DIMENSIONAL REDUCTION IN NONLINEAR FILTERING: A HOMOGENIZATION APPROACH

Transcription

1 DIMNSIONAL RDUCION IN NONLINAR FILRING: A HOMOGNIZAION APPROACH PR IMKLLR 1, N. SRI NAMACHCHIVAYA 2, NICOLAS PRKOWSKI 1, AND HOONG C. YONG 2 Abrac. We propoe a homogenized filer for mulicale ignal, which allow o reduce he dimenion of he yem. We prove ha he nonlinear filer converge o our homogenized filer wih rae. hi i achieved by a uiable aympoic expanion of he dual of he Zakai equaion, and by probabiliically repreening he correcion erm wih he help of BDSD. 1. Inroducion Filering heory i an eablihed field in applied probabiliy and deciion and conrol yem, which i imporan in many pracical applicaion from inerial guidance of aircraf and pacecraf o weaher and climae predicion. I provide a recurive algorihm for eimaing a ignal or ae of a random dynamical yem baed on noiy meauremen. More preciely, filering problem coni of an unobervable ignal proce X def = {X : } and an obervaion proce Y def = {Y : } ha i a funcion of X corruped by noie. he main objecive of filering heory i o ge he be eimae of X baed on he informaion Y = σ{y : }. def hi i given by he condiional diribuion π of X given Y or equivalenly, he condiional expecaion f(x ) Y for a rich enough cla of funcion. Since hi eimae minimize he mean quare error lo, we call π he opimal filer. he goal of filering heory i o characerize hi condiional diribuion effecively. In implified problem where he ignal and he obervaion model are linear and Gauian, he filering equaion i finie-dimenional, and he oluion i he well-known Kalman-Bucy filer. In more realiic problem, nonlineariie in he model lead o more complicaed equaion for π, defined by Zakai (1969) and Fujiaki e al. (1972), which decribe he evoluion of he condiional diribuion in he pace of probabiliy meaure (ee, for example, Bain and Crian (29), Kallianpur (198), Liper and Shiryaev (21)). I i impracical o implemen a numerical oluion o uch infinie dimenional ochaic evoluion equaion of he general nonlinear filering problem by finie difference or finie elemen approximaion. herefore, exended Kalman filer algorihm, which ue linear approximaion o he ignal dynamic and obervaion, have been ued exenively in everal applicaion. hee provide eenially a fir order approximaion o an infinie dimenional problem and can perform quie poorly in problem wih rong nonlineariie. Paricle filer have been well eablihed for he implemenaion of nonlinear filering in cience and engineering applicaion. Douce e al. (21) and Arulampalam e al. (22) provide comprehenive inigh ino paricle filering. However, due o dimenionaliy iue (ee, for example, Snyder e al. (28)) and compuaional complexiie ha arie in repreening he ignal deniy uing a high number of paricle, he problem of paricle filering in high dimenion i ill no compleely reolved. A a reul of hee difficulie, we have eablihed a novel paricle filering mehod Park e al. (211) for mulicale ignal and obervaion procee ha combine he homogenizaion wih filering echnique. he heoreical bai for hi new capabiliy i preened in hi paper. 21 Mahemaic Subjec Claificaion. 6G35, 35B27, 6H15, 6H35. Key word and phrae. nonlinear filering; dimenional reducion; homogenizaion; paricle filering; aympoic expanion; SPD; BDSD. 1

2 2 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG he reul preened here are e wihin he conex of low-fa dynamical yem, where he rae of change of differen variable differ by order of magniude. Muliple ime cale occur in model hroughou he cience and engineering field. For example, climae evoluion i governed by fa amopheric and low oceanic dynamic and ae dynamic in elecric power yem coni of fa- and lowly-varying elemen. hi paper addree he effec of he mulicale ignal and obervaion procee via he udy of he Zakai equaion. We conruc a lower dimenional Zakai equaion in a canonical way. hi problem ha alo been udied in Park e al. (21) uing a differen approach from wha i preened here. In moderae dimenional problem, paricle filer are an aracive alernaive o numerical approximaion of he ochaic parial differenial equaion (SPD) by finie difference or finie elemen mehod. For he reduced nonlinear model an appropriae form of paricle filer can be a viable and ueful cheme. Hence, Lingala e al. (212) preen he numerical oluion of he lower dimenional ochaic parial differenial equaion derived here, a i i applied o a chaoic high-dimenional mulicale yem. In general, hi paper provide rigorou mahemaical reul ha uppor he numerical algorihm baed on he idea ha ochaically averaged model provide qualiaively ueful reul which are poenially helpful in developing inexpenive lower-dimenional filering a demonraed by Park e al. (211) in he conex of homogenized paricle filer and by Harlim and Kang (212) in he conex of averaged enemble Kalman filer. he convergence of he opimal filer o he homogenized filer i hown uing backward ochaic differenial equaion (BSD) and aympoic echnique. Le u decribe he main reul. We aume he ignal i given a oluion of he wo ime cale ochaic differenial equaion (SD) dx = b(x, Z )d + σ(x, Z )dv dz = 1 f(x, Z )d + 1 g(x, Z )dw. Here X i he low componen and Z i he fa componen. We aume ha for every fixed x, he oluion Z x of dz x = f(x, Z x )d + g(x, Z x )dw i ergodic and converge rapidly o i unique aionary diribuion. In hi cae i i well known ha X converge in diribuion o a diffuion X which i governed by an SD dx = b(x )d + σ(x )dv. hi X i ued o conruc an averaged filer π. We denoe he opimal filer for he full yem by π. Define he x-marginal of π a π,x, i.e. ϕ(x)π,x (dx) = ϕ(x)π (dx, dz). Our main reul i hen heorem. Under he aumpion aed in heorem 3.1, for every p 1 and here exi C >, uch ha for every ϕ C 4 b ( Q π,x (ϕ) π (ϕ) p ) 1/p C ϕ 4,. In paricular, here exi a meric d on he pace of probabiliy meaure, uch ha d generae he opology of weak convergence, and uch ha for every here exi C > uch ha Q d(π,x, π ) C. We begin in Secion 2 by preening he general formulaion of he mulicale nonlinear filering problem. Here we decribe he meaure-valued Zakai equaion and inroduce he homogenized equaion ha we eek o derive for he reduced dimenion unnormalized filer. Secion 3 preen

3 DIMNSIONAL RDUCION IN NONLINAR FILRING 3 he formal aympoic expanion of he muli cale Zakai equaion ha reul in everal SPD. We alo preen he main reul of hi paper in hi ecion. Secion 4 provide he probabiliic repreenaion of he SPD, ha i, we decribe he oluion of he infinie dimenional SPD by finie dimenional backward doubly ochaic differenial equaion (BDSD). We reae ome of he reul in hi conex due o Rozovkii (199) and Pardoux and Peng (1994) a he end of hi ecion. We preen ome of he preliminary reul of Pardoux and Vereennikov (23) on convergence of he raniion funcion of Z x in ecion 5. hee eimae are ued in he proof of he main reul preened in ecion Formulaion of mulicale nonlinear filering problem Le (Ω, F, (F ), Q) be a filered probabiliy pace ha uppor a (k + l + d)-dimenional andard Brownian moion (V, W, B). Le he ignal (X, Z ) be a wo ime cale diffuion proce wih a fa componen Z and a low componen X : (1) dx = b(x, Z )d + σ(x, Z )dv dz = 1 f(x, Z )d + 1 g(x, Z )dw, where X R m, Z R n, W R l and V R k are independen andard Brownian moion, 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 he funcion above are aumed o be Borel-meaurable. For fixed x R m, define (2) dz x = f(x, Z x )d + g(x, Z x )dw. Aume ha for all x R m, Z x i ergodic and converge rapidly oward i aionary meaure µ(x, ). We will make hi precie laer. he d-dimenional obervaion Y i given by Y = h(x, Z )d + B wih Borel-meaurable h : R m+n R d. B i aumed o be a d-dimenional andard Brownian moion ha i independen of W and V. Define Y = σ(y : ) N, where N are he Q-negligible e. For a finie meaure π on R m+n and for a bounded meaurable funcion ϕ on R m+n denoe π(ϕ) = ϕ(x, z)π(dx, dz). hen our aim i o calculae he meaure-valued proce (π, ) deermined by Define he Giranov ranform dp ( dq = D = exp F π (ϕ) = ϕ(x, Z ) Y. h(x, Z ) db 1 2 ) h(x, Z) 2 d. Under P, he obervaion proce, Y, i a Brownian moion and independen of (X, Z ). By he Kallianpur-Sriebel formula, P ϕ(x Q ϕ(x, Z ) Y, Z ) dq F dp Y = dq F P dp Y wih ( dq dp = D = exp F h(x, Z ) dy 1 2 ) h(x, Z) 2 d.

4 4 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG So if we define hen ρ (ϕ) = P ( ϕ(x, Z ) exp h(x, Z) dy 1 2 π (ϕ) = ρ (ϕ) ρ (1). ) h(x, Z) 2 d Y, Denoe by L = 1 L F + L S he differenial operaor aociaed o (X, Z ). ha i, n L F = f i (x, z) + 1 n (gg 2 ) ij (x, z) z i 2 z i z j L S = b i (x, z) x i (σσ 2 ) ij (x, z) x i x j where denoe he ranpoe of a marix or a vecor. hen he unnormalized meaure-valued proce, ρ, aifie he Zakai equaion: (3) dρ (ϕ) = ρ (L ϕ)d + ρ (hϕ)dy ρ (ϕ) = Q ϕ(x, Z) for every ϕ Cb 2(Rm+n, R) (ee, for example, Bain and Crian (29)). For k, Cb k i he pace of k ime coninuouly differeniable funcion f, uch ha f and all i parial derivaive up o order k are bounded. he heory of ochaic averaging (ee, for example, Papanicolaou e al. (1977)) ell u ha under uiable condiion, X converge in law o X a, where X i he oluion of an SD dx = b(x )d + σ(x )dw for uiably averaged b and σ. Denoe he generaor of X by L. We wan o how ha a long a we are only inereed in eimaing he low componen, we can ake advanage of hi fac. More preciely, we wan o find a homogenized (unnnormalized) filer ρ, uch ha for mall, ρ,x which i he x-marginal of ρ, i cloe o ρ. he x-marginal of ρ i defined a ρ,x (ϕ) = ϕ(x)ρ (dx, dz) R m+n for every meaurable bounded ϕ : R m R, and ρ i he oluion of (4) dρ (ϕ) = ρ ( Lϕ)d + ρ ( hϕ)dy ρ (ϕ) = Q ϕ(x ), where h i a uiably averaged verion of h. he meaure-valued procee π and π,x are hen defined in erm of ρ and ρ,x a π wa defined in erm of ρ : π (ϕ) = ρ (ϕ) ρ (1) and π,x (ϕ) = ρ,x (ϕ) ρ,x (ϕ). Noe ha he homogenized filer i ill driven by he real obervaion Y and no by a homogenized obervaion, which i pracical for implemenaion of he homogenized filer in applicaion ince uch homogenized obervaion i uually no available. However, hould uch homogenized obervaion be available, uing i would lead o lo of informaion for eimaing he ignal compared o uing he acual obervaion.

5 DIMNSIONAL RDUCION IN NONLINAR FILRING 5 In hi paper, we will prove L 1 -convergence of he acual filer o he homogenized filer, i.e. we will how ha for any >, lim d(π,x, π ) =, where d denoe a uiable diance on he pace of probabiliy meaure ha generae he opology of weak convergence. hi convergence reul i hown in Park e al. (21) for a wo-dimenional mulicale ignal proce wih no drif in he fa componen SD. Here, we exend he reul o an R m+n -dimenional ignal proce wih drif and diffuion coefficien of he fa and low componen dependen on boh componen. he proof of Park e al. (21) i baed on repreening he low componen a a ime-changed Brownian moion under a uiable meaure, which canno be exended eaily o he mulidimenional eing we aume here. Baed on (3) and (4), he filer convergence problem i a problem of homogenizaion of a SPD. In Papanicolaou e al. (1977), homogenizaion of diffuion procee wih periodic rucure i done uing he maringale problem approach. In Papanicolaou and Kohler (1975) and Chaper 2 of Benouan e al. (1978), limi behavior of ochaic procee i udied uing aympoic analyi. Benouan e al. (1978) udie linear SPD wih periodic coefficien and alo ued a probabiliic approach in Chaper 3. Homogenizaion in he nonlinear filering problem framework ha been udied in Benouan and Blankenhip (1986) and Ichihara (24) via aympoic analyi on a dual repreenaion of he nonlinear filering equaion. A far a we are aware, Ichihara (24) ha ued BSD for udying homogenizaion of Zakai-ype SPD for he fir ime. Our convergence proof applie BSD echnique by invoking he dual repreenaion of he filering equaion and uing aympoic analyi o deermine he limi behavior of he oluion of he backward equaion. Pardoux and Vereennikov (23) give precie eimae for he raniion funcion of an ergodic SD of he ype (2), and hee reul are ued in our proof. o our knowledge, uch mehod of homogenizaion for SPD combining BSD and aympoic mehod ha no been done before. o our knowledge, a reul preened in Chaper 6 of Kuhner (199) i he cloe o he reul preened in hi paper. In heorem of Kuhner (199) i i hown ha for a fixed e funcion, he difference of he unnormalized acual and homogenized filer for mulicale jumpdiffuion procee converge o zero in diribuion. Sandard reul hen give convergence in probabiliy of he fixed ime marginal. Kuhner (199) mehod of proof i by averaging he coefficien of he SD for he unnormalized filer and howing ha he limi of boh filer aify he ame SD ha poee a unique oluion. We obain L p convergence of he meaure valued proce, no ju for fixed e funcion, and we are able o quanify he rae of convergence, which, o he be of our knowledge, ha no been achieved before in homogenizaion of nonlinear filer.. In Klepina e al. (1997), convergence of he nonlinear filer i hown in a very general eing, baed on convergence in oal variaion diance of he law of (X, Y ). hi i hen applied o wo example. Since he diffuion marix of our low componen i allowed o depend on he fa componen, our reul are no a pecial cae. In he example of Klepina e al. (1997), X converge o X in probabiliy, which i no longer he cae in our eing. However i migh be poible o apply he oal variaion echnique developed in Klepina e al. (1997) o obain convergence in our eing. Only he rae of convergence canno be deermined wih hee echnique. For a given bounded e funcion ϕ and erminal ime, we follow Pardoux (1979) in inroducing he aociaed dual proce v,,ϕ (x, z), which i a dynamic verion of P ϕ(x ) D Y : v,,ϕ (x, z) = P,x,z ϕ(x ) D, Y, where P,x,z i he meaure under which X and Z are governed by he ame dynamic a under P, bu (X, Z ) ay in (x, z) unil ime, hen i ar o follow he SD dynamic. D, = D ( D ) 1 ; and Y, = σ(y r Y : r ) N (recall ha N denoe he

6 6 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG Q-negligible e). From he Markov propery of (X, Z ) i follow ha for any, : ρ (v,,ϕ ) = ρ,x (ϕ). In paricular (becaue a ime, ρ i ju he aring diribuion of (X, Z )): ρ,x (ϕ) = v,,ϕ (x, z)q (X,Z ) (dx, dz). Similarly inroduce where v,,ϕ (x) = P,x ϕ(x ) D, Y,, ( D, = exp h(x r ) dyr 1 ) h(x r ) 2 dr 2 and P,x i he meaure under which X i governed by he ame dynamic a under P, bu ay in x unil ime. We can alo how ha for any, : ρ (v,,ϕ ) = ρ (ϕ), o ha ρ (ϕ) = v,,ϕ (x)q X (dx). Noe ha Q X = Q X, becaue he homogenized proce ha he ame aring diribuion a he unhomogenized one. Now fix and ϕ Cb 2(Rm, R) and wrie v = v,,ϕ and v = v,,ϕ. Our aim i o how ha for nice e funcion ϕ, and for he dual procee v and v defined above, v (x, z) v (x) p i mall (in a way ha will depend on x and z). hen ρ,x (ϕ) ρ (ϕ) p = (v(x, z) v(x))q p (X,Z ) (dx, dz) v(x, z) v(x) p Q (X,Z ) (dx, dz) = v(x, z) v(x) p Q (X,Z ) (dx, dz) will alo be mall a long a Q (X,Z ) i well behaved. 3. Formal expanion of he filering equaion and he main reul Before we coninue, le u change noaion: For large par of hi aricle we will only work under P, and he proce Y i a Brownian moion under P which i independen of (X, Z, X ). herefore from now on we wrie P inead of P and B inead of Y o faciliae he reading. he diribuion and noaion for he Markov procee (X, Z, X ) do no change. he key poin i now ha v and v olve backward SPD: (5) and (6) dv (x, z) = L v (x, z)d + h(x, z) v (x, z)db v (x, z) = ϕ(x) dv (x) = Lv (x, z)d + h(x) v (x)db v (x) = ϕ(x). Here and everywhere in hi aricle, d B denoe Iô backward inegral. We formally expand v a v (x, z) = u (x, z) + u 1 / (x, z) + 2 u 2 / (x, z). Noe ha rigorouly hi doe no make any ene, becaue:

7 DIMNSIONAL RDUCION IN NONLINAR FILRING 7 We work wih equaion wih erminal condiion. Bu when we end, hen / converge o infiniy. So for which ime hould he erminal condiion of e.g. u 1 be defined? he erm in hi expanion will all be ochaic. hen if u 1 i adaped o F B, he ochaic inegral u 1 / (x, z)d B a priori doe no make any ene for < 1. However if we do uch a formal aympoic expanion, and hen call v (, x) = u (, x), ψ 1 (, x, z) = u 1 / (x, z), R(, x, z) = 2 u 2 / (x, z) (of coure all erm excep v depend on, which we omi in he noaion o faciliae he reading), hen hee erm have o olve he following equaion: dv (x) = Lv (x, z)d + h(x) v (x)d B (7) (8) dψ 1 (x, z) = 1 L F ψ 1 (x, z)d + (L S L)v (x)d + ( h(x, z) h(x) ) v (x)d B dr (x, z) = L R (x, z)d + L S ψ 1 (x, z)d wih erminal condiion + h(x, z) ( ψ 1 (x, z) + R (x, z) ) d B v (, x) = ϕ(x), ψ 1 (, x, z) = R(, x, z) =. Noe ha he equaion for v i exacly he deired equaion (6). By exience and uniquene of he oluion o hee linear equaion, we can apply uperpoiion o obain ha hen indeed v (x, z) = v (x) + ψ 1 (x, z) + R (x, z). herefore he problem of howing L p -convergence of v o v reduce o howing L p -convergence of ψ 1 + R o. o achieve hi, we will give probabiliic repreenaion of ψ 1 and R in erm of backward doubly ochaic differenial equaion. hi will allow u o apply he exiing eimae for he raniion funcion of Z x from Pardoux and Vereennikov (23). I will be convenien for u o work wih funcion ha are mooher in heir x-componen han hey are in heir z-componen or vice vera. o do o, inroduce he funcion pace C k,l (R m R n, R d ): For θ : R m R n R d, θ = θ(x, z), wrie θ C k,l (R m R n, R d ), if θ i k ime coninuouly differeniable in i x-componen and l ime coninuouly differeniable in i z-componen. If θ a well a i parial derivaive up o order (k, l) are bounded, wrie θ C k,l b (Rm R n, R d ). Inroduce he following aumpion: (H a ) For he exience of a aionary diribuion µ(x, dz) for Z x, we uppoe ha here exi M >, α >, uch ha for all z M up f(x, z), z C z α. x For he uniquene of he aionary diribuion µ(x, dz) of Z x, we uppoe uniform ellipiciy, i.e. ha here are < λ Λ <, uch ha λi gg (x, y) ΛI in he ene of poiive emi-definie marice (I i he uni marix). (HF k,l ) he coefficien of he fa diffuion aify f C k,l b (Rm R n, R n ) and g C k,l b (Rm R n, R n k ). (HS k,l ) he coefficien of he low diffuion aify b C k,l b (Rm R n, R m ) and σ C k,l b (Rm R n, R m k ). (HO k,l ) he obervaion funcion h aifie h C k,l b (Rm R n, R d ).

8 8 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG We will uually wrie p (x, dz) inead of µ(x, dz). Alo inroduce he noaion p (z, θ; x) := θ(x, z )p (z, z ; x)dz := z θ(z x ) R n where z denoe he aring poin of Z x, and z p (z, z ; x) i he deniy of Z x if a ime i i ared in z. Noe ha he deniy exi for all > under he condiion (H a ), becaue of he uniform ellipiciy of gg. Similarly p (θ; x) = θ(x, z)p (x, dz). R n Le he differenial operaor L be defined a L = bi (x) + 1 x i 2 2 ā ij (x, z) x i x j where b(x) = p (b; x) and ā = p (σσ ; x). Alo define h(x) = p (h; x). We inroduce he following noaion: A muliindex α = (α 1,..., α m ) N n α = α α m. Given uch a muliindex, define he differenial operaor α D α = x α xαm m Finally inroduce he following norm for f Cb k(rm, R n ): f k, = D α f where i he uual upremum norm. Our main reul i α k i of order heorem 3.1. Aume (H a ), (HF 8,4 ), (HS 7,4 ), (HO 8,4 ), and ha he iniial diribuion Q (X,Z ) ha finie momen of every order. hen for every p 1 and here exi C >, uch ha for every ϕ Cb 4 ( Q π,x (ϕ) π (ϕ) p ) 1/p C ϕ 4,. In paricular, here exi a meric d on he pace of probabiliy meaure, uch ha d generae he opology of weak convergence, and uch ha for every here exi C >, uch ha Q d(π,x, π ) C. hi reul will be proven in Secion 6. In paricular we can ue Borel-Canelli o conclude ha if ( n ) converge quickly enough o, hen π n will a.. converge weakly o π. he idea are raher imple: We repreen he backward SPD by finie-dimenional ochaic equaion (hi will be BDSD). he diffuion operaor ge replaced by he aociaed diffuion. We are able o olve hoe finie-dimenional equaion explicily, or a lea give explici eimae up o an applicaion of Gronwall. hi allow u o eimae ψ 1 and R in erm of he raniion funcion of he fa diffuion. Bu Pardoux and Vereennikov (23) proved very precie eimae for hi raniion funcion. hee eimae allow u o obain he convergence. While he idea are imple, he precie formulaion and he acual proof are quie echnical. We ar by decribing he probabiliic repreenaion.

9 (9) DIMNSIONAL RDUCION IN NONLINAR FILRING 9 4. Probabiliic repreenaion of SPD In hi ecion, we derive probabiliic repreenaion for SPD of he form dψ(ω,, x) = Lψ(ω,, x)d + f(ω,, x)d ψ(, x) = ϕ(ω, x), + (g(ω,, x) + G(ω,, x)ψ(ω,, x))d B, where ψ : Ω, R m R, f : Ω, R m R, g : Ω, R m R 1 d, and G : Ω, R m R 1 d, ϕ : Ω R m R are all joinly meaurable, and (B :, ) i a d-dimenional andard Brownian moion under he meaure P. quaion (9) repreen he general form of he equaion (7) and (8) for he correcor ψ 1 (x, z) and error R (x, z), repecively. he differenial operaor L i given by L = b i (x) x i a ij (x) x i x j for meaurable b : R m R m and a : R m S m m (S m m denoe poiive emidefinie ymmeric marice). We will repreen hee equaion in erm of BDSD a inroduced by Pardoux and Peng (1994). Noe ha for hee linear equaion i i poible o give a Feynman-Kac ype repreenaion wihou uing BDSD. hi i done, for example, in Rozovkii (199) ( he Mehod of Sochaic Characeriic ). However he BDSD-repreenaion ha he advanage ha i permi u o apply Gronwall lemma. hi would no be poible wih he mehod of ochaic characeriic. A BDSD i an inegral equaion of he form Y = ξ + f(, Y, Z )d + g(, Y, Z )db Z dw where B and W are independen Brownian moion. he oluion (Y, Z ) will be F, B F W - meaurable. Saring from he noion of BDSD, we can define forward-backward doubly ochaic differenial equaion. Le σ = a 1/2 and X,x = x + b(x,x )d + X,x = x for We hen define he following BDSD dy,x Y,x = f(, X,x = ϕ(x,x ) )d + (g(, X,x )d + G(, X,x σ(x,x )dw for )Y,x )db Z,x dw I urn ou ha Y give a finie-dimenional probabiliic repreenaion for equaion (9), more preciely we have Y,x = ψ(, x). hi i no compleely covered by Pardoux and Peng (1994), becaue we have random unbounded coefficien, and becaue we do no aume he diffuion marix a o have a mooh quare roo. On he oher ide, he equaion i of a paricularly imple linear ype. In he remainder of hi ecion, we give he precie aemen and proof for hi repreenaion. hi can be kipped a fir reading. We will no be able o ge an exience reul for claical oluion of he above SPD from he heory of BDSD: hi i due o he fac ha for hi we would need moohne properie of a quare roo of a. Bu even when a i mooh, in he degenerae ellipic cae i doe no need o have a mooh quare roo (ee, for example, Sroock (28), Chaper 2.3). Inead we will ue he exience reul of Rozovkii (199) and only reprove he uniquene reul of Pardoux and Peng (1994) in our eing. hi will work under Lipchiz coninuiy of a 1/2.

10 1 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG Define for F,B, = σ(b u B : u ) and F, B a he compleion of F,B, polynomial growh: under P. Inroduce he pace of adaped random field of Definiion. P (R m, R n ) i he pace of random field H : Ω, R m R n ha are joinly meaurable in (ω,, x), and for fixed (, x), ω H(ω,, x) i F, B -meaurable. Furher for fixed ω ouide a null e, H ha o be joinly coninuou in (, x), and i ha o aify he following inequaliy: For every p 1 here i C p >, q >, uch ha for all x R m up H(, x) p C p (1 + x q ) We make he following aumpion on he coefficien of he SPD: (S k ) f and g are k ime coninuouly differeniable and he parial derivaive up o order k are all in P. G i (k + 1) ime coninuouly differeniable and he parial derivaive up o order (k + 1) are all uniformly bounded in (ω,, x). ϕ i k ime coninuouly differeniable, and all parial derivae of order o k grow a mo polynomially. We make he following aumpion on he coefficien of he differenial operaor L: (D k ) b C k b (Rm, R m ), a C k b (Rm, S m m ), and a i degenerae ellipic: For every ξ R m and every x R m, hen we have he following reul: a(x)ξ, ξ = a ij (x)ξ i ξ j. Propoiion 4.1. Aume (S k ) and (D k ) for ome k 3. hen he equaion (9) ha a unique claical oluion ψ in he ene ha for every fixed ω ouide a null e, ψ(ω,, ) C,k 1 (, R d, R), ψ and i parial derivaive are in P (R m, R), and ψ olve he inegral equaion. If ψ i any oher oluion of he inegral equaion, hen ψ and ψ are indiinguihable. If furher f, g and ϕ a well a heir derivaive up o order k are uniformly bounded in (ω,, x), hen for any p > here exi C p, q > (only depending on p, he dimenion involved, he bound on a, b and G, and on ), uch ha for all α k 1 and x R m : up D α ψ(, x) p C(1 + x q ) ϕ p k, + up f(, ) p k, + up g(, ) p k, Proof. hi i a combinaion of heorem and Corollary of Rozovkii (199) (he claimed bound i only given for he equaion in unweighed Sobolev pace, in Corollary Bu from ha we can deduce he reul for he weighed Sobolev cae). he only hing we need o verify i ha our polynomial growh aumpion on he coefficien i compaible wih he Sobolev norm condiion here. Bu if θ P (R m, R n ), hen for any p 1 here cerainly i an.

11 DIMNSIONAL RDUCION IN NONLINAR FILRING 11 r < uch ha θ ake i value in he weighed L p -pace wih weigh (1 + x 2 ) r/2 : θ(, x) p (1 + x 2 ) r 2 dx up θ(, x) p (1 + x 2 ) r 2 dx up = up θ(, x) p (1 + x 2 ) r 2 dx C p (1 + x q )(1 + x 2 ) r 2 dx < for mall enough r. Now we combine hi reul wih he heory of BDSD: Le (W :, ) be an n-dimenional andard Brownian moion ha i independen of B. For, F W, i defined analogouly o F B,. For we e F = F B, F W. Noe ha hi i no a filraion, a i i neiher decreaing nor increaing in. Inroduce he following noaion: H 2 (Rm ) i he pace of meaurable R m -valued procee Y.. Y i F -meaurable and Y 2 d <. S 2 (Rm ) i he pace of coninuou adaped R m -valued procee Y.. Y F and A BDSD i an inegral equaion of he form (1) Y = ξ + f(,, Y, Z )d + up Y 2 <. g(,, Y, Z )db Z dw, where f :, Ω R R 1 n R, g :, Ω R R 1 n R 1 l, and for fixed y R, z R 1 n he procee (ω, ) f(, ω, x, z) and (ω, ) g(, ω, x, z) are (F B, F W ) B(R)-meaurable, and for every, f(,, x, z) and g(,, x, z) are F -meaurable. (Y, Z) will be called oluion of (1) if (Y, Z) S 2 (R) H2 (R1 n ) and if he couple olve he inegral equaion. We will alo wrie he equaion in differenial form: dy = f(, Y, Z )d + g(, Y, Z )d B Z dw. Oberve ha wih uiable adapaion, all of he following reul alo hold in he mulidimenional cae, i.e. for Y R m. We reric o one-dimenional Y for impliciy and becaue ulimaely we are only inereed in ha cae. Pardoux and Peng (1994) how ha under he following condiion, equaion (1) ha a unique oluion: ξ L 2 (Ω, F, P; R) for any (y, z) R R 1 n : f(,, y, z) H 2 (R) and g(,, y, z) H2 (R1 k ) f and g aify Lipchiz condiion and g i a conracion in z: here exi conan L > and < α < 1.. for any (ω, ) and y 1, y 2, z 1, z 2 : f(, ω, y 1, z 1 ) f(, ω, y 2, z 2 ) 2 L( y 1 y z 1 z 2 2 ) g(, ω, y 1, z 1 ) g(, ω, y 2, z 2 ) 2 L y 1 y α z 1 z 2 2. and

12 12 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG Now we wan o aociae a diffuion X o he differenial operaor L. o do o, aume ha (D k ) i aified for ome k 2. hen σ := a 1/2 i Lipchiz coninuou by Lemma of Sroock (28). Hence for every (, x), R m, here exi a rong oluion of he SD X,x = x + b(x,x )d + X,x = x for. Aociae he following BDSD o (9): (11) dy,x Y,x = f(, X,x = ϕ(x,x ). )d + (g(, X,x ) + G(, X,x σ(x,x )dw for, )Y,x )db Z,x dw, Under he aumpion (S k ) and (D k ) for k 2, hi equaion ha a unique oluion. Propoiion 4.2. Aume (S k ) and (D k ) for ome k 3. hen he unique claical oluion ψ of he SPD (9) i given by ψ(, x) = Y,x, where (Y,x, Z,x ) i he unique oluion of he BDSD (11). We can give exacly he ame proof a in Pardoux and Peng (1994), heorem 3.1, aking advanage of he independence of B and W. For he reader convenience, we include i here. Proof. Le ψ be a claical oluion of (9). I uffice o how ha (ψ(, X,x ), Dψ(, X,x )σ(x,x ) : ) olve he BDSD (11). Here Dψ i he gradien of ψ. For hi purpoe, conider a pariion = < 1 < < n = of,. hen and ψ(, X,x n 1 ) = ψ(, X,x ) + (ψ( i, X,x ) ψ( i+1, X,x i+1 )) = ϕ(x,x i= i n 1 ) + (ψ( i, X,x ) ψ( i+1, X,x i+1 )) i= ψ( i,x,x i ) ψ( i+1, X,x i+1 ) = (ψ( i, X,x ) ψ( i, X,x i+1 )) + (ψ( i, X,x i+1 ) ψ( i+1, X,x = i ( i i i+1 i i+1 i Lψ( i, X,x )d + i i+1 (Lψ(, X,x i+1 ) + f(, X,x i+1 ))d i i+1 )) ) Dψ( i, X,x )σ(x,x )dw (g(, X,x i+1 ) + G(X,x i+1 )ψ(, X,x i+1 ))d B. hi i juified becaue X,x and ψ are independen and becaue ψ grow polynomially, hence we can apply Iô formula. We alo ued he fac ha ψ i a claical oluion o (9). If we le he meh ize end o, hen by coninuiy of X,x and ψ, he reul follow. 5. Preliminary eimae he noaion D α x indicae ha he differenial operaor D α i only acing on he x-variable. he following reul will help u o juify he BDSD-repreenaion on he deeper level. Recall ha p (z, θ; x) = θ(x, Z x ) Z x = z.

13 DIMNSIONAL RDUCION IN NONLINAR FILRING 13 Propoiion 5.1. Aume (HF k,l ). Le θ C k,l (R m R n, R) aify for ome C, p > Dx α Dz β θ(x, z) C(1 + x p + z p ). hen α k β l (, x, z) p (z, θ; x) C,k,l (R + R m R n, R) and here exi C 1, p 1 >, uch ha for all (, x, z), ) R m R n Dx α Dz β p (z, θ; x) C 1 e C1 (1 + x p 1 + z p 1 ). α k β l If he bound on he derivaive of θ can be choen uniformly in x, i.e. up Dx α Dz β θ(x, z) C(1 + z p ), x α k β l hen he bound on he derivaive of p (z, θ; x) i alo uniform in x: up Dx α Dz β p (z, θ; x) C 1 e C1 (1 + z p 1 ). x Proof. Noe ha α k β l p (z, θ; x) = θ(x, Z x ) Z x = z = (θ(x, Z ) (X, Z ) = (x, z) i he oluion of Kolmogorov backward equaion aociaed o (X, Z), where X = X, Z = Z + f(x, Z )d + g(x, Z )dw. In hi formulaion, he fir reul i andard. Cf. e.g. Sroock (28), Corollary he econd aemen can be proven in he ame way a Sroock (28), Corollary Some reul from Pardoux and Vereennikov (23) are colleced in he following Propoiion: Propoiion 5.2. Aume (H a ) and (HF k,3 ). Le θ C k, (R m R n, R) aify for ome C, p > : up Dx α θ(x, z) C(1 + z p ). x hen α k (1) x p (θ; x) C k b (Rm, R). (2) Aume addiionally ha θ aifie he cenering condiion R n θ(x, z)p (x, dz) = for all x, and ha θ C k,1 (R m R n, R) and up Dz β Dx α θ(x, z) C(1 + z p ). x hen α k β 1 (x, z) p (z, θ; x)d C k,1 (R m R n, R),

14 14 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG and for every q > here exi C 1, q 1 >, uch ha for every z R n up Dz β Dx α p (z, θ; x) q d C 1 (1 + z q 1 ). x α k β 1 Proof. he aemen in he Propoiion are aken from heorem 1, heorem 2 and Propoiion 1 of Pardoux and Vereennikov (23): (1) We ge from heorem 1 of Pardoux and Vereennikov (23), ha for any q > here exi C q >, uch ha for any (x, z, z ) R m R n R n : α k up x D x α p (z ; x) C q 1 + z q. So if we chooe q large enough and differeniae p (θ; x) under he inegral ign, hen we obain he fir claim. (Of coure here we have o ue he growh conrain on θ and i derivaive). (2) hi follow from he bound on he derivaive of p (z, θ; x) ha are given in Pardoux and Vereennikov (23), heorem 2, formulae (14) and (15): For any k > here exi C k, m k >, uch ha for any (, x, z) 1, ) R m R n Dz β Dx α 1 + z m k p (z, θ; x) C k (1 + ) k. α k β 1 We combine hi eimae wih Propoiion 5.1, from where we obain for (, x, z) R + R m R n up Dx α Dz β p (z, θ; x) C 1 e C1 (1 + z p 1 ). x α k β l We chooe k uch ha qk > 1 and ue he fir eimae on 1, ) and he econd eimae on, 1). he reul follow. We will alo need ome momen bound for he diffuion X and Z. Propoiion 5.3. Aume (H a ) and ha he coefficien b and σ and f and g of he fa and low moion are bounded and globally Lipchiz coninuou. hen for any p 1 here exi C p >, uch ha up Z p (X, Z) = (x, z) C p (1 + z p ). (,,x), ),1 R m Alo, for every > and every p 1 here exi C(p, ), q >, uch ha up X p (X, Z) = (x, z) C(p, )(1 + x p ). (,),,1 Proof. he fir claim can be proven exacly a in Vereennikov (1997): Fir wrie Z := Z 2. hen d Z = f(x 2, Z )d + g(x 2, Z )d W where W := 1/W 2 i a Wiener proce. Nex, inroduce he ame ime change a in Pardoux and Vereennikov, page 163: κ(x, z) := g(x, z) z / z, γ () := Define Z := Z τ (). hen, κ 2 (X 2, Z )d, τ () := (γ ) 1 (). d Z = κ 2 (X 2, Z )f(x 2, Z )d + κ 1 (X 2, Z )g(x 2, Z )d W

15 DIMNSIONAL RDUCION IN NONLINAR FILRING 15 wih a new andard Brownian moion W. Now we are in a poiion o ju copy he proof of Lemma 1 in Vereennikov (1997) (which we do no do here) o ge he fir reul. he econd claim i obviou, becaue he coefficien of X are bounded. Now we we are able o impoe condiion on he coefficien of he diffuion ha guaranee moohne of he coefficien of L. Recall ha L wa defined a L = bi (x) ā ij (x, z) x i 2 x i x j where b = p (b; x) and ā = p (σσ ; x). Propoiion 5.4. Aume (HF k,3 ), (HS k, ), and (HO k, ). hen b C k b (R m, R m ), ā C k b (Rm, S m m ), h C k b (Rm, R k ) Proof. All he erm of b, ā and h are of he form p (θ; x). So by Propoiion 5.2, we only need o verify ha he repecive θ are in C k, and aify he polynomial bound up Dx α θ(x, z) C(1 + z p ) x α k for ome C, p >. Bu we even aumed hem o be in C k, b, o he reul follow. 6. Proof of he main reul We will find convergence rae for he correcor and remainder erm ha are expreed in erm of v and i derivaive. So now we give bound on v and i derivaive in erm of he e funcion ϕ. hi i neceary, becaue we do no only wan o how convergence of he filer inegraing fixed e funcion, bu wih repec o a uiable diance on he pace of probabiliy meaure. Lemma 6.1. Le k 2 and aume b, ā, ϕ C k+1 b, and h C k+2 b. hen v C,k (, R m, R), and for any p 1 here exi C p, q >, independen of ϕ, uch ha for all x R m : up D α v (x) p C p (1 + x q ) ϕ p k,. α k In paricular, v and all i parial derivaive up o order (, k) are in P (R m, R). Proof. hi i a imple applicaion of Propoiion 4.1, noing ha he equaion (6) for v i of he ype (9) wih f =, g =, and G = h. We will prove L p -convergence of ψ 1 and R eparaely: Lemma 6.2. Le k, l 2. Aume (H a ), (HF k+1,l+1 ), (HS k+1,l+1 ), and (HO k+1,l+1 ). Alo aume v C,k+1 (, R m, R), and ha all i parial derivaive in x up o order k + 1 are in P (R m, R). Finally aume ā, b, h Cb k. hen ψ1 C,k,l (, R m R n, R), and ψ 1 a well a i parial derivaive up o order (, k, l) are in P (R m R n, R). For any p 1 here exi C p, q >, independen of ϕ, uch ha for any (x, z) R m+n and any (, 1) up Dx α ψ 1 (x, z) p α k 1 p 2 Cp (1 + z q ) α k+1 up D α x v (x) p.

16 16 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG Proof. ψ 1 (x, z) olve he BSPD (12) dψ 1 (x, z) = ψ 1 (x, z) =. 1 L F ψ 1 (x, z) + (L S L)v (x) d + h(x, z) h(x) v (x)d B, xience of he oluion ψ 1 and i derivaive a well a he polynomial growh all follow from Propoiion 4.1. Wrie Z,x,(,z) for he oluion of he SD dz,x,(,z) = 1 Z,x,(,z) = z,. f(x, Z,x,(,z) )d + 1 g(x, Z,x,(,z) )dw, We conider (x, Z,x,(,z) ) a a join diffuion, ju a in he proof of Propoiion 5.1 (x ha generaor ). By Propoiion 4.2, he oluion of (12) i given by θ (,x,z)(1), he unique oluion o he BDSD dθ (,x,z)(1) θ (,x,z)(1) =. We will drop upercrip (, x, z) for θ (,x,z)(1) inead of Z,x,(,z) θ 1 F B, = (L S (, Z,x,(,z) ) L)v (x)d ( + h(x, Z,x,(,z) ) h(x) ) v (x)db + γ,x,z dw, and wrie θ 1 inead. Similarly, we wrie Z,x. ψ 1 (x, z) i F, B -meaurable, hence, o i θ1. We can hen wrie θ 1 =, where θ 1 F, B = (L S L)v (x)d F, B + h(x, Z,x ) h(x) v (x)db F, B γ,x,z dw F, B. W and B are independen, herefore W i a Brownian moion in he large filraion (F W F, B :, ), hence γ,x,z dw F W F, B =, and by he ower propery γ,x,z dw F, B =. v i F B, -meaurable and L ha deerminiic coefficien. hu = = Lv (x)d F B, Lv (x) F, B d p (b i ; x) x i v (x) + p ((σσ 2 ) ij ; x) v x i x (x) j d.

17 Since Z,x i independen of B, DIMNSIONAL RDUCION IN NONLINAR FILRING 17 = = L S (, Z,x { m { m )v(x)d F, B = b i (x, Z,x ) x i v (x) (σσ ) ij (x, Z,x ) p (z, b i ; x) v x (x) i p L S (, Z,x )v (x) F B, d 2 v x i x (x) j d (z, (σσ 2 ) ij ; x) v x i x (x) j d, o (L S L)v (x)d F, B { m = p (z, b i p (b i ; x); x) v x (x) i + 1 p (z, (σσ ) ij p ((σσ 2 ) ij ; x); x) v 2 x i x (x) j d (he p (.; x) erm have been brough inide he inegral p (z, ; x) ince hey no depend on z) p u (z, b i p (b i ; x); x) v x u+(x)du i + 2 p u (z, (σσ ) ij p ((σσ 2 ) ij ; x); x) v x i x u+(x)du j p u (z, b i p (b i ; x); x) du up v x (x) i + p u (z, (σσ ) ij p ((σσ ) ij ; x); x) du up 2 2 v x i x (x) j (f p (f; x) i cenered, o by Propoiion 5.2, (2):) C 1 (1 + z q 1 ) up v x (x) m i + up 2 v x i x (x) j

18 18 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG and herefore finally p (13) (L S L)v (x)d F, B p C 2 (1 + z q 2 ) v x (x) i up p + up Nex, uing again v F, B and ha Z,x i independen of B, h(x, Z,x ) h(x) v (x)db F, B = = h(x, Z,x ) h(x) v (x) F B, d B p (z, h h; x) v(x)d B. For r, r r p ime i run backward. Hence by he Burkholder-Davi-Gundy inequaliy, where 2 p v x i x (x) j. (z, h h; x) v (x)d B, i a maringale w.r.. (F B r, p (z, h h; x) v(x)d B p C p p (z, h h; x) v(x)d B = p (z, h h; x) v(x)d B p 2 p, (z, h h; x) v(x) 2 d p u (z, h h; x) 2 du up C 3 (1 + z q 3 ) up v (x) 2, v (x) 2 where he la inequaliy i by Propoiion 5.2, (2), ince h h i cenered. herefore, p (z, h h; x) v(x)d p B p 2 C4 (1 + z q 4 ) up v (x) (14) p. Combining (13) and (14), θ 1 p p C 4 (1 + z q 4 ) α 2 Nex, conider a fir order x-derivaive of θ 1 : x k θ 1 = x k + x k L S L v (x)d up Dx α v(x) p. h(x, Z,x ) h(x) v (x)d B. : r, ) if A before, he forward Ió inegral erm vanihed afer aking he (condiional) expecaion.

19 DIMNSIONAL RDUCION IN NONLINAR FILRING 19 Inerchanging order of differeniaion and inegraion, L S x L v (x)d k { p u (z, b i p (b i ; x); x) v x k x u+(x) i 2 } +p u (z, b i p (b i ; x); x) v x k x u+(x) du i + { p 2 u (z, (σσ ) ij p ((σσ 2 ) ij ; x); x) v x k x i x u+(x) j +p u (z, (σσ ) ij p ((σσ 3 } ) ij ; x); x) v x i x j x u+(x) du k { p u (z, b i p (b i ; x); x) x k du up v x (x) i + p u (z, b i p (b i ; x); x) du up 2 } v x k x (x) i { + 2 p u (z, (σσ ) ij p ((σσ ) ij ; x); x) x k du up 2 v x i x (x) j + p u (z, (σσ ) ij p ((σσ ) ij ; x); x) du up 3 } v x i x j x (x) k. hen, from Propoiion 5.2, (2) again, x k L S L v(x)d C 5(1 + z q 5 ) up 1 β 3 Dxv β (x). ince he quaniie b b and σσ σσ are cenered. aking expecaion, (15) x k L S L v(x)d p p C 6 (1 + z q 6 ) 1 β 3 up Dxv β (x) p.

20 2 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG Nex, by (HO k,l ), we can inerchange he order of ordinary differeniaion and ochaic inegraion (cf. Karandikar (1983)): ( h(x, Z,x ) x h(x) ) v (x)d p B k ( = h(x, Z,x ) x h(x) v (x) ) db p k ( C p ( h(x, Z,x ) x h(x) v (x) ) ) 2 p/2 d, k where ( h(x, Z,x ) x h(x) v (x) ) 2 d k = p u (z, h x h; x)vu+(x) + p u (z, h h; x) v k x u+(x) k { 2 p u (z, h x h; x) 2 vu+(x) 2 du k + pu (z, h h; x) } 2 v 2 x u+(x) k du { C 7 (1 + z q 7 ) up v (x) 2 + up v 2} x (x). k he la ep follow once again from Propoiion 5.2, (2). So, ( h(x, Z,x ) x h(x) ) v (x)d p (16) B k { p 2 C8 (1 + z q 8 ) up v (x) p + Combining (15) and (16) θ 1 p x k p 2 C9 (1 + z q 9 ) α 3 up up Ieraing hee argumen for he higher order derivaive of θ 1, D x α θ 1 p p 2 C1 (1 + z q 1 ) α k 1 α k+1 2 v p} x (x). k D x α v(x) p. up du D x α v(x) p. Lemma 6.3. Le k, l 3. Aume (HF k,l ), (HS k,l ), and (HO k+1,l+1 ). Alo aume ψ 1 C,k+2,l (, R m R n, R) and ha all i parial derivaive up o order (, k + 2, l) are in P (, R m, R). hen for any p 1 here exi C p >, independen of ϕ, uch ha for any (x, z) R m+n, any (, 1), and any, R (x, z) p C p D x α ψ(x 1, z ) p d. α 2 (x,z )=(X,(,x),Z,(,z) )

21 Proof. R (x, z) olve he BSPD DIMNSIONAL RDUCION IN NONLINAR FILRING 21 (17) dr (x, z) = ( L R (x, z) + L S ψ 1 (x, z) ) d R (x, z) =. + h(x, z) ( ψ 1 (x, z) + R (x, z) ) d B, xience of he oluion R and i derivaive, a well a he polynomial growh all follow from Propoiion 4.1. By Propoiion 4.2, he oluion of (17) i given by θ (,x,z)(2), he oluion o he BDSD dθ (,x,z)(2) θ (,x,z)(2) =. = L S ψ 1 (X,(,x) + h(x,(,x) + h(x,(,x), Z,(,z) )d, Z,(,z), Z,(,z) ) ψ 1 (X,(,x) ) θ (,x,z)(2), Z,(,z) )db d B γ,x,z dw δ,x,z dv We will drop upercrip (, x, z) for θ (,x,z)(2), (, z) for Z,(,z), and (, x) for X,(,x). R (x, z) i F B, -meaurable, hence, o i θ2. A before, he ochaic inegral over dv and dw vanih when we ake condiional expecaion wih repec o F B,. hu (18) θ 2 = L S ψ(x 1, Z)d F, B + h(x, Z) ψ(x 1, Z)d B F, B + h(x, Z) θd 2 B F, B. Conider each erm eparaely: L S ψ(x 1, Z)d p p F, B L S ψ(x 1, Z)d ( m ( ) p 1 b i (X, Z) x i + 1 p (σσ ) ij (X 2, Z) 2 ψ 1 x i x (X, Z) i d ( m C 1 b ψ 1 x (X, Z) p i C σσ m 1 α 2 2 p ψ 1 x i x (X, Z) d i D α x ψ 1 (X, Z ) p d.

22 22 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG Noe ha Z and X are F W independen. hu o ha (19) F V -meaurable, ψ 1 i F, B -meaurable, and B and (V, W ) are D x α ψ(x 1, Z) p = Dx α ψ(x 1, Z) p F V = D α x ψ(x 1, z ) p L S ψ(x 1, Z)d p F, B C 2 1 α 2 F W (x,z )=(X,Z ), D x α ψ(x 1, z ) p (x z )=(X,Z ) d. Nex, by Jenen inequaliy, he ower propery, and he Burkholder-Davi-Gundy inequaliy, h(x, Z) ψ(x 1, Z)d B p F, B C p h(x, Z ) ψ 1 (X, Z )d B p h(x, Z ) ψ 1 (X, Z )d B p 2 where by Hölder inequaliy and he Cauchy-Schwarz inequaliy h(x, Z,x ) ψ 1 (X, Z )d B p 2 = ( h(x, Z,x, ) ψ 1 (X, Z ) 2 d C 3 h(x, Z) p ψ(x 1, Z) p d. So by he ame argumen a for he fir erm, h(x, Z) ψ(x 1, Z)d B p (2) F, B C 4 ψ(x 1, z ) p d. (x,z )=(X,Z ) Finally, uing Burkholder-Davi-Gundy in he econd line, and Cauchy-Schwarz in he hird line h(x, Z) θd 2 B p F, B h(x, Z) θd 2 B p ( ) p C p h(x, Z) θ d (21) C p ( C 5 h p ) p h(x, Z) 2 θ d θ 2 p d. ) p 2

23 DIMNSIONAL RDUCION IN NONLINAR FILRING 23 Combining (18) wih (19), (2), and (21) By Gronwall, θ 2 p C6 θ 2 p C 6 α 2 C 7 α 2 α 2 + C 5 h p D α x ψ(x 1, z ) p (x,z )=(X,Z ) d θ 2 p d. D x α ψ(x 1, z ) p (x,z )=(X,Z ) D x α ψ(x 1, z ) p (x,z )=(X,Z ) d e ( )C 5 h p d. where we chooe C 7 o ha he inequaliy hold for every, (replace e ( )C 5 h e C 5 h ). by Now we can collec all hee reul, o obain he fir ep oward heorem 3.1. Lemma 6.4. Aume (H a ), (HF 8,4 ), (HS 7,4 ), (HO 8,4 ), and ha ϕ C 7 b (Rm, R). hen for every p 1 here exi C, q 1, q 2 >, independen of ϕ, uch ha up v (x, z) v (x) p p/2 C (1 + x q 1 + z q 2 ) ϕ p 4,. Proof of heorem 3.1. We rack he neceary condiion backward from Lemma For he oluion R given in Lemma 6.3 o exi and aify he aed bound, we need (HF 3,3 ), (HS 3,3 ), (HO 4,4 ), and ψ 1 C,5,3 (, R m R n, R). he polynomial growh condiion will be aified anyway. 2. For ψ 1 o be in C,5,3 (, R m R n, R), we need (H a ), (HF 6,4 ), (HS 6,4 ), (HO 6,4 ) and ā, b, h Cb 5. We alo need v C,6 (, R m, R). Again, he polynomial growh condiion will be aified. 3. For v o be in C,6 (, R m, R) we need ā, b, ϕ Cb 7 and h Cb For ā, b o be in Cb 7 we need (HF 7,3) a well a (HS 7, ) by Propoiion 5.4. Similarly we need (HF 8,3 ) a well a (HO 8, ) for h o be in Cb So ufficien condiion are (H a ), (HF 8,4 ), (HS 7,4 ), (HO 8,4 ). In ha cae we obain from Lemma 6.1 (22) α 2 α 4 up D α v (x) p C 1 (1 + x q 1 ) ϕ p 4,. From Lemma 6.2 we obain up Dx α ψ 1 (x, z) p p 2 C2 (1 + z q 2 ) (23) From Lemma 6.3 we ge (24) R (x, z) p C 3 α 2 α 4 D x α ψ(x 1, z ) p up D x α v (x) p. (x,z )=(X,(,x),Z,(,z) ) d.

24 24 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG Combining (22), (24), (24), we ge for any, (by ime-homogeneiy of X and Z ) R (x, z) p + ψ 1 (x, z) p ) (25) p/2 C 4 (1 + up From Propoiion 5.3 we obain X q 1 + Z,x q 2 (X, Z) = (x, z) ϕ p 4,. up X q 1 + Z,x q 2 (X, Z) = (x, z) C 5 (1 + x q 3 + z q 4 ). Noing ha he righ hand ide in (25) doe no depend on,, up R (x, z) p + up ψ 1 (x, z) p Finally which complee he proof. p/2 C 6 (1 + x q 3 + z q 4 ) ϕ p 4,. up v (x, z) v (x) p ( C 7 up R (x, z) p + up ψ 1 (x, z) p) p/2 C 8 (1 + x q 3 + z q 4 ) ϕ p 4,, Now we recall ha all he calculaion up unil now were under he changed meaure P. We only wroe P and B o faciliae he reading. So le u ranfer he reul o he original meaure Q. Lemma 6.5. Aume (H a ), (HF 8,4 ), (HS 7,4 ), (HO 8,4 ), and ha ϕ C 7 b (Rm, R). hen for every p 1 here exi C, q 1, q 2 >, independen of ϕ, uch ha up Q v (x, z) v (x) p p/2 C (1 + x q 1 + z q 2 ) ϕ p 4,. Proof. hi i a imple applicaion of he Cauchy-Schwarz inequaliy in combinaion wih Gronwall lemma: Q v (x, z) v (x) p = P v (x, z) v (x) p dq dp ( ) dq 2 1/2 P v (x, z) v (x) 2p 1/2 P dp, o we ee ha he reul i rue by Lemma 6.4 a long a he econd expecaion i finie. Recall ha we had defined he noaion ( dq dp = D = exp h(x, Z) dy 1 ) h(x F 2, Z) 2 d. So D aifie he SD d D = D h(x, Z ) dy, D = 1. Since under P, Y i a Brownian moion, we ge by Iô-iomery P ( D ) 2 = P ( D ) 2 h(x, Z) 2 d h 2 P ( D ) 2 d, o ha by Gronwall P ( D )2 <

25 DIMNSIONAL RDUCION IN NONLINAR FILRING 25 Lemma 6.6. Aume (H a ), (HF 8,4 ), (HS 7,4 ), (HO 8,4 ), ha ϕ Cb 7, and ha he iniial diribuion Q (X,Z ) ha finie momen of every order. hen for every p 1 here exi C >, independen of ϕ, uch ha Q ρ,x (ϕ) ρ (ϕ) p p/2 C ϕ p 4,. Proof. A we already decribed in he inroducion, we obain from Lemma 6.5 Q ρ,x (ϕ) ρ (ϕ) p = Q (v(x, z) v(x))q p (X,Z ) (dx, dz) Q v(x, z) v(x) p Q (X,Z ) (dx, dz) p/2 C 1 (1 + x q 1 + z q 2 ) Q (X,Z ) (dx, dz) ϕ p 4, p/2 C 2 ϕ p 4,. he convergence of he acual filer, i.e. of π,x o π, now follow exacly a in Chaper 9.4 of Bain and Crian (29). For he ake of compleene, we include he argumen. Lemma 6.7. Le p 1. hen a long a h i bounded. up (,1,, { Q ρ,x (1) p + Q ρ (1) p } < Proof. We give he argumen for Q ρ,x (1) p, Q ρ (1) p being compleely analogue. We have Q ρ,x (1) p = P ρ,x p dq (1) P dp ρ,x (1) 2p 1/2 P ( ) dq 2 1/2 dp We howed in he proof of Lemma 6.5 ha he econd expecaion i finie. Noe ha x x 2p i convex. herefore by Jenen inequaliy, P ρ,x (1) 2p ( = P P exp h(x, Z) dy 1 ) 2p h(x 2, Z) 2 d Y ( P exp h(x, Z) dy 1 ) 2p h(x 2, Z) 2 d dq 2p P dp 2p+1 dp = Q. dq he reul now follow exacly a in he proof of Lemma 6.5, becaue for D = dp /dq F have dd = h(x, Z ) db, D = 1 and B i a Brownian moion under Q. we

26 26 P. IMKLLR, N.S. NAMACHCHIVAYA, N. PRKOWSKI, AND H.C. YONG Define for any meaurable and bounded e funcion ϕ : R m R Recall ha π,x π (ϕ) = ρ (ϕ) ρ (1). wa defined analogouly wih ρ,x inead of ρ. We hen have Lemma 6.8. Aume (H a ), (HF 8,4 ), (HS 7,4 ), (HO 8,4 ), and ha he iniial diribuion Q (X,Z ) ha finie momen of every order. Le p 1. hen here exi C > uch ha for every ϕ Cb 7 Q π,x (ϕ) π (ϕ) p p/2 C ϕ p 4,. Proof. In he hird line we ue ha π,x i a.. equal o a probabiliy meaure. Q π,x (ϕ) π (ϕ) p ρ,x = (ϕ) Q ρ,x (1) ρ (ϕ) p ρ (1) ρ,x = (ϕ) ρ (ϕ) Q ρ (1) π,x (ϕ)ρ,x (1) ρ (1) p ρ (1) ( ρ,x C p (ϕ) ρ (ϕ) p Q ρ (1) + ϕ p ρ,x (1) ρ (1) p) Q ρ (1) ( C p Q ρ (1) 2p ) ( 1/2 Q ρ,x (ϕ) ρ (ϕ) 2p 1/2 p/2 C 1 ϕ 4,, where he la ep follow from Lemma 6.6 and Lemma ϕ p Q ρ,x (1) ρ (1) ) 2p 1/2 Since he bound only depend on ϕ 4,, we can replace he aumpion ϕ Cb 7 by ϕ C4 b : Ju approximae ϕ Cb 4 by ϕn Cb 7 in he 4, -norm, and ake advanage of he fac ha and π are a.. equal o probabiliy meaure. herefore we have π,x Corollary 6.9. Aume (H a ), (HF 8,4 ), (HS 7,4 ), (HO 8,4 ), and ha he iniial diribuion Q (X,Z ) ha finie momen of every order. Le p 1. hen here exi C > uch ha for every ϕ C 4 b, Q π,x (ϕ) π (ϕ) p p/2 C ϕ p 4,. Now noe ha here exi a counable algebra (ϕ i ) i N of Cb 4 funcion ha rongly eparae poin in R m. ha i, for every x R m and δ >, here exi i N, uch ha inf y: x y >δ ϕ i (x) ϕ i (y) >. ake e.g. all funcion of he form n exp q j (x x j ) 2 j=1 wih n N, q j Q +, x j Q m. By heorem of hier and Kurz (1986), he equence (ϕ i ) i convergence deermining for he opology of weak convergence of probabiliy meaure. ha i, if µ n and µ are probabiliy meaure on R m, uch ha lim n µ n (ϕ i ) = µ(ϕ i ) for every i N, hen µ n converge weakly o µ. Define he following meric on he pace of probabiliy meaure on R m : d(ν, µ) = d (ϕi )(ν, µ) = ν(ϕ i ) µ(ϕ i ) 2 i.

27 DIMNSIONAL RDUCION IN NONLINAR FILRING 27 Becaue (ϕ i ) i convergence deermining, he meric d generae he opology of weak convergence. herefore he proof of heorem 3.1 i complee. 7. Concluion and fuure direcion hi paper preened he heoreical bai for he developmen of a lower-dimenional paricle filering algorihm for he ae eimaion in complex mulicale yem. o hi end, we combined ochaic homogenizaion wih nonlinear filering heory o conruc a homogenized SPD which i he approximaion of a lower-dimeional nonlinear filer for he coare-grained proce. he convergence of he opimal filer of he coare-grained proce o he oluion of he homogenized filer i hown uing BSD and aympoic echnique. hi homogenized SPD can be ued a he bai for an efficien muli-cale paricle filering algorihm for eimaing he low dynamic of he yem, wihou direcly accouning for he fa dynamic. In Lingala e al. (212) we preen a numerical algorihm baed on hi cheme, ha enable efficien incorporaion of obervaion daa for eimaion of he coare-grained ( low ) dynamic, and we apply he algorihm o a high-dimenional chaoic mulicale yem. ven hough hi paper deal wih ju one widely eparaed characeriic ime cale, one can exend hi work o incorporae a more realiic eing where he ignal ha more han one ime cale eparaion. A before we le be a mall parameer ha meaure he raio of low and fa ime cale. Conider he ignal and obervaion procee governed by: dz = 1 2 f(z, X ) + 1 g(z, X ) dw, Z = z, (26) dx = 1 bi (Z, X ) + b(z, X ) + σ(z, X ) dv, X = x, dy = h(z, X ) d + db, Y =, where W, V and B are independen Wiener procee and x and z are random iniial condiion which are independen of W, V and B. I i imporan o realize ha here are everal cale in (26), even he low proce X ha a fa varying componen. hi cae i imporan, in paricular, for applicaion in geophyical flow and climae dynamic. he drif erm b and he diffuion σ caue flucuaion of order order 1, and he drif erm f and he diffuion g caue flucuaion of order order 2, wherea he drif erm b I caue flucuaion a an inermediae order 1. I wa found ha when he average of b I wih repec o he invarian meaure of he fa componen Z (for he fixed low componen) i zero, he limi diribuion of he low componen (away from he iniial layer) can alo be obained in erm of he oluion of ome auxiliary Poion equaion in he homogenizaion heory. However, a unified framework o deal wih 1 erm in developing a lower-dimenional nonlinear filer for he coare-grained proce i ill no available. Our condiion on he coefficien are very rericive and exclude for example linear model. hi i due o he fac ha we are uing homogenizaion of SPD o obain convergence of he filer, and ha for exience of oluion o he SPD, he coefficien need o be bounded and ufficienly mooh. Working wih weak oluion in place of claical oluion would no improve he condiion much. Uing vicoiy oluion or enirely relying on probabiliic argumen migh be a way o ge le rericive condiion, however wih hee mehod we do no expec ha a rae of convergence can be obained. While we were able o obain he explici rae of convergence, he conan C in heorem 3.1 depend on he erminal ime. I would be inereing o find condiion under which hi can be avoided. hi migh be achieved by building on abiliy reul for nonlinear filer, ee e.g. Crian and Rozovkii (211), Chaper 4, Sabiliy and aympoic analyi.