Conditional distributions, exchangeable particle systems, and stochastic partial differential equations

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1 Codiioal diribuio, exchageable paricle yem, ad ochaic parial differeial equaio Da Cria, Thoma G. Kurz, Yoojug Lee 23 July 2 Abrac Sochaic parial differeial equaio whoe oluio are probabiliy-meaurevalued procee are coidered. Meaure-valued procee of hi ype arie aurally a de Fiei meaure of ifiie exchageable yem of paricle ad a he oluio for filerig problem. I boh hee cae, he oluio i he codiioal diribuio of he oluio of a ochaic differeial equaio. The mai reul ae ha, uder mild odegeeracy codiio o he coefficie of he ochaic differeial equaio, he codiioal diribuio of i oluio charge ay ope e. Uder roger codiio we how ha i i aboluely coiuou wih repec o Lebegue meaure ad i deiy i poiive almo everywhere. A applicaio we how he exiece of a oluio of a yem of ieracig diffuio ad udy he properie of he oluio of he oliear filerig equaio wihi a framework ha allow for he igal oie ad he obervaio oie o be correlaed. The work wa moivaed by a model of ae price deermiaio i which he price i give a a quaile of he valuaio of ifiiely may idividual iveor. MSC 2 ubjec claificaio: 6H5, 6G9, 6G35, 6J25 Keyword: codiioal diribuio, ochaic parial differeial equaio, quaile procee, filerig equaio, meaure-valued procee Iroducio Le Ω, F, P be a probabiliy pace ad E, r a complee eparable meric pace. Le B ad W be d ad d -dimeioal adard Browia moio, ad le V be a cadlag E-valued Deparme of Mahemaic, Imperial College Lodo, 8 Quee Gae, Lodo SW7 2AZ, UK, d.cria@imperial.ac.uk Deparme of Mahemaic, Uiveriy of Wicoi-Madio, 48 Licol Drive, Madio, WI , USA, kurz@mah.wic.edu Deparme of Saiic, Harvard Uiveriy, 65 Sciece Ceer, Oe Oxford Sree, Cambridge, MA , USA, ylee@a.harvard.edu

2 proce. We aume ha B i idepede of W, V ad ha W i compaible wih V i he ee ha for each, W + W i idepede of F W,V, where F W,V = σw, V,. Le X be a d-dimeioal ochaic proce aifyig he equaio X = X + f X, V d + σ X, V dw + σ X, V db.. We aume ha, give V, X i codiioally idepede of W, V ad B, ha i, EfX F W,V,B = EfX V..2 For reao ha we will make clear below, we are iereed i he PR d -valued proce π = {π, }, where π i he codiioal diribuio of X give F W,V π ϕ = E ϕ X F W,V for ay ϕ BR d, he bouded, Borel-meaurable fucio o R d. The fir reul of he paper ae ha, uder very geeral odegeeracy ad regulariy codiio Aumpio A below, for >, π charge ay ope e A R d almo urely ad he ull e ca be choe idepede of A. Furher, uder addiioal codiio o he coefficie of., π i aboluely coiuou wih repec o Lebegue meaure o R d ad, wih probabiliy oe, i deiy i ricly poiive. Our primary iere i hee reul i o rea ifiie yem of ochaic differeial equaio X i = X i +, f X, i V d + σ X, i V dw +, σ X i, V db i,.3 where he B i are idepede adard Browia moio, {X i } i a exchageable equece ha i idepede of W ad {B i }, ad V = lim δ X i..4 We require he oluio {X i } o be exchageable o ha he limi i.4 exi by defiei heorem. See Theorem A.. I paricular, if he oluio of he yem i weakly uique, he {X i } mu be exchageable, o.4 mu exi. Uder a Lipchiz codiio o he fir variable ad a coiuiy codiio o he ecod, weak exiece of a exchageable oluio ca be how for which he B i are idepede of W ad V. By exchageabiliy, i= π ϕ = EϕX F W,V = E ϕx F i W,V i= 2

3 ad ice V i meaurable wih repec o F W,V V ϕ = E lim If rog uiquee hold, i= ϕx i F W,V i follow ha = lim E i= V ϕ = E ϕ X F W σv. ϕx i F W,V = π ϕ. I.3, he proce W i commo o all diffuio, while he procee B i, i are muually idepede Browia moio. Syem of hi ype have bee coidered by Kurz ad Proer 6 ad Kurz ad Xiog 7, 8 uder he aumpio ha he coefficie are Lipchiz fucio of V i he Waerei meric o PR d. Thi aumpio exclude a variey of iereig example. I paricular, for d =, we are iereed i equaio whoe coefficie are fucio of quaile of V, V α = if x R {x R V, x α}, ad he reul o π play a ceral role i provig exiece of oluio of a yem i which he coefficie are coiuou fucio of he quaile. Thi ype of applicaio arie aurally i he ae pricig lieraure. For iace, Lee 9, adopig he microecoomic approach formalized i Föllmer ad Schweizer 2, derive he equilibrium price a a quaile of he valuaio of ifiiely may iveor who compee for a ae hrough a coiuouime aucio yem. A ecod applicaio of he uppor reul i o he oluio of ochaic filerig problem. Le X, Y be he oluio of X = X + Y = f X, Y d + h X, Y d + k Y dw. σ X, Y dw + Here Y play he role of V, o B i o idepede of W, Y. iverible ad eig we have X = X + Y = + W = W + ky hx, Y d, σ X, Y db fx, Y + σx, Y ky hx, Y d σx, Y d W + k Y d W, 3 σ X, Y db Aumig ha ky i

4 ad uder mode aumpio o hx, y/ky, a Giraov chage of meaure give a equivale probabiliy meaure uder which B i idepede of W, Y. I hi framework we how ha he codiioal diribuio of X give F Y charge ay ope e. Moreover, uder addiioal codiio, i i aboluely coiuou wih repec o Lebegue meaure o R d, ad wih probabiliy oe i deiy i ricly poiive. The mai reul are proved uder he followig codiio o he coefficie of.. A f : R d R m R d, σ : R d R m R d R d, σ : R d R m R d R d are coiuou fucio, uiformly Lipchiz i he fir argume. Tha i, here exi a coa c uch ha f x, y f x, y c x x 2 for all x, x 2 R d ad y R m wih a imilar iequaliy holdig for σ ad σ. σ i poiive defiie, i.e., for ay y R m ad ξ, x R d wih ξ. ξ σ x, y ξ > For d >, almo urely, V ha pah wih fiie lef limi. I oher word, for all >, def V = lim V,< exi ad i fiie. A2 f, σ ad σ are coiuouly differeiable i he fir compoe. Theorem. Uder aumpio A, here exi a e Ω F of full meaure uch ha for every ω Ω, π ω charge every ope e, i.e., π ω A > for every oempy, ope e A. Theorem.2 Uder aumpio A+A2, here exi a e Ω F of full meaure uch ha for every ω Ω, π ω i aboluely coiuou wih repec o Lebegue meaure. Moreover if y ρ ω y i he deiy of π ω wih repec o Lebegue meaure, ρ ω i ricly poiive. The addiioal codiio i A required o rea he muli-dimeioal cae i o eeded whe d = becaue we are able o exploi he order rucure of R. For d >, he iegral of a oigular, marix-valued fucio may be igular, while for d =, he iegral of a o-zero real-valued fucio i alway o-zero, provided i doe o chage ig. Ackowledgeme. Thi work wa compleed while he fir wo auhor were viiig he Iaac Newo Iiue i Cambridge, UK. The hopialiy ad uppor provided by he Iiue i graefully ackowledged. The reearch of he fir auhor wa parially uppored by he EPSRC gra EP/H55/. The reearch of he ecod auhor wa uppored i par by NSF gra DMS

5 2 Proof of he properie of he codiioal diribuio Le Ϝ be a fucio Ϝ :, R d R d wih he followig properie: For each z R d, he fucio Ϝ, z i a meaurable, locally-bouded fucio. For each,, he fucio z Ϝ, z i differeiable. Ϝ, z will deoe he marix of parial derivaive Ϝ, z ij = j Ϝ i, z. For each z R d, he fucio Ϝ, z i a meaurable, locally-bouded fucio. Now coider a ew probabiliy meaure P z, aboluely coiuou wih repec o P, defied by dp z dp = exp Ϝ, z db Ϝ, z 2 d, F 2 where Ϝ, z i he row vecor Ϝ, z, Ϝ, z 2,..., Ϝ, z d. The, by Giraov heorem, he proce B z = {B z, } B z = B + Ϝ, z d i a Browia moio uder P z, idepede of W ad V. Sice B z, W, V ha he ame law uder P z a B, W, V ha uder P, i follow ha Xz give by dx z = f X z, V d + σ X z, V dw + σ X z, V db z 2. = f X z, V d + σ X z, V dw + σ X z, V db + σ X z, V Ϝ, z d. ha he ame law uder P z a X ha uder P, ad for ϕ BR d, E ϕ X F W,V = E z ϕ X z F W,V = E ϕ X z M z F W,V where M z i defied a M z = exp Ϝ, z db 2 Ϝ, z 2 d,

6 I he followig, we will ue a Fubii argume for he fucio ι, where z, ω ι ϕ X z M z e 2 z 2 2π d 2 i defied o he produc pace R d Ω. Coequely, we eed o kow ha ι i BR d F W,V -meaurable. Meaurabiliy i o immediae a X z i iiially defied for each z idividually. However, oe ca prove he exiece of a proce X z uch ha for each z, X z ad X z are idiiguihable ad z, ω ῑ X z i BR d F W,V -meaurable. More preciely, we ca aume ha X i opioal, ha i, he mappig, z, ω, R d Ω X z i meaurable wih repec o he σ-algebra geeraed by procee of he form ξi f i z i, i+, where = < <, f i CR d, ad ξ i i F W,V i -meaurable. To avoid furher meaurabiliy complicaio, from ow o, we will ue hi verio of he oluio of 2.. Hece, if ϕ : R d Ω R i a o-egaive, BR d F W,V -meaurable fucio, he codiioal verio of Fubii heorem for oegaive fucio give e 2 z z E ϕ X, F W,V = = E R d E ϕ X z, M z F W,V R d ϕ X z, M z e 2 z z 2π d 2 2π d 2 dz F W,V We rea he oe-dimeioal ad he muli-dimeioal cae eparaely. 2. The oe-dimeioal cae For >, le Ϝ, z = { z for, oherwie where a > i a arbirary poiive coa. I hi cae, 2. become dz. 2.3, 2.4 dx z = f X z, V d + σ X z, V dw + σ X z, V db σ X z, V z, d. Sice σ i poiive, wih probabiliy, he fucio z X z i a ricly icreaig, coiuou fucio ad lim z X z = ad lim z X z =. I paricular, z 6

7 X z i a coiuou bijecio, o if β, β i a o-empy ope ierval, he X β, β i a o-empy ope ierval. I paricular, X β, β ha poiive Lebegue meaure. Hece, if we chooe ϕ i 2.3 o be he idicaor fucio of a ope ierval β, β, he P X β, β F W,V = E e zb z2 + 2 dz F W,V. 2π X β,β Sice z e zb z2 + 2 i poiive o X β, β, i follow ha X z 2 + β,β e zb 2 dz i poiive wih probabiliy a i i codiioal expecaio. Thi prove Theorem. i he cae d =. Aumig ha f, σ ad σ are differeiable, z X z i differeiable wih probabiliy. I poiive derivaive i give by where i z = + J z def = dx z dz = σ X z, V exp i z d, 2.6 f X z, V 2 σ X z, V 2 2 σ X z, V 2 d σ X z, V dw + σ X z, V db + σ X z, V azd. Now, ice z X z i a bijecio, i i iverible, ad we ca defie { } exp X y B X y ν y = J X y. Takig ϕ = A, A BR, i 2.3 ad uig he chage of variable y = X z, P X A F W,V = E ν ydy 2π F W,V = E ν y F W,V dy. 2π Hece, he codiioal diribuio of X give F W,V o Lebegue meaure wih deiy A ρ y = E ν y F W,V. 2π A i aboluely coiuou wih repec Sice ν y i ricly poiive, by Lemma A.2, here exi a verio of ρ y uch ha wih probabiliy oe, ρ y > for all y R ad. Thi prove Theorem.2 i he cae d =. 7

8 Corollary 2. Uder aumpio A+A2, here exi a radom variable c,, k poiive almo urely uch ha if ρ ry c,, k. 2.7 r,y, k,k I paricular, he e Ω F of full meaure appearig i he aeme of Theorem.2 o which π ω i aboluely coiuou wih repec o Lebegue meaure ad he deiy of π ω wih repec o Lebegue meaure i ricly poiive ca be choe idepede of he ime variable,. Proof. Uig he idepedece properie of X, B, W, ad V, we have EfX, B F W,V = EfX, B V, for ay reaoable fucio f, hece here exi h f uch ha EfX, B, W, V F W,V = h f V, W, V. Sice ν y i a fucio of X, B, W ad V, hi implie ha ρ y = E ν y F W,V = E ν y F W,V 2π 2π Chooe m o be a arbirary poiive coa. Sice he fucio, x miν x, m i bouded, poiive ad joily coiuou i, x i follow ha i codiioal expecaio ρ m y = E miν x, m F W,V 2π ha a verio which i bouded, poiive ad joily coiuou i, x. Hece, 2.7 hold rue wih c,, k = if r,y, k,k ρ m r y >. Lemma 2.2 Uder codiio A+A2, he deiy fucio y ρ y i aboluely coiuou. Moreover, i i differeiable almo everywhere ad dρ dy y = dν E 2π dy y F W,V. 2.8 More geerally, if f, σ ad σ are m-ime coiuouly differeiable i he fir compoe, he he deiy fucio y ρ y i m -ime coiuouly differeiable ad m-ime differeiable almo everywhere. A imilar formula o 2.8 hold for higher derivaive of ρ a well.. 8

9 Proof. The fucio y ν y i coiuouly differeiable uder codiio A+A2 ad where ι x = ι 2 x = { exp { exp X X dν y dy = ι x ι 2 x, 2.9 y y B X y B X J X y B X J X y y } } y + y J X dj dx X y J X y 2 We wa o prove ha E dν y R dx dy <. I order o do ha, we how ha he propery hold for boh fucio o he righ had ide of 2.9. We how how hi i doe for he fir fucio. We have ha { } exp zb z2 + E ι 2 y dy = E B z + dz 2. R R J z = e z2 + 2 E e pzb p E J z q q E B z + r r dz 2. R e z2 + p 2 Q r z r E J z q q dz, 2.2 R where p, q, r, are choe o ha p < + ad + + = ad Q p q r r i a uiably choe polyomial o ha E B z + r Q r z for ay z R. To ge 2., we ued he chage of variable z = X y ha ad applied Hölder iequaliy o obai 2.. From 2.6 i follow ha where C i he marigale C = J ac σ exp σ X z, V dw + c f,σ, σ c σ z 2 up C, σ X z, V db, I 2.3 we ued he fac ha c σ def = if x,y σx, y > ad ha, 2.3,. c f,σ, σ c σ def = up x,y def = up x,y f x, y 2 σ x, y 2 2 σ x, y 2 σ x, y 9

10 are fiie quaiie. Thi follow from codiio A+A2. Hece, immediaely, where k = ac σ q exp qc f,σ, σ E E J z q ke qc σ z, 2.4 exp 2q up C, Noe ha k i fiie a he ruig maximum of he marigale C ha expoeial mome of all order. From 2.2 ad 2.4 we deduce immediaely he iegrabiliy of ι. The iegrabiliy of ι 2 follow i a imilar maer a all he erm ivolved a imilar o hoe appearig i ι. The oly erm ha i differe dj dj. Explicily i give by dz dz dj dz z = σ X z, V exp i z σ X z, V J z + di dz z d,. ad oe prove i a imilar maer ha dj E dz k e k z, 2.5 where k ad k are ome uiably choe coa. I follow ha ρ y ρ y 2 = E ν y ν y 2 F W,V 2π y dν = E y dy 2π y 2 dy F W,V y dν = E 2π dy ydy F W,V dy 2.6 y 2 ad we deduce from he above he abolue coiuiy of ρ ad, herefore, i differeiabiliy almo everywhere. We oe ha he la ideiy follow by he codiioal Fubii heorem a we have proved he iegrabiliy of dνy over he produc pace Ω R. The dy mehodology o how ha ρ ha higher derivaive i imilar. Oberve fir ha { } d m exp X y B X y 2 + ν y dy = 2 m J X y T, B, X y, dx dx X X y,... dxm X dx m y X y i a radom variable which ha mo- where T, B, X y, dx y,... dxm dx dx m me of all order corolled by a upper boud of he ype 2.5. Oe he how he iegrabiliy of dm ν y over he produc pace Ω R which implie he m-ime differeiabiliy dy m of ρ.

11 2.2 The mulidimeioal cae For α, =, 2,..., defie Le X α, z = {X α, z be i Jacobia J α, Ϝ α,, z = α,αz. 2.7 z, } be he oluio of 2. wih Ϝ replaced by Ϝ α,, ad le J α, z ij = j X α, i z. The J α, z = {J α, z, } i zero for α, ad for α, J α, aifie he followig ochaic differeial equaio J α, z = α f X α, z, V J α, z d + d i= α d i= α α σ i X α, m i= α z, V J α, z db i σ i X α, z, V α σ i X α,,αzi J α, z d z, V J α, z dw i σ X α, z, V α,αd, 2.8 where f : R d R m R d d i he marix-valued fucio defied a f x, v ij def = jf x, v i x j ad σ i, i =,..., d σ i, i =,..., m are fucio defied i he ame maer σ i, i =,..., d σ i, i =,..., m are he colum vecor of σ, repecively σ, σ = σ, σ 2,..., σ d, σ = σ, σ 2,..., σ m. Le Φ α, z = {Φ α, z, } ad Υ α, z = {Υ α, z, } be he oluio of he followig marix ochaic differeial equaio Φ α, z = I m ϖ z, X α, α i= d i= Υ α, z = I σ α i X α, σ α i X α, Υ α, α z, V Φ α, z d z, V Φ α, z dw i z, V Φ α, z db i, z κ z, X α, z, V d

12 where m i= d i= ϖ z, X α, z, V = f X α, z, V + Υ α, α Υ α, α κ z, X α, z, V = ϖ z, X α, z, V I i eay o check ha d i= z σ i X α, z, V dw i z σ i X α, z, V db i, σ i X α, z, V α,αzi m i= d Υ α, z Φ α, z =, σ i X α, z, V 2 d i= σ i X α, z, V 2. ad ice Υ α, z Φ α, z = I, i follow ha Υ α, z Φ α, z = I, for all, i.e., Φ α, z ad Υ α, z are o-igular ad ivere o each oher. The we ca wrie he oluio of 2.8 explicily a α J α, z = Φ α, z z σ X α, z, V d. Φ α, Υ α, α Ulike he oe-dimeioal cae, he Jacobia J α, z i o igular, J α, i oigular. Wrie z may be igular. However, ice z i oigular for α if ad oly if Γ α, z = α Υ α, α Γ α, z = σ Xα, α z, V α + + α α Υ α, α α z σ X α, z, V d σ X α, z I σ X α, z, V d. z, V σ Xα α, z, V α d Sice X α, z ad Υ α, z are joily coiuou i ad z, x, v σ x, v i coiuou, ad lim V = V, i follow ha, for almo all ω Ω ad each compac K R d, lim up α z K α α lim up z K α σ X α, z, V σ X α, z, V d = Υ α, z I σ X α, z, V d =. 2

13 Hece, Ω lim = { ω Ω lim up z K } J α, z σ X, V = for each compac K ha probabiliy. Le K K 2 be compac ube of R d wih R d = k K k, ad defie Lemma 2.3 Ω k, = {ω Ω J α,m z i oigular for m, z K k }. ad, i paricular, P k Ω k, =. Ω lim k Ω k, Proof. I i eough o prove ha Ω lim Ω k, for each k. Le ω Ω lim bu o i Ω k,. I follow ha here exi i, z i, i ad z i K k uch ha, for hi paricular ω, he correpodig Jacobia J i z i = J i z i ω are igular. Hece here exi correpodig λ i R d wih λ i = ad J i z i λ i =. If λ i a limi poi of {λ i }, he uiformiy over compac i he defiiio of Ω lim implie σ X, V λ =. Sice σ X, V i oigular, we have a coradicio. From 2.3 we ge ha, for ay e A ad ay k > π A = P X A F W,V = E r z dz F W,V {z X z A} Ωk, E {z R d X z A} r z dz F W,V, where r z def = exp z B B z z +. 2 Now for ω Ω k, he applicaio z K k X z i a coiuou bijecio. I i ijecive ice i Jacobia i alway o-igular. The urjeciviy follow by mea of he Ivere Fucio Theorem: Sice lim z X z =, he image of z X z i a cloed e. However, he image of z X z i a ope e, oo. Tha i becaue ay z R d ha he propery ha i ha a ope eighborhood U z o ha he fucio rericed o U z i a coiuou bijecio from U z o V Xz where V Xz i a ope eighborhood of 3

14 X z. Hece for ay z R d, he ope e V Xz i i he image of z X z. Sice R d ha o proper ube which i boh cloed ad ope, we ge he urjeciviy of z X z. So for ay ω Ω k ad A a ope e, he e { z R d X z A } ha poiive Lebegue meaure, hu ω {z R d X k z A} rk z dz i poiive o Ω k. Le ow Ω p = {ω Ω π A > }. The, for all k >, P Ω p P Ω k. Tha i becaue P Ω\Ω p Ω k =. If o Ω\Ωp Ωk = E Ω\Ω pπ A E {z R d X k z A} rk z dz F W,V Hece from he previou lemma we deduce ha π charge ay ope e A. Moreover he ull e ca be choe o be idepede of he e A, ice he opology R d ha a couable bae. Now if A i a e of Lebegue meaure, he π A = P X A F W,V = E Ω {z R d X z A} r z dz F W,V +E Ω\Ω {z R d X z A} r z dz F W,V. 2.9 Sice ω r z dz R d i uiformly iegrable, i follow ha he ecod erm i 2.9 coverge o. Hece π A = lim E Ω {z R d X z A} r z dz F W,V = lim E Ω r X y A de J X y dy F W,V >. Sice A r X y de J X y dy 4

15 i alway a iegral over a ull e, i follow ha he codiioal diribuio of X give F W,V i aboluely coiuou wih repec o he Lebegue meaure. Thi prove he fir par Theorem 2 For he ecod par of Theorem 2 le a i he oe-dimeioal cae A be he followig radom e A = {x R ρ y = }. The, from 2.9 we have ha = P X A F W,V Ω = E R A y {z R d X z A} r z dz F W,V 2π E Ω y r X y de J X y F W,V Hece A y E Ω y r X y R 2π de J X y F W,V dy = ad herefore for ay ω Ω, A i a e of Lebegue meaure where Ω i he followig icreaig equece { Ω = ω Ω E Ω y r X y de J X y F W,V dy > } >, y a.e.. Sice P Ω P Ω ad P lim Ω = i follow ha A i a e of Lebegue meaure wih probabiliy. Thi prove he ecod par of Theorem 2 for he mulidimeioal cae. 3 Weak exiece for SPDE wih coefficie depedig o quaile A decribed i he iroducio, we ow coider a ifiie yem of oe-dimeioal ieracig diffuio X i = X i + f X i, V α d + σ X, i V α dw + 5 σ X, i V α db i, 3.

16 where ad V α = if {x R v, x α} v = lim m m m δ X i. 3.2 We aume ha {X} i i exchageable ad require he oluio {X i } o be exchageable o ha he limi i 3.2 exi by defiei heorem. See Theorem A.. A i Kurz ad Xiog 7, v will be a oluio of he ochaic parial differeial equaio φ, v = φ, v + where φ, v deoe i= LV α φ, v d + φ, v = R φxv, dx σ, V α φ, v dw, 3.3 ad LV α φ = σx, V α 2 + σx, V α 2 d 2 φ 2 dx + fx, V α dφ 2 dx. I 3., he proce W i commo o all diffuio, while he procee B i, i are muually idepede Browia moio. We will aume he followig o he coefficie of he equaio. Q f : R R R, σ : R R R, σ : R R R are bouded, coiuou fucio, uiformly Lipchiz i boh argume ad coiuouly differeiable i he fir compoe ad σ i poiive defiie. The we have he followig: Theorem 3. There exi a weak oluio for he yem ad, hece, for he ochaic parial differeial equaio 3.3. Proof. Coider he Euler-ype approximaio of defied a follow: X i, where = X i, + f X i,, V α, d + V α, { = if σ X i, x R v, V α, dw + }, x α σ X i,, V α db i, 3.4 ad v i defied a i 3.2. The yem 3.4 ha a uique rog oluio. The exiece ad uiquee of he oluio i obaied progreively o ierval k, k+. We oe ha, o each uch ierval, he proce V α, i coa ad equal o he quaile of he empirical 6

17 meaure of he yem a he begiig of he ierval. Exiece ad uiquee of he oluio follow from he aumpio ha f, σ, ad σ are Lipchiz i he fir compoe. We alo have v ϕ = E ϕ X i, F W = E R ϕ X i, z M i, z e 2 z z 2π d 2 dz F W, 3.5 where X z i defied a i 2., ad i follow ha v charge every ope e ad, hece, ha V α, = if {x R v, x α} = up {x R v, x < α}. For each i, he boudede of he coefficie implie he equece {X i, } > i relaively compac i diribuio i D R,. Thi relaive compace ogeher wih he coiuiy of he procee eure relaive compace of {X } > i D R,. Takig a ubequece, if eceary, we ca aume ha {X } > coverge i diribuio o a coiuou proce X = X i i. By Lemma A.3 i he Appedix v coverge i diribuio o v defied by v = lim δ X i. To complee he proof, we eed he followig wo lemma. I he fir oe, we drop he aumpio ha V be a quaile, allowig i o ake value i a complee, eparable meric pace E, ad oly require ha he coefficie be bouded, Lipchiz coiuou i he fir variable, ad coiuou i he ecod. Lemma 3.2 Suppoe {V } are E-valued procee ad X z aifie dx z = f X z, V d + σ X z, V dw + σ X z, V db + σ X z, V zd. Defie ad Γ C, = M Bϕ, = M W ϕ, = i= C V d, C BE, ϕv db, ϕv dw, ϕ C b E, C b E. Suppoe ha Γ Γ, i L m E. See Appedix A.3. The for ϕ B,..., ϕ B k, ϕw,..., ϕ W l C b E, {Γ, MB ϕb,..., MB ϕb k, M W ϕw,..., MW ϕw l } i relaively compac i L m E D R k+l,, ad a ubequece ca be eleced alog which covergece hold for all choice 7

18 of ϕ B,..., ϕ B k, ϕw,..., ϕ W l C b E. For ay limi poi, M B ad M W are orhogoal marigale radom meaure aifyig M B ϕ, M B ϕ 2 = ϕ yϕ 2 yγdy, E M W ϕ, M W ϕ 2 = ϕ yϕ 2 yγdy, M B ϕ, M W ϕ 2 =, ad X z Xz aifyig X z = X z + f X z, v Γdv d + E, + σ X z, v M B dv d + E, where he ochaic iegral are defied a i 6. E E, E, σ X z, v M W dv d σ X z, v zγdv d, 3.6 Proof. Relaive compace follow from he fac ha EMBϕ, + h MBϕ, 2 F = E ϕy 2 Γ dy, + h F ϕ 2 h for each ϕ C b E ad imilarly for {MW }. Alog ay coverge ubequece, {Γ, MB, M W } aifie he covergece codiio i Theorem 4.2 of Kurz ad Proer 6. See Example 2. of 6. Uder he boudede ad Lipchiz codiio o f, σ, ad σ, X z coverge o he oluio of 3.6 by Theorem 7.4 of Kurz ad Proer 6. E Lemma 3.3 Le {X }be a equece of uiformly iegrable radom variable covergig i diribuio o a radom variable X ad {D } be a equece of σ-field defied o he probabiliy pace where {X } reide. Le {Y } be a equece of S-valued radom variable uch ha E X D = GY, where G : S R i coiuou. Suppoe X, Y X, Y. The E X Y = GY. Proof. Sice {X } i uiformly iegrable, i follow by Jee iequaliy ha {GY } i uiformly iegrable. The, employig he covergece i diribuio ad he uiform iegrabiliy, E GY gy = lim E GY gy = lim E X gy = E XgY, for every g C b S, ad he lemma follow. 8

19 We reur ow o he proof of Theorem 3.. Le ρ ϕ = R ϕ X i, z M i, z e 2 z z 2π d 2 From 3.5 ad he defiiio of Γ ad MW, for ay e fucio ϕ E ρ ϕ F W = E ρ ϕ F Γ,MW = v ϕ. Hece, leig Γ,, Γ ad W deoe he rericio of Γ, Γ, ad W o he ime ierval,, ρ, v, Γ,, W ρ, v, Γ, W, where By Lemma 3.3 ρ ϕ = Rd ϕ X i z M i z e 2 z z 2π d 2 E ρ ϕ F Γ,M W = v ϕ. A i he proof of Theorem., v charge ay ope e, ad by Lemma A.8, V α, coverge i diribuio o V α, where I ur, i follow ha M W ad M B aify ad V α = if {x R v, x α}. M B ϕ, = M α W ϕ, = ϕv α db, ϕv α dw, dz. dz. ϕ C b E, C b E. Applyig Theorem 7.4 of Kurz ad Proer 6, i follow ha X, V α,, v coverge i diribuio o X, V α, v which i a weak oluio of Quaile Proce Nex, we fid a equaio for he quaile proce V α = if{x R, v, x α}. Recall ha we coidered a ifiie yem of oe-dimeioal ieracig diffuio X i = X i + f X i, V α d + σ X, i V α dw + 9 σ X, i V α db i, 4.

20 where ad V α = if {x R v, x α} v = lim v where v = δ X i. 4.2 To prove he followig reul we chooe a bouded, mooh, ricly poiive fucio q : R R wih bouded fir ad ecod derivaive uch ha q x dx = ad R i= q x up x R qx <. 4.3 Defie he fucio, v,ɛ, v, ɛ F,ɛ, F ɛ : R R a follow x = i= q ɛ x X i F,ɛ x = x v,ɛ y dy v ɛ x = q R ɛ x y v dy F ɛ x = x vɛ y dy v,ɛ,, x R. The, he fucio v,ɛ = v, i follow ha v,ɛ of he probabiliy meaure wih deiie v,ɛ where q ɛ : R R, q ɛ x = q x ɛ ɛ mooh fucio ad, ice lim v Hece he quaile V Lebegue meaure uiquely defied by he formula F,ɛ, V = α are uiformly bouded coverge poiwie o v. ɛ wih repec o he coverge o he quaile V α,ɛ of he meaure wih deiy v ɛ wih repec o he Lebegue meaure, lim V = V α,ɛ. Moreover, ice alo he derivaive of he fucio v,ɛ coverge o he derivaive of he fucio v ɛ ad are uiformly bouded, i follow ha v,ɛ coverge o v ɛ uiformly o compac. I paricular hi implie ha lim v,ɛ V = dv. Similarly, lim,ɛ vv ɛ α,ɛ followig propoiio. x dx x=v = dvɛ x dx x=v α,ɛ Thi wo fac will be ued i he Propoiio 4. Aumig ha A+A2 hold rue ad ha f, σ ad σ are wice coiuouly differeiable i he fir compoe, he he quaile V α aify he followig evoluio equaio for ay > >. V α = V α + fvr α, Vr α dr + 2vr, V α r σvr α, Vr α dw r x σx, V r α v r x x=v α d. 4.4 I uffice o chooe q uch ha qx = c q exp x for x, where c q i he ormalizaio coa. 2

21 Proof. Fir, oe ha, by he defiiio of he quaile, Υ V, X,..., X =, where Υ : R + R i he mooh fucio Υ v, x,..., x = i= v q ɛ y x i dy α. Sice Υ v, x v,..., x = i= q ɛ v x i >, by he implici fucio heorem here exi a couable e of ball B x j, r j R j uch ha B x j, r j = R ad a couable e of mooh fucio Q,j : B x j, r j R uch ha V = Q,j X,..., X, if X,..., X B xj, r j. I paricular i follow ha V i a emi-marigale. Thi fac allow u o deduce he evoluio equaio for he emimarigale V. By applyig he geeralized Iô formula ee, for example, Kuia 4 we have = dυ V, X,..., X = Υ v + 2 Υ 2 v 2 + j= V, X,..., X Υ x j v which implie ha = v,ɛ dv + j= Υ x j V, X,..., X d V + 2 V, X,..., X dx j 2 Υ j= V, X,..., X d V, X j. V dv j= + 2 j= + 2 j= j= j= x 2 j f X j, V α qɛ V X j d σ X j, V α qɛ V X j dw q ɛ σ 2 X j, V α q ɛ V X j d + 2 q ɛ V V X j d V X j j= q ɛ j= j= V, X,..., X d X j σ X j, V α qɛ V X j db j V σ X i, V α d B j, V 2 σ 2 X j, V α q ɛ V X j d X j σ X i, V α d W, V

22 From hi ideiy i follow ha Therefore V = W, V = B i, V = dv = v,ɛ v,ɛ v,ɛ 2v,ɛ 2v,ɛ v,ɛ v,ɛ V 2 v,ɛ v,ɛ v,ɛ V j= V 2 V V j= V V V V V q ɛ Oberve ha he erm V v,ɛ V j= 2 j= σ 2 X, j V α qɛ V X j d j= σ X, j V α σ X, j V α 2 qɛ V X j 2 d σ X, j V α qɛ V X j d qɛ V X j d f X j, V α qɛ V X j d j= j= j= j= j= σ X j, V α qɛ V X j dw σ X j, V α qɛ V X j db j σ 2 X j, V α q ɛ q ɛ X j + σ 2 X j, V α q ɛ V X j d V X j d V V X j σ X i, V α d W, V σ X i, V α d B j, V 4.5 j= f X j, V α qɛ V X j i bouded by f, he upremum orm of f, wih imilar boud holdig for he ecod ad he hird erm i 4.5 ad for he erm appearig i he expreio for V, W, V, B i, V. The erm x 2v,ɛ x j= σ 2 x, V α + σ 2 x, V α q ɛ x X j i uiformly bouded by ɛ σ 2 + σ 2 followig propery 4.3 of he fucio q. A imilar boud ca be proved for all he remaiig erm i 4.5 are uiformly bouded o compac 22

23 a if if r, v,ɛ x i ricly poiive o compac uig he ighe of he equece v ad σ, σ, ad q ɛ are bouded. Uig hee boud, we ake he limi i 4.5 a ed o ifiiy o obai ha dv α,ɛ = v ɛ V α,ɛ R + v ɛ V α,ɛ R 2v ɛ V α,ɛ 2v ɛ V α,ɛ + v ɛ V α,ɛ f x, V α q ɛ V α,ɛ x v dx σ x, V α q ɛ V α,ɛ x v dx R σ 2 x, V α + σ 2 x, V α q ɛ V α,ɛ q ɛ V α,ɛ x v dx R v ɛ V α,ɛ R 2 q ɛ V α,ɛ v ɛ V α,ɛ d dw x v dx d 2 σ x, V α q ɛ V α,ɛ x v dx d R x σ x, V α v dx σ x, V α q ɛ V α,ɛ x v dx d 4.6 R Nex ice v x = lim ɛ v ɛ x a ɛ ed o, i follow ha V α = lim ɛ V α,ɛ. Followig from Corollary 2. ad he boudede of boh v r x ad v x rx o e of he form, k, k, we ca ake he limi i 4.6 a ɛ ed o o obai ha 2v, V α, V α x v x dv α = fv α, V α d + σv α, V α dw 2v, V α σ2 V α x=v α x σ 2 x, V α + σ 2 x, V α d + σv α, V α v, V α v x x=v d α x σx, V α v x d x=v α which give 4.4. Remark 4.2 See alo 9 for he equaio 4.4. Remark 4.3 Uder addiioal aumpio o he iiial diribuio of X for example if he diribuio of X i aboluely coiuou wih repec o he Lebegue meaure ad i deiy i wice coiuouly differeiable oe ca how ha 4.4 hold rue alo for =. 5 Applicaio o oliear filerig Le Ω, F, P be a probabiliy pace o which we have defied wo idepede d-dimeioal, repecively m -dimeioal adard Browia moio B = {B i d i=, } ad W = 23

24 {W i m i=, } Le X, Y be he oluio of he followig ochaic yem X = X + Y = f X, Y d + h X, Y d + k Y dw. σ X, Y dw + σ X, Y db Le F Y be σ-field geeraed by he proce Y ad π be he codiioal diribuio of X give he σ-field geeraed by he proce Y. We how ha π charge ay ope e. Moreover, uder addiioal codiio, we how ha i i aboluely coiuou wih repec o he Lebegue meaure o R d ad ha a poiive deiy. Here are he required codiio: F f : R d R m R d, h : R d R m R m, σ : R d R m R m R d, ad σ : R d R m R d R d are coiuou fucio, uiformly Lipchiz i he fir argume. We aume ha σ i poiive defiie, k : R m R m R m i iverible, k i bouded ad σk h : R d R m R d i a coiuou fucio, uiformly Lipchiz i he fir argume. The radom variable X ha fiie ecod mome. F2 f, σk h, σ ad σ are coiuouly differeiable i he fir compoe. We have he followig Corollary 5. Uder aumpio F, here exi a e Ω F of full meaure uch ha for every ω Ω, π ω charge ay ope e. Moreover uder aumpio A+A2, here exi a e Ω F of full meaure uch ha for every ω Ω, π ω i aboluely coiuou wih repec o he Lebegue meaure ad he deiy of π ω wih repec o he Lebegue meaure i poiive almo everywhere. Proof. Le Z = {Z, } be defied a Z = exp k Y h X, Y dw k Y h X, Y k Y h X, Y d. 2 Uder codiio F, Z i a marigale. Coider he probabiliy meaure P aboluely coiuou wih repec o P defied a d P = Z. dp F The, by Giraov heorem, he proce W } = { W, defied by W = W k Y h X, Y d 24

25 for i a Browia moio uder P idepede of B formula, E ϕ X F Y = Ẽ ϕ X ζ F Y ad, by Kalliapur-Sriebel, 5. where ζ = Z Ẽ Z F Y ad X = X + f + σk h X, Y d + σ X, Y d W + We oe ha, uder P, Y aifie he SDE hece Y = W = k Y d W, k Y dy σ X, Y db. ad i paricular F Y = F W,Y for all. From 5. we obai ha a i 2.3 ha e 2 z z π ϕ = E ϕ X z M z ζ F Y R d 2π d dz 2 where M z i he marigale defied i 2.2. The aalyi he proceed i a ideical fahio o ha i he proof of Theorem. ad.2. Remark 5.2 Noe ha we cao apply he reul of he Theorem. ad.2 uder he origial meaure P a he Browia moio B i o idepede of Y uder P. 25

26 A Appedix A. Covergece of equece of exchageable familie. Le S be a complee, eparable meric pace. A family of S-valued radom variable {ξ,..., ξ m } i exchageable if for every permuaio σ,..., σ m of,..., m, {ξ σ,..., ξ σm } ha he ame diribuio a {ξ,..., ξ m }. A equece ξ, ξ 2,... i exchageable if every fiie ubfamily ξ,..., ξ m i exchageable. Theorem A. defiei Le ξ, ξ 2,... be a exchageable equece of S-valued radom variable. The here i a PS-valued radom variable Ξ uch ha m Ξ = lim δ ξi m m ad, codiioed o Ξ, ξ, ξ 2,... are iid wih diribuio Z, ha i, for each f BS m, m =, 2,..., Efξ,..., ξ m Ξ = f, Ξ m. We will refer o Ξ a he defiei meaure for ξ, ξ 2,.... Proof of he followig lemma ca be foud i he Appedix of 3. Lemma A.2 For =, 2,..., le {ξ,..., ξ N } be exchageable, S-valued radom variable. We allow N =. Le Ξ be he correpodig empirical meaure, where if N =, we mea i= Ξ = N δ ξ N i, i= Ξ = lim m m m δ ξ i. Aume ha N ad ha for each m =, 2,..., {ξ,..., ξm} {ξ,..., ξ m } i S m. The {ξ i } i exchageable ad eig ξi = S for i > N, {Ξ, ξ, ξ2...} {Ξ, ξ, ξ 2,...} i PS S, where Ξ i he defiei meaure for {ξ i }. If for each m, {ξ,..., ξm} {ξ,..., ξ m } i probabiliy i S m, he Ξ Ξ i probabiliy i PS. The covere alo hold i he ee ha Ξ Ξ implie {ξ,..., ξm} {ξ,..., ξ m }. We are iereed i applyig he above lemma i he cae S = D E,. I ha eig, i addiio o he PD E, -valued radom variable Ξ, i i aural o coider he PE-valued procee Z = N N i= i= δ X i, where N may be ifiie which will have ample pah i D PE,. Ulike Ξ, covergece of Z i o alway aured. 26

27 Lemma A.3 For =, 2,..., le X = X,..., XN be exchageable familie of D E, - valued radom variable uch ha N ad X X i D E,. Defie Ξ = N N i= δ Xi PD E,, Ξ = lim m m m i= δ X i, Z = N N i= δ Xi PE, ad Z = lim m m m i= δ X i. a Le D Ξ = { : EΞ{x : x x } > }. The for,..., l / D Ξ, Ξ, Z,..., Z l Ξ, Z,..., Z l. b If X X i D E,, he X, Z X, Z i D E PE,. If X X i probabiliy i D E,, he X, Z X, Z i probabiliy i D E PE,. Remark A.4 a The e D Ξ i a mo couable. b If for i j, wih probabiliy oe, X i ad X j have o imulaeou dicoiuiie, he D Ξ = ad covergece of X o X i D E, implie covergece i D E,. I paricular, hi cocluio hold if he X i are coiuou. c If {X } i relaively compac i D E,, he {X, Z } i relaively compac i D E PE,. Lemma A.5 If X = X, X 2,... i a exchageable equece i D E,, he Z i coiuou if ad oly if for i j, wih probabiliy oe X i ad X j have o imulaeou dicoiuiy. A.2 Covergece of quaile For < α <, ad for µ PR, defie q α µ = if{x : µ, x α}. Noe ha µ i a poi of coiuiy for q α if ad oly if µq α µ, q α µ + ɛ > ad µq α µ ɛ, q α µ > for every ɛ >. Lemma A.6 Le {Y } be a equece of PR-valued radom variable uch ha Y Y. Suppoe ha wih probabiliy, he meaure Y charge every ope e. The q α Y q α Y for each < α <. Proof. The lemma follow by he coiuou mappig heorem. Lemma A.7 Suppoe z D PR, ad for each, z ad z charge every ope e. The if < α < ad z z i D PR,, q α z q α z i D R,. Proof. The lemma follow by Propoiio of Ehier ad Kurz ad he coiuiy properie of q α. The coiuou mappig heorem give he followig. Lemma A.8 Suppoe {Z } i a equece of procee i D PR, uch ha Z Z. If, wih probabiliy, Z ad Z charge every ope e for all, he for < α <, q α Z q α Z. 27

28 A.3 Covergece of radom meaure The followig reul are from Kurz 5. Le LS be he pace of meaure µ o, S uch ha µ, S < for each >, ad le L m S LS be he ubpace o which µ, S =. For µ LS, le µ deoe he rericio of µ o, S. Le ρ deoe he Prohorov meric o M, S, ad defie ρ o LS by ρµ, ν = e ρ µ, ν d, ha i, {µ } coverge i ρ if ad oly if {µ } coverge weakly for almo every. Lemma A.9 A equece of L m S, ρ-valued radom variable {Γ } i relaively compac if ad oly if for each ɛ > ad each >, here exi a compac K S uch ha if EΓ, K ɛ. Lemma A. Le {x, µ } D E, LS, ad x, µ x, µ. Le h CE S. Defie u = hx, yµ d dy, u = hx, yµd dy, S z = µ, S, ad z = µ, S., S a If x i coiuou o, ad lim z = z, he lim u = u. b If x, z, µ x, z, µ i D E R, LS, he x, z, u, µ x, z, u, µ i D E R R, LS. I paricular, lim u = u a all poi of coiuiy of z. c The coiuiy aumpio o h ca be replaced by he aumpio ha h i coiuou a.e. ν for each, where ν ME S i he meaure deermied by ν A B = µ{, y : x A,, y B}. d I boh a ad b, he boudede aumpio o h ca be replaced by he aumpio ha here exi a oegaive covex fucio ψ o, aifyig lim r ψr/r = uch ha up ψ hx, y µ d dy < A. for each >., S 28

29 A.4 Meaurabiliy ad poiiviy of radom fucio give by codiioal expecaio Lemma A. Le Ω, F, P be a complee probabiliy pace, E a complee, eparable meric pace, ad {F x, x E} a collecio of complee ub-σ-algebra of F. Suppoe ha for each A F, here exi a BE F meaurable proce X A idexed by E uch ha for each x, P A F x = X A x The for each bouded, BE F-meaurable proce Y here exi aoher BE F- meaurable proce Ŷ uch ha EY x Fx = Ŷ x a.. Proof. If Y x = B x A for B BE ad A F, he Ŷ x = BxX A x aifie he requireme of he lemma. Sice {B A : B BE, A F} i cloed uder ierecio ad geerae BE F ad he collecio of Y for which he cocluio of he lemma hold i cloed uder bouded moooe icreaig limi, he lemma follow by he moooe cla heorem for fucio. See Theorem 4.3 i he Appedix of Ehier ad Kurz. Lemma A.2 Suppoe ha he cocluio of Lemma A. hold ad ha Y i BE F- meaurable ad ricly poiive. The Ŷ ca be ake o be ricly poiive. Proof. Le A = {x, ω : Y x, ω } ad A = {x, ω : 2 Y x, ω < 2 }, =, 2,.... The =A = E Ω, ad we ca aume ha E A F x for all x, ω. Noe ha = lim a.. E Ak F x a.. k= for all x. If eceary, we ca replace E A F x by E Ak F x E Ak F x k= o eure k= E A k F x ad he replace E A F x by E Ak F x o eure k= E A k F x = for all x, ω. The 2 E A F x Ŷ x a.. = k= ad we ca replace Ŷ x by Ŷ x = 2 E A F x o be aured ha Ŷ x > for all x, ω. k= 29

30 Referece Sewar N. Ehier ad Thoma G. Kurz. Markov Procee: Characerizaio ad Covergece. Wiley Serie i Probabiliy ad Mahemaical Saiic: Probabiliy ad Mahemaical Saiic. Joh Wiley & So Ic., New York, 986. ISBN Ha Föllmer ad Mari Schweizer. A microecoomic approach o diffuio model for ock price. Mahemaical Fiace, 3: 23, Peer M. Koeleez ad Thoma G. Kurz. Macrocopic limi for ochaic parial differeial equaio of McKea-Vlaov ype. Probab. Theory Relaed Field, 46-2: , 2. ISSN doi:.7/ URL hp://dx.doi. org/.7/ H. Kuia. Sochaic differeial equaio ad ochaic flow of diffeomorphim. I École d éé de probabilié de Sai-Flour, XII 982, volume 97 of Lecure Noe i Mah., page Spriger, Berli, Thoma G. Kurz. Averagig for marigale problem ad ochaic approximaio. I Applied Sochaic Aalyi New Bruwick, NJ, 99, volume 77 of Lecure Noe i Corol ad Iform. Sci., page Spriger, Berli, Thoma G. Kurz ad Philip E. Proer. Weak covergece of ochaic iegral ad differeial equaio. II. Ifiie-dimeioal cae. I Probabiliic Model for Noliear Parial Differeial Equaio Moecaii Terme, 995, volume 627 of Lecure Noe i Mah., page Spriger, Berli, Thoma G. Kurz ad Jie Xiog. Paricle repreeaio for a cla of oliear SPDE. Sochaic Proce. Appl., 83:3 26, 999. ISSN Thoma G. Kurz ad Jie Xiog. Numerical oluio for a cla of SPDE wih applicaio o filerig. I Sochaic i Fiie ad Ifiie Dimeio, Tred Mah., page Birkhäuer Boo, Boo, MA, 2. 9 Yoojug Lee. Modelig he radom demad curve for ock: A ieracig paricle repreeaio approach. 24. URL hp:// reearch.hml. 3