 s ( x ) d ← B s |F B t,T  − E  Z T t γ t,x,z s dW s |F B t,T  . W and B are in dep end en t; therefore W is a Brownian motion in the large filtration ( F W s ∨ F B t,T : s ∈ [0 , T ]), hence E [ R T t γ t,x,z s dW s |F W t ∨ F B t,T ] = 0, and b y th e to wer p rop erty E  Z T t γ t,x,z s dW s |F B t,T  = 0 . v 0 s is F B s,T -measurable, and ¯ L has d etermin istic co efficien ts. Thus E  Z T t ¯ L v 0 s ( x ) ds |F B t,T  = Z T t E [ ¯ L v 0 s ( x ) |F B s,T ] ds = Z T t ( m X i =1 p ∞ ( b i ; x ) ∂ ∂ x i v 0 s ( x ) + m X i,j =1 p ∞ (( σ σ ∗ ) ij ; x ) ∂ 2 ∂ x i x j v 0 s ( x ) ) ds. Since Z ε,x is indep end en t of B , E  Z T t L S ( · , Z ε,x s ) v 0 s ( x ) ds |F B t,T  DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 23 = Z T t E [ L S ( · , Z ε,x s ) v 0 s ( x ) |F B s,T ] ds = Z T t ( m X i =1 E [ b i ( x, Z ε,x s )] ∂ ∂ x i v 0 s ( x ) + 1 2 m X i,j =1 E [( σ σ ∗ ) ij ( x, Z ε,x s )] ∂ 2 ∂ x i x j v 0 s ( x ) ) ds = Z T t ( m X i =1 p ( s − t ) /ε ( z , b i ; x ) ∂ ∂ x i v 0 s ( x ) + 1 2 m X i,j =1 p ( s − t ) /ε ( z , ( σ σ ∗ ) ij ; x ) ∂ 2 ∂ x i x j v 0 s ( x ) ) ds, so     E  Z T t ( L S − ¯ L ) v 0 s ( x ) ds |F B t,T      =      Z T t ( m X i =1 p ( s − t ) /ε ( z , b i − p ∞ ( b i ; x ); x ) ∂ ∂ x i v 0 s ( x ) + 1 2 m X i,j =1 p ( s − t ) /ε ( z , ( σ σ ∗ ) ij − p ∞ (( σ σ ∗ ) ij ; x ); x ) ∂ 2 ∂ x i x j v 0 s ( x ) ) ds      [the p ∞ ( · ; x ) terms hav e b een b rought insid e the in tegral p ( s − t ) /ε ( z , · ; x ) since they not dep end on z ] ≤ ε      m X i =1 Z ( T − t ) /ε 0 p u ( z , b i − p ∞ ( b i ; x ); x ) ∂ ∂ x i v 0 εu + t ( x ) du      + ε 2      m X i,j =1 Z ( T − t ) /ε 0 p u ( z , ( σ σ ∗ ) ij − p ∞ (( σ σ ∗ ) ij ; x ); x ) ∂ 2 ∂ x i x j v 0 εu + t ( x ) du      ≤ ε m X i =1 Z ∞ 0 | p u ( z , b i − p ∞ ( b i ; x ); x ) | du sup t ≤ s ≤ T     ∂ ∂ x i v 0 s ( x )     + ε 2 m X i,j =1 Z ∞ 0 | p u ( z , ( σ σ ∗ ) ij − p ∞ (( σ σ ∗ ) ij ; x ); x ) | du sup t ≤ s ≤ T     ∂ 2 ∂ x i x j v 0 s ( x )     24 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G [ f − p ∞ ( f ; x ) is cente red, so by Prop ositio n 5.2 , ( 5.2 )] ≤ εC 1 (1 + | z | q 1 ) ( m X i =1 sup t ≤ s ≤ T     ∂ ∂ x i v 0 s ( x )     + m X i,j =1 sup t ≤ s ≤ T     ∂ 2 ∂ x i x j v 0 s ( x )     ) and th er efore fin ally E      E  Z T t ( L S − ¯ L ) v 0 s ( x ) ds |F B t,T      p  ≤ ε p C 2 (1 + | z | q 2 ) (13) × E " m X i =1 sup t ≤ s ≤ T     ∂ ∂ x i v 0 s ( x )     p + m X i,j =1 sup t ≤ s ≤ T     ∂ 2 ∂ x i x j v 0 s ( x )     p # . Next, using again v 0 s ∈ F B s,T and th at Z ε,x is indep end en t of B , E  Z T t [ h ( x, Z ε,x s ) − ¯ h ( x )] ∗ v 0 s ( x ) d ← B s |F B t,T  = Z T t E [[ h ( x, Z ε,x s ) − ¯ h ( x )] ∗ v 0 s ( x ) |F B s,T ] d ← B s = Z T t p ( s − t ) /ε ( z , h − ¯ h ; x ) ∗ v 0 s ( x ) d ← B s . F or t ≤ r ≤ T , r 7→ R T r p ( s − t ) /ε ( z , h − ¯ h ; x ) ∗ v 0 s ( x ) d ← B s , is a martingale w .r.t. ( F B r,T : r ∈ [ t, T ]) if time is ru n b ac kwards. Hence b y the Burkholder–Da vis– Gundy inequ alit y , E      Z T t p ( s − t ) /ε ( z , h − ¯ h ; x ) ∗ v 0 s ( x ) d ← B s     p  ≤ C p E  Z T t p ( s − t ) /ε ( z , h − ¯ h ; x ) ∗ v 0 s ( x ) d ← B s  p/ 2  , where  Z T t p ( s − t ) /ε ( z , h − ¯ h ; x ) ∗ v 0 s ( x ) d ← B s  = Z T t     p ( s − t ) /ε ( z , h − ¯ h ; x ) ∗ v 0 s ( x )     2 ds ≤ ε Z ∞ 0 | p u ( z , h − ¯ h ; x ) | 2 du sup t ≤ s ≤ T | v 0 s ( x ) | 2 ≤ εC 3 (1 + | z | q 3 ) sup t ≤ s ≤ T | v 0 s ( x ) | 2 , DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 25 where the last in equ alit y is by Prop ositio n 5.2 , ( 5.2 ), since h − ¯ h is cente red. Therefore, E      Z T t p ( s − t ) /ε ( z , h − ¯ h ; x ) ∗ v 0 s ( x ) d ← B s     p  (14) ≤ ε p/ 2 C 4 (1 + | z | q 4 ) E h sup t ≤ s ≤ T | v 0 s ( x ) | p i . Com b ining ( 13 ) an d ( 14 ), E [ | θ 1 t | p ] ≤ ε p C 4 (1 + | z | q 4 ) X | α |≤ 2 E h sup t ≤ s ≤ T | D α x v 0 s ( x ) | p i . Next, consider a first-order x -der iv ativ e of θ 1 t , ∂ ∂ x k θ 1 t = ∂ ∂ x k Z T t E [ L S − ¯ L ] v 0 s ( x ) ds + ∂ ∂ x k Z T t E [ h ( x, Z ε,x s ) − ¯ h ( x )] ∗ v 0 s ( x ) d ← B s . As b efore, the forward It´ o in tegral term v anish ed after taking the (condi- tional) exp ectati on. In terc hanging order of d ifferen tiation and int egration,     ∂ ∂ x k Z T t E [ L S − ¯ L ] v 0 s ( x ) ds     ≤ ε m X i =1     Z ( T − t ) /ε 0  ∂ ∂ x k p u ( z , b i − p ∞ ( b i ; x ); x ) ∂ ∂ x i v 0 εu + t ( x ) + p u ( z , b i − p ∞ ( b i ; x ); x ) ∂ 2 ∂ x k x i v 0 εu + t ( x )  du     + ε 2 m X i,j =1     Z ( T − t ) /ε 0  ∂ ∂ x k p u ( z , ( σ σ ∗ ) ij − p ∞ (( σ σ ∗ ) ij ; x ); x ) × ∂ 2 ∂ x i x j v 0 εu + t ( x ) + p u ( z , ( σ σ ∗ ) ij − p ∞ (( σ σ ∗ ) ij ; x ); x ) × ∂ 3 ∂ x i x j x k v 0 εu + t ( x )  du     ≤ ε m X i =1  Z ∞ 0     ∂ ∂ x k p u ( z , b i − p ∞ ( b i ; x ); x )     du sup t ≤ s ≤ T     ∂ ∂ x i v 0 s ( x )     26 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G + Z ∞ 0 | p u ( z , b i − p ∞ ( b i ; x ); x ) | du sup t ≤ s ≤ T     ∂ 2 ∂ x k x i v 0 s ( x )      + ε 2 m X i,j =1  Z ∞ 0     ∂ ∂ x k p u ( z , ( σ σ ∗ ) ij − p ∞ (( σ σ ∗ ) ij ; x ); x )     du × sup t ≤ s ≤ T     ∂ 2 ∂ x i x j v 0 s ( x )     + Z ∞ 0 | p u ( z , ( σ σ ∗ ) ij − p ∞ (( σ σ ∗ ) ij ; x ); x ) | du × sup t ≤ s ≤ T     ∂ 3 ∂ x i x j x k v 0 s ( x )      . Then, from Prop osition 5.2 , ( 5.2 ) again,     ∂ ∂ x k Z T t E [ L S − ¯ L ] v 0 s ( x ) ds     ≤ εC 5 (1 + | z | q 5 ) X 1 ≤ β ≤ 3 sup t ≤ s ≤ T | D β x v 0 s ( x ) | , since th e qu an tities b − ¯ b and σ σ ∗ − ¯ σ σ ∗ are centered. T aking exp ectat ion, E      ∂ ∂ x k Z T t E [ L S − ¯ L ] v 0 s ( x ) ds     p  (15) ≤ ε p C 6 (1 + | z | q 6 ) X 1 ≤ β ≤ 3 E h sup t ≤ s ≤ T | D β x v 0 s ( x ) | p i . Next, b y (HO k ,l ), we can in terchange the order of ordinary d ifferen tiation and sto c hastic in tegration [cf. Karandik ar ( 1983 )], E      ∂ ∂ x k  Z T t E [ h ( x, Z ε,x s ) − ¯ h ( x )] ∗ v 0 s ( x ) d ← B s      p  = E      Z T t ∂ ∂ x k ( E [ h ( x, Z ε,x s ) − ¯ h ( x )] ∗ v 0 s ( x )) d ← B s     p  ≤ C p E  Z T t     ∂ ∂ x k ( E [ h ( x, Z ε,x s ) − ¯ h ( x )] ∗ v 0 s ( x ))     2 ds  p/ 2  , where Z T t     ∂ ∂ x k ( E [ h ( x, Z ε,x s ) − ¯ h ( x )] ∗ v 0 s ( x ))     2 ds = ε Z ( T − t ) /ε 0     ∂ ∂ x k p u ( z , h − ¯ h ; x ) v 0 εu + t ( x ) DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 27 + p u ( z , h − ¯ h ; x ) ∂ ∂ x k v 0 εu + t ( x )     2 du ≤ 2 ε  Z ∞ 0     ∂ ∂ x k p u ( z , h − ¯ h ; x )     2 | v 0 εu + t ( x ) | 2 du + Z ∞ 0 | p u ( z , h − ¯ h ; x ) | 2     ∂ ∂ x k v 0 εu + t ( x )     2 du  ≤ εC 7 (1 + | z | q 7 )  sup t ≤ s ≤ T | v 0 s ( x ) | 2 + sup t ≤ s ≤ T     ∂ ∂ x k v 0 s ( x )     2  . The last step follo ws once again from Pr op osition 5.2 , ( 5.2 ). So, E      ∂ ∂ x k  Z T t E [ h ( x, Z ε,x s ) − ¯ h ( x )] ∗ v 0 s ( x ) d ← B s      p  (16) ≤ ε p/ 2 C 8 (1 + | z | q 8 )  E h sup t ≤ s ≤ T | v 0 s ( x ) | p i + E  sup t ≤ s ≤ T     ∂ ∂ x k v 0 s ( x )     p  . Com b ining ( 15 ) an d ( 16 ) E      ∂ ∂ x k θ 1 t     p  ≤ ε p/ 2 C 9 (1 + | z | q 9 ) X α ≤ 3 E h sup t ≤ s ≤ T | D α x v 0 s ( x ) | p i . Iterating these arguments for th e h igher order deriv ativ es of θ 1 , X | α |≤ k − 1 E [ | D α x θ 1 t | p ] ≤ ε p/ 2 C 10 (1 + | z | q 10 ) X | α |≤ k +1 E h sup t ≤ s ≤ T | D α x v 0 s ( x ) | p i .  Lemma 6.3. L et k, l ≥ 3 . Assume (HF k ,l ) , (HS k ,l ) and (HO k +1 ,l +1 ) . Also assume ψ 1 ∈ C 0 ,k +2 ,l ([0 , T ] × R m × R n , R ) and that al l its p artial deriva- tives up to or der (0 , k + 2 , l ) ar e i n P T ([0 , T ] × R m , R ) . Then for any p ≥ 1 ther e exists C p > 0 , indep endent of ϕ , such that for any ( x, z ) ∈ R m + n , any ε ∈ (0 , 1) and any t ∈ [0 , T ] , E [ | R t ( x, z ) | p ] ≤ C p X | α |≤ 2 Z T t E [ E [ | D α x ψ 1 s ( x ′ , z ′ ) | p ] ( x ′ ,z ′ )=( X ε, ( t,x ) s ,Z ε, ( t,z ) s ) ] ds. Pr oof. R t ( x, z ) solv es the BSPDE − dR t ( x, z ) = ( L ε R t ( x, z ) + L S ψ 1 t ( x, z )) dt + h ( x, z ) ∗ ( ψ 1 t ( x, z ) + R t ( x, z )) d ← B t , (17) R T ( x, z ) = 0 . 28 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G Existence of the solution R and its deriv ativ es, as we ll as the p olynomial gro wth all follo w from Prop osition 4.1 . By Prop osition 4.2 , the solution of ( 17 ) is give n b y θ ( t,x,z )(2) t , th e s olution to the BDSDE − dθ ( t,x,z )(2) s = L S ψ 1 s ( X ε, ( t,x ) s , Z ε, ( t,z ) s ) ds + h ( X ε, ( t,x ) s , Z ε, ( t,z ) s ) ∗ ψ 1 s ( X ε, ( t,x ) s , Z ε, ( t,z ) s ) d ← B s + h ( X ε, ( t,x ) s , Z ε, ( t,z ) s ) ∗ θ ( t,x,z )(2) s d ← B s − γ t,x,z s dW s − δ t,x,z s dV s , θ ( t,x,z )(2) T = 0 . W e will drop su p erscrip ts ( t, x, z ) for θ ( t,x,z )(2) t , ( t, z ) for Z ε, ( t,z ) and ( t, x ) for X ε, ( t,x ) . R t ( x, z ) is F B t,T -measurable, hence, so is θ 2 t . As b efore, the sto c h astic in - tegrals o ver dV a nd dW v anish w hen w e take conditional exp ectati on with resp ect to F B t,T . Thus θ 2 t = E  Z T t L S ψ 1 s ( X ε s , Z ε s ) ds    F B t,T  + E  Z T t h ( X ε s , Z ε s ) ∗ ψ 1 s ( X ε s , Z ε s ) d ← B s    F B t,T  (18) + E  Z T t h ( X ε s , Z ε s ) ∗ θ 2 s d ← B s    F B t,T  . Consider eac h term separately: E      E  Z T t L S ψ 1 s ( X ε s , Z ε s ) ds    F B t,T      p  ≤ E      Z T t L S ψ 1 s ( X ε s , Z ε s ) ds     p  ≤ ( T − t ) p − 1 Z T t E "      m X i =1 b i ( X ε s , Z ε s ) ∂ ∂ x i + 1 2 m X i,j =1 ( σ σ ∗ ) ij ( X ε s , Z ε s ) ∂ 2 ∂ x i x i ! ψ 1 s ( X ε s , Z ε s )     p # ds ≤ C 1 Z T t k b k ∞ m X i =1 E      ∂ ∂ x i ψ 1 s ( X ε s , Z ε s )     p  + 1 2 k σ σ ∗ k ∞ m X i,j =1 E      ∂ 2 ∂ x i x i ψ 1 s ( X ε s , Z ε s )     p  ! ds DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 29 ≤ C 2 Z T t X 1 ≤| α |≤ 2 E [ | D α x ψ 1 s ( X ε s , Z ε s ) | p ] ds. Note that Z ε s and X ε s are F W s ∨ F V s -measurable, ψ 1 s is F B s,T -measurable and B and ( V , W ) are indep enden t. Thus E [ | D α x ψ 1 s ( X ε s , Z ε s ) | p ] = E [ E [ | D α x ψ 1 s ( X ε s , Z ε s ) | p |F V s ∨ F W s ]] = E [ E [ | D α x ψ 1 s ( x ′ , z ′ ) | p ] ( x ′ ,z ′ )=( X ε s ,Z ε s ) ] , so th at E =      E  Z T t L S ψ 1 s ( X ε s , Z ε s ) ds    F B t,T      p  (19) ≤ C 2 X 1 ≤| α |≤ 2 Z T t E [ E [ | D α x ψ 1 s ( x ′ , z ′ ) | p ] ( x ′ z ′ )=( X ε s ,Z ε s ) ] ds. Next, by Jen s en’s inequalit y , the to wer pr op erty and the Bur kholder– Da vis–Gundy inequ ality , E      E  Z T t h ( X ε s , Z ε s ) ∗ ψ 1 s ( X ε s , Z ε s ) d ← B s    F B t,T      p  ≤ E      Z T t h ( X ε s , Z ε s ) ∗ ψ 1 s ( X ε s , Z ε s ) d ← B s     p  ≤ C p E  Z T t h ( X ε s , Z ε s ) ∗ ψ 1 s ( X ε s , Z ε s ) d ← B s  p/ 2  , where by H¨ older’s inequalit y and the Cauch y–Sc hw arz inequalit y ,  Z T t h ( X ε s , Z ε,x s ) ∗ ψ 1 s ( X ε s , Z ε s ) d ← B s  p/ 2 =  Z T t | h ( X ε s , Z ε,x s ) ∗ ψ 1 s ( X ε s , Z ε s ) | 2 ds  p/ 2 ≤ C 3 Z T t | h ( X ε s , Z ε s ) | p | ψ 1 s ( X ε s , Z ε s ) | p ds. So b y the same arguments as for the first term, E      E  Z T t h ( X ε s , Z ε s ) ∗ ψ 1 s ( X ε s , Z ε s ) d ← B s    F B t,T      p  (20) ≤ C 4 Z T t E [ E [ | ψ 1 s ( x ′ , z ′ ) | p ] ( x ′ ,z ′ )=( X ε s ,Z ε s ) ] ds. 30 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G Finally , using Burkholder–Da vis–Gund y in the second line, and Cauch y- Sc hw arz in the thir d line, E      E  Z T t h ( X ε s , Z ε s ) ∗ θ 2 s d ← B s    F B t,T      p  ≤ E      Z T t [ h ( X ε s , Z ε s )] ∗ θ 2 s d ← B s     p  (21) ≤ C p E  Z T t | h ( X ε s , Z ε s ) ∗ θ 2 s | 2 ds  p/ 2  ≤ C p E  Z T t | h ( X ε s , Z ε s ) | 2 | θ 2 s | 2 ds  p/ 2  ≤ C 5 k h k p ∞ Z T t E [ | θ 2 s | p ] ds. Com b ining ( 18 ) w ith ( 19 ), ( 20 ) and ( 21 ), E [ | θ 2 t | p ] ≤ C 6 X | α |≤ 2 Z T t E [ E [ | D α x ψ 1 s ( x ′ , z ′ ) | p ] ( x ′ ,z ′ )=( X ε s ,Z ε s ) ] ds + C 5 k h k p ∞ Z T t E [ | θ 2 s | p ] ds. By Gr onw all, E [ | θ 2 t | p ] ≤ C 6  X | α |≤ 2 Z T t E [ E [ | D α x ψ 1 s ( x ′ , z ′ ) | p ] ( x ′ ,z ′ )=( X ε s ,Z ε s ) ] ds  e ( T − t ) C 5 k h k p ∞ ≤ C 7  X | α |≤ 2 Z T t E [ E [ | D α x ψ 1 s ( x ′ , z ′ ) | p ] ( x ′ ,z ′ )=( X ε s ,Z ε s ) ] ds  , where we choose C 7 so that the in equ alit y holds for every t ∈ [0 , T ] (replace e ( T − t ) C 5 k h k ∞ b y e T C 5 k h k ∞ ).  No w we can collect all these r esults to obtain the fir st step to wa rds The- orem 3.1 . Lemma 6.4. Assume (H stat ) , (HF 8 , 4 ) , (HS 7 , 4 ) , (HO 8 , 4 ) and that ϕ ∈ C 7 b ( R m , R ) . Then for every p ≥ 1 ther e exists C, q 1 , q 2 > 0 , indep endent of ϕ , such that sup 0 ≤ t ≤ T E [ | v ε t ( x, z ) − v 0 t ( x ) | p ] ≤ ε p/ 2 C (1 + | x | q 1 + | z | q 2 ) k ϕ k p 4 , ∞ . DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 31 Pr oof of Theorem 3.1 . W e trac k the necessary conditions backw ards from Lemm a 6.3 : (1) F or the solution R giv en in Lemma 6.3 to exist and satisfy the stated b ound , w e need (HF 3 , 3 ), (HS 3 , 3 ), (HO 4 , 4 ) and ψ 1 ∈ C 0 , 5 , 3 ([0 , T ] × R m × R n , R ) . The p olynomial gro wth cond ition will b e satisfied an ywa y . (2) F or ψ 1 to b e in C 0 , 5 , 3 ([0 , T ] × R m × R n , R ) , we need (H stat ), (HF 6 , 4 ), (HS 6 , 4 ), (HO 6 , 4 ) and ¯ a, ¯ b, ¯ h ∈ C 5 b . W e also need v 0 ∈ C 0 , 6 ([0 , T ] × R m , R ) . Again, the p olynomial gro wth condition will b e satisfied. (3) F or v 0 to b e in C 0 , 6 ([0 , T ] × R m , R ) we need ¯ a, ¯ b, ϕ ∈ C 7 b and ¯ h ∈ C 8 b . (4) F or ¯ a, ¯ b to b e in C 7 b w e need (HF 7 , 3 ) as well as (HS 7 , 0 ) b y Pr op osition 5.4 . S imilarly w e need (HF 8 , 3 ) as well as (HO 8 , 0 ) f or ¯ h to b e in C 8 b . (5) So su fficien t conditions are (H stat ), (HF 8 , 4 ), (HS 7 , 4 ), (HO 8 , 4 ). In that case w e obtain f r om Lemma 6.1 X | α |≤ 4 E h sup 0 ≤ t ≤ T | D α v 0 t ( x ) | p i ≤ C 1 (1 + | x | q 1 ) k ϕ k p 4 , ∞ . (22) F rom Lemma 6.2 we obtain X | α |≤ 2 sup 0 ≤ t ≤ T E [ | D α x ψ 1 t ( x, z ) | p ] (23) ≤ ε p/ 2 C 2 (1 + | z | q 2 ) X | α |≤ 4 E h sup 0 ≤ t ≤ T | D α x v 0 t ( x ) | p i . F rom Lemma 6.3 we get E [ | R t ( x, z ) | p ] (24) ≤ C 3 X | α |≤ 2 Z T t E [ E [ | D α x ψ 1 s ( x ′ , z ′ ) | p ] ( x ′ ,z ′ )=( X ε, ( t,x ) s ,Z ε, ( t,z ) s ) ] ds. Com b ining ( 22 ), ( 24 ), ( 24 ), we get for any t ∈ [0 , T ] (b y time-homogeneit y of X ε and Z ε ) E [ | R t ( x, z ) | p ] + E [ | ψ 1 t ( x, z ) | p ] (25) ≤ ε p/ 2 C 4  1 + sup 0 ≤ s ≤ T E [ | X ε s | q 1 + | Z ε,x s | q 2 | ( X ε 0 , Z ε 0 ) = ( x, z )]  k ϕ k p 4 , ∞ . F rom Prop osition 5.3 we obtain sup 0 ≤ s ≤ T E [ | X ε s | q 1 + | Z ε,x s | q 2 | ( X ε 0 , Z ε 0 ) = ( x, z )] ≤ C 5 (1 + | x | q 3 + | z | q 4 ) . Noting that the righ t-hand side in ( 25 ) do es not dep end on t ∈ [0 , T ], sup 0 ≤ t ≤ T E [ | R t ( x, z ) | p ] + su p 0 ≤ t ≤ T E [ | ψ 1 t ( x, z ) | p ] ≤ ε p/ 2 C 6 (1 + | x | q 3 + | z | q 4 ) k ϕ k p 4 , ∞ . 32 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G Finally sup 0 ≤ t ≤ T E [ | v ε t ( x, z ) − v 0 t ( x ) | p ] ≤ C 7  sup 0 ≤ t ≤ T E [ | R t ( x, z ) | p ] + su p 0 ≤ t ≤ T E [ | ψ 1 t ( x, z ) | p ]  ≤ ε p/ 2 C 8 (1 + | x | q 3 + | z | q 4 ) k ϕ k p 4 , ∞ , whic h completes the pro of.  No w w e r ecall that all the calculations up un til now w ere u nder the c hanged measure P ε . W e only wrote P and B to facilitate the reading. So let us transfer the results to the original measur e Q . Lemma 6.5. Assume (H stat ) , (HF 8 , 4 ) , (HS 7 , 4 ) , (HO 8 , 4 ) and that ϕ ∈ C 7 b ( R m , R ) . Then for every p ≥ 1 ther e exist C, q 1 , q 2 > 0 , indep endent of ϕ , such that sup 0 ≤ t ≤ T E Q [ | v ε t ( x, z ) − v 0 t ( x ) | p ] ≤ ε p/ 2 C (1 + | x | q 1 + | z | q 2 ) k ϕ k p 4 , ∞ . Pr oof. This is a simple application of the Cauc h y–Sch wa r z inequalit y in combinatio n with Gr onw all’s lemma, E Q [ | v ε t ( x, z ) − v 0 t ( x ) | p ] = E P ε  | v ε t ( x, z ) − v 0 t ( x ) | p d Q d P ε  ≤ E P ε [ | v ε t ( x, z ) − v 0 t ( x ) | 2 p ] 1 / 2 E P ε  d Q d P ε  2  1 / 2 , so we see that the resu lt is true by Lemma 6.4 as long as the second exp ec- tation is finite. Recall that we had defined th e notation d Q d P ε    F t = ˜ D ε t = exp  Z t 0 h ( X ε s , Z ε s ) ∗ d Y ε s − 1 2 Z t 0 | h ( X ε s , Z ε s ) | 2 ds  . So ˜ D ε satisfies th e SDE d ˜ D ε t = ˜ D ε t h ( X ε t , Z ε t ) ∗ d Y ε t , ˜ D ε 0 = 1 . Since under P ε , Y ε is a Bro wn ian motion, we get by Itˆ o-isometry E P ε [( ˜ D ε t ) 2 ] = E P ε  Z t 0 ( ˜ D ε s ) 2 | h ( X ε s , Z ε s ) | 2 ds  ≤ k h k 2 ∞ E P ε  Z t 0 ( ˜ D ε s ) 2 ds  , so th at by Gron wall E P ε [( ˜ D ε T ) 2 ] < ∞ .  DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 33 Lemma 6.6. A ssume ( H stat ) , (HF 8 , 4 ) , (HS 7 , 4 ) , (HO 8 , 4 ) , that ϕ ∈ C 7 b and that the initial distribution Q ( X ε 0 ,Z ε 0 ) has finite moments of every or der. Then for every p ≥ 1 ther e exists C > 0 , indep endent of ϕ , such that E Q [ | ρ ε,x T ( ϕ ) − ρ 0 T ( ϕ ) | p ] ≤ ε p/ 2 C k ϕ k p 4 , ∞ . Pr oof. As we already describ ed in the In tro duction , we obtain fr om Lemma 6.5 E Q [ | ρ ε,x T ( ϕ ) − ρ 0 T ( ϕ ) | p ] = E Q      Z ( v ε 0 ( x, z ) − v 0 0 ( x )) Q ( X ε 0 ,Z ε 0 ) ( dx, dz )     p  ≤ Z E Q [ | v ε 0 ( x, z ) − v 0 0 ( x ) | p ] Q ( X ε 0 ,Z ε 0 ) ( dx, dz ) ≤ ε p/ 2 C 1 Z (1 + | x | q 1 + | z | q 2 ) Q ( X ε 0 ,Z ε 0 ) ( dx, dz ) k ϕ k p 4 , ∞ ≤ ε p/ 2 C 2 k ϕ k p 4 , ∞ .  The con vergence of the actual filter, that is, of π ε,x to π 0 , now follo ws exactly as in Chapter 9.4 of Bain and Crisan ( 2009 ). F or the s ake of com- pleteness, we include the argumen ts. Lemma 6.7. L e t p ≥ 1 . Then sup ε ∈ (0 , 1] ,t ∈ [0 ,T ] { E Q [ | ρ ε,x t (1) | − p ] + E Q [ | ρ 0 t (1) | − p ] } < ∞ as long as h is b ounde d. Pr oof. W e give the argument for E Q [ | ρ ε,x t (1) | − p ], E Q [ | ρ 0 t (1) | − p ] b eing completely analogue. W e ha ve E Q [ | ρ ε,x t (1) | − p ] = E P ε  | ρ ε,x t (1) | − p d Q d P ε  ≤ E P ε [ | ρ ε,x t (1) | − 2 p ] 1 / 2 E P ε  d Q d P ε  2  1 / 2 . W e sho w ed in the pro of of Lemma 6.5 that th e second exp ectation is finite. Note th at x 7→ x − 2 p is con ve x. Therefore b y Jens en’s inequalit y , E P ε [ | ρ ε,x t (1) | − 2 p ] = E P ε      E P ε  exp  Z t 0 h ( X ε s , Z ε s ) ∗ d Y ε s − 1 2 Z t 0 | ¯ h ( X ε s , Z ε s ) | 2 ds     Y ε t      − 2 p  34 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G ≤ E P ε      exp  Z t 0 h ( X ε s , Z ε s ) ∗ d Y ε s − 1 2 Z t 0 | ¯ h ( X ε s , Z ε s ) | 2 ds      − 2 p  ≤ E P ε      d Q d P ε     − 2 p  = E Q      d P ε d Q     2 p +1  . The r esu lt now follo ws exactly as in the pro of of Lemma 6.5 b ecause for D ε t = d P ε /d Q | F t , we ha ve dD ε t = − h ( X ε t , Z ε t ) ∗ dB t , D ε 0 = 1 , and B is a Brownian motion u nder Q .  Define for any m easurable and b ounded test fu nction ϕ : R m → R , π 0 t ( ϕ ) = ρ 0 t ( ϕ ) ρ 0 t (1) . Recall that π ε,x t w as defined analogously with ρ ε,x t instead of ρ 0 t . W e then ha ve Lemma 6.8. Assume (H stat ) , (HF 8 , 4 ) , (HS 7 , 4 ) , (HO 8 , 4 ) and that the initial distribution Q ( X ε 0 ,Z ε 0 ) has finite moments of e very or der. L et p ≥ 1 . Then ther e exists C > 0 such that for every ϕ ∈ C 7 b E Q [ | π ε,x T ( ϕ ) − π 0 T ( ϕ ) | p ] ≤ ε p/ 2 C k ϕ k p 4 , ∞ . Pr oof. In the third lin e w e use that π ε,x is a.s. equal to a pr obabilit y measure, E Q [ | π ε,x T ( ϕ ) − π 0 T ( ϕ ) | p ] = E Q      ρ ε,x T ( ϕ ) ρ ε,x T (1) − ρ 0 T ( ϕ ) ρ 0 T (1)     p  = E Q      ρ ε,x T ( ϕ ) − ρ 0 T ( ϕ ) ρ 0 T (1) − π ε,x T ( ϕ ) ρ ε,x T (1) − ρ 0 T (1) ρ 0 T (1)     p  ≤ C p  E Q      ρ ε,x T ( ϕ ) − ρ 0 T ( ϕ ) ρ 0 T (1)     p  + k ϕ k p ∞ E Q      ρ ε,x T (1) − ρ 0 T (1) ρ 0 T (1)     p  ≤ C p ( E Q [ | ρ 0 T (1) | − 2 p ]) 1 / 2 ( E Q [ | ρ ε,x T ( ϕ ) − ρ 0 T ( ϕ ) | 2 p ] 1 / 2 + k ϕ k p ∞ E Q [ | ρ ε,x T (1) − ρ 0 T (1) | 2 p ] 1 / 2 ) ≤ ε p/ 2 C 1 k ϕ k 4 , ∞ , where th e last step follo ws from Lemmas 6.6 and 6.7 .  DIMENSION A L REDU CTION IN N ONLINEAR FI L TERING 35 Since the b oun d only d ep ends on k ϕ k 4 , ∞ , w e can replace th e assump tion ϕ ∈ C 7 b b y ϕ ∈ C 4 b : Just appr oximate ϕ ∈ C 4 b b y ϕ n ∈ C 7 b in the k · k 4 , ∞ -norm, and tak e adv an tage of the fact that π ε,x T and π 0 T are a.s. equal to probabilit y measures. Therefore we hav e: Corollar y 6.9. Assume (H stat ) , (HF 8 , 4 ) , (HS 7 , 4 ) , (HO 8 , 4 ) and that the initial distribution Q ( X ε 0 ,Z ε 0 ) has finite moments of every or der. L et p ≥ 1 . Then ther e exists C > 0 such that for every ϕ ∈ C 4 b , E Q [ | π ε,x T ( ϕ ) − π 0 T ( ϕ ) | p ] ≤ ε p/ 2 C k ϕ k p 4 , ∞ . No w note that there exists a count able algebra ( ϕ i ) i ∈ N of C 4 b functions that strongly separates p oints in R m . T hat is, for ev ery x ∈ R m and δ > 0, there exists i ∈ N , su c h that inf y : | x − y | >δ | ϕ i ( x ) − ϕ i ( y ) | > 0. T ak e, for example, all fu nctions of the f orm exp − n X j =1 q j ( x − x j ) 2 ! with n ∈ N , q j ∈ Q + , x j ∈ Q m . By Theorem 3.4.5 of Ethier and K urtz ( 1986 ), the sequence ( ϕ i ) is conv ergence determining for the top ology of wea k con- v ergence of probabilit y measures. That is, if µ n and µ are probabilit y mea- sures on R m , such that lim n →∞ µ n ( ϕ i ) = µ ( ϕ i ) f or ev ery i ∈ N , th en µ n con verge s wea kly to µ . Define the follo wing metric on the s p ace of probabilit y measur es on R m : d ( ν, µ ) = d ( ϕ i ) ( ν, µ ) = ∞ X i =1 | ν ( ϕ i ) − µ ( ϕ i ) | 2 i . Because ( ϕ i ) is con v ergence determin in g, the metric d generates the top ology of weak con vergence. Therefore the pr o of of Theorem 3.1 is complete. 7. Conclusion and future d irections. This pap er pr esen ted the theoret- ical b asis for the dev elopment of a low er-dimensional particle filtering al- gorithm for the state estimation in complex m ultiscale systems. T o this end, w e com bin ed sto c hastic homogenization with nonlinear filtering theory to constr u ct a h omogenized S P DE which is the app ro ximation of a low er- dimesional nonlinear fi lter for the “coarse-grained” p r o cess. The conv ergence of the optimal filter of the “co arse-grained” pro cess to the solution of the homogenized fi lter is sho wn us ing BSDEs and asymptotic tec hniques. This homogenized SPDE can b e u s ed as the basis for an efficien t multi-sc ale particle filtering algorithm for estimating the slo w dynamics of the system, without directly accounti ng for the fast dynamics. In L in gala et al. ( 2012 ) we 36 IMKELLER, NAMACHCHIV A Y A, PERKO WSK I AND YEON G present a numerical algorithm based on th is sc h eme that enab les efficien t in- corp oration of observ ation d ata for estimation of th e coarse-grained (“slo w”) dynamics, and we apply the algorithm to a high-dimensional c haotic multi- scale s ystem. Ev en thou gh this pap er deals with just one widely separated c haracteristic time scale, on e can extend this wo rk to incorp orate a more realistic setting where the signal has more than one time scale separation. As b efore w e let ε b e a small parameter that measures the r atio of slo w and fast time scales. Consider the signal and obs erv ation pro cesses go verned by d Z ε t = 1 ε 2 f ( Z ε t , X ε t ) + 1 ε g ( Z ε t , X ε t ) dW t , Z ε 0 = z , dX ε t = 1 ε b I ( Z ε t , X ε t ) + b ( Z ε t , X ε t ) + σ ( Z ε t , X ε t ) dV t , X ε 0 = x, (26) d Y ε t = h ( Z ε t , X ε t ) dt + dB t , Y ε 0 = 0 , where W , V and B are indep enden t Wiener pro cesses and x an d z are r an- dom initial conditions which are indep en den t of W , V and B . It is imp ortan t to realize that there are sev eral s cales in ( 26 ), ev en the slo w p r o cess X ε t has a fast v arying comp onent. This case is imp ortan t, in particular, for applica- tions in geoph ys ical fl ows and cli mate dynamics. T he drift term b and th e diffusion σ cause fl uctuations of ord er order 1, and th e drift term f and the diffusion g ca use fluctuations of order ord er ε − 2 , wh ereas the drift term b I causes fluctuations at an intermediate order ε − 1 . It w as found that when the a v erage of b I with resp ect to the in v arian t measure of the fast comp o- nen t Z ε t (for the fixed slo w comp onent ) is zero, the limit d istr ibution of the slo w comp onent (a wa y from the initial la y er) can also b e obtained in terms of th e solution of some auxiliary P oisson equation in the homogenization theory . Ho w ever, a unifie d fr amework to deal with ε − 1 term in d ev eloping a lo w er-dimen sional nonlinear filter for the “coarse-grained” p r o cess is still not av ailable. Our conditions on the co efficien ts are very r estrictive and exclude, f or ex- ample, linear mo dels. This is due to the fact that we are usin g homogeniza- tion of SPDEs to obtain con ve rgence of the filter, and that for existence of solutions to the SPDEs, the co efficien ts need to b e b oun ded and sufficient ly smo oth. W orking with wea k solutions in place of classical solutions would not impro ve the conditions m uch. Using viscosit y solutions or en tirely relying on pr obabilistic arguments migh t b e a wa y to get less restrictiv e conditions ho wev er, with these method s we do not expect that a rate of conv ergence can b e obtained. While w e w ere able to obtain the explicit r ate of con vergence √ ε , the constan t C in Theorem 3.1 dep end s on the terminal time T . It w ould b e in teresting to find conditions under whic h this can b e av oided. 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Perkowski Institut f ¨ ur Ma themat ik Humboldt-Universit ¨ at zu Berlin R udower Ch aussee 25 12489 Berlin Germany E-mail: imkeller@math.h u-b erli n.de perko ws k@math.h u- ber lin.de N. S. Nam achchiv a y a H. C. Yeong Dep ar tment of Aerosp ace Engineering University of Illinois a t Urba n a-Champ aig n 306 T alb ot Labora tor y, MC-236 104 South Wrigh t Street Urbana, Illinois 61801 USA E-mail: na v am@illinois. edu h y eong2@illinois.edu