Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs

Transcription

1 Penalizaion mehod fo a nonlinea Neumann PDE via weak oluion of efleced SDE Khaled Bahlali a,,1, Lucian Maiciuc b,2, Adian Zălinecu b,2, a Univeié de Toulon, IMATH, EA 2134, La Gade Cedex, Fance. b Faculy of Mahemaic, Alexandu Ioan Cuza Univeiy, Caol I Blvd., no. 9, Iai, 756, Romania. July 31, 213 Abac In hi pape we pove an appoximaion eul fo he vicoiy oluion of a yem of emi-linea paial diffeenial equaion wih coninuou coefficien and nonlinea Neumann bounday condiion. The appoximaion we ue i baed on a penalizaion mehod and ou appoach i pobabiliic. We pove he weak uniquene of he oluion fo he efleced ochaic diffeenial equaion and we appoximae i in law by a equence of oluion of ochaic diffeenial equaion wih penalized em. Uing hen a uiable genealized backwad ochaic diffeenial equaion and he uniquene of he efleced ochaic diffeenial equaion, we pove he exience of a coninuou funcion, given by a pobabiliic epeenaion, which i a vicoiy oluion of he conideed paial diffeenial equaion. In addiion, hi oluion i appoximaed by oluion of penalized paial diffeenial equaion. AMS Claificaion ubjec: 6H99, 6H3, 35K61. Keywod o phae: Reflecing ochaic diffeenial equaion; Penalizaion mehod; Weak oluion; Jakubowki S-opology; Backwad ochaic diffeenial equaion. 1 Inoducion Le G be a C 2 convex, open and bounded e fom R d and fo, x [, T ] Ḡ we conide he following eflecing ochaic diffeenial equaion SDE fo ho X + K = x + bx d + σx dw, [, T ]. Coeponding auho. CNRS, LATP, CMI, Aix Maeille Univeié, 39 ue Jolio-Cuie,13453 Maeille The wok of hi auho wa paially uppoed by PHC Volubili MA/1/224 and PHC Taili 13MDU The wok of hi auho wa uppoed by IDEAS no. 241/ and POSDRU/89/1.5/S/ addee: bahlali@univ-ln.f Khaled Bahlali, lucian.maiciuc@ymail.com Lucian Maiciuc, adian.zalinecu@gmail.com Adian Zălinecu 1

2 wih K a bounded vaiaion poce uch ha fo any [, T ], K = and K [,] = lx d K [,] 1 {X G}d K [,], whee he noaion K [,] and fo he oal vaiaion of K on he ineval [, ]. The coefficien b and σ ae uppoed o be only bounded coninuou on R d and σσ unifomly ellipic. The fi main pupoe i o pove ha he weak oluion X, K i appoximaed in law in he pace of coninuou funcion by he oluion of he non-eflecing SDE X n = x + [bx n nx n π Ḡ Xn ] d + σx n dw, [, T ], whee π Ḡ i he pojecion opeao. Since fo n he em Kn := n Xn π Ḡ Xn d foce he oluion X n o emain nea he domain, he above equaion i called SDE wih penalizaion em. The cae whee b and σ ae Lipchiz ha been conideed by Lion, Menaldi and Szniman in [1] and by Menaldi in [13] whee hey have poven ha Eup [, T ] X n X, a n. Noe ha Lion and Szniman have hown, uing Skoohod poblem, he exience of a weak oluion fo he SDE wih nomal eflecion o a non-neceaily convex domain. The cae of eflecing SDE wih jump ha been eaed by Łaukajy and Słomińki in [7] in he Lipchiz cae; he ame auho have exended in [8] hee eul o he cae whee he coefficien of he eflecing equaion i only coninuou. In hee wo pape i i poven ha he appoximaing equence X n n i igh wih epec o he S-opology, inoduced by Jakubowki in [5] on he pace D R+, R d of càdlàg R d -valued funcion. Auming he weak in law uniquene of he limiing efleced diffuion X, hey pove in [8] ha X n S-convege weakly o X. We menion ha X n n may no be elaively compac wih epec o he Skoohod opology J 1. In cona o [8], we can no imply aume he uniquene in law of he limi X, and he weak S-convegence of X n o X i no ufficien o ou goal. In ou famewok, we need o how he uniquene in law of he couple X, K and ha he convegence in law of he equence X n, K n o X, K hold wih epec o unifom opology. The fi main eul of ou pape will be he weak uniquene of he oluion X, K, ogehe wih he convegence in law in he pace of coninuou funcion of he penalized diffuion o he efleced diffuion X and he coninuiy wih epec o he iniial daa. Subequenly, uing a pope genealized BSDE, we deduce a a econd main eul an appoximaion eul fo a coninuou vicoiy oluion of he yem of emi-linea paial diffeenial equaion PDE fo ho wih a nonlinea Neumann bounday condiion u i, x + Lu i, x + f i, x, u, x =,, x [, T ] G, u i n, x + h i, x, u, x =,, x [, T ] G, u i T, x = g i x, x G, i = 1, k, whee L i he infinieimal geneao of he diffuion X, defined by L = 1 σ σ 2 ij + 2 x i,j i x j i 2 b i x i,

3 and u i / n i he ouwad nomal deivaive of u i on he bounday of he domain. Boufoui and Van Caeen have eablihed in [2] a imila eul, bu in he cae whee he coefficien b and σ ae unifomly Lipchiz. We menion ha he cla of BSDE involving a Sielje inegal wih epec o he coninuou inceaing poce K [,] wa udied fi in [15] by Padoux and Zhang; he auho povided a pobabiliic epeenaion fo he vicoiy oluion of a Neumann bounday paial diffeenial equaion. I hould be menioned ha he coninuiy of he vicoiy oluion i ahe had o pove in ou fame. In fac, hi popey eenially ue he coninuiy wih epec o iniial daa of he oluion of ou BSDE. We develop hee a moe naual mehod baed on he uniquene in law of he oluion X, K, Y of he efleced SDE-BSDE and on he coninuiy popey. Simila echnique wee developed, in he non efleced cae, in [1], bu in ou iuaion he poof i moe delicae. The difficuly i due o he peence of he eflecion poce K in he fowad componen and he genealized pa in he backwad componen. Thoughou hi pape we ue diffeen ype of convegence defined a follow: fo he pocee Y n n and Y, by Y n Y we denoe he convegence in law wih epec o he unifom opology, by Y n u J1 Skoohod opology J 1 and by Y n Y n we mean he convegence in law wih epec o he Y we undeand he weak convegence conideed S in S-opology. The pape i oganized a follow: in he nex ecion we give he aumpion, we fomulae he poblem and we ae he wo main eul. The hid ecion i devoed o he poof of he fi main eul poof of he convegence in law of X n, K n o X, K a n and he coninuou dependence wih epec o he iniial daa. In Secion 4 he genealized BSDE ae inoduced, he coninuiy wih epec o he iniial daa i obained and we pove he appoximaion eul fo he PDE inoduced above. 2 Fomulaion of he poblem; he main eul Le G be a C 2 convex, open and bounded e fom R d and we uppoe ha hee exi a funcion l C 2 b Rd uch ha G = {x R d : l x < }, G = {x R d : l x = }, and, fo all x G, l x i he uni ouwad nomal o G. In ode o define he appoximaion pocedue we hall inoduce he penalizaion em. Le p : R d R+ be given by p x = di 2 x, Ḡ. Wihou eicion of genealiy we can chooe l uch ha l x, δ x, x R d, whee δ x := p x i called he penalizaion em. I can be hown ha p i of cla C 1 on R d wih 1 2 δ x = 1 2 di2 x, Ḡ = x πḡ x, x Rd, 3

4 whee π Ḡ x i he pojecion of x on Ḡ. I i clea ha δ i a Lipchiz funcion. On he ohe hand, x di 2 x, Ḡ i a convex funcion and heefoe Le T > and uppoe ha: z x, δ x, x R d, z Ḡ. 1 A 1 b : R d R d and σ : R d R d ae bounded coninuou funcion. Remak 1 In fac we can aume ha he funcion b and σ have ublinea gowh bu, fo he impliciy of he calculu, we will wok wih aumpion A 1. A 2 he maix σσ i unifomly ellipic, i.e. hee exi α > uch ha fo all x R d, σσ x α I. Moeove, hee exi ome poiive conan C i, i = 1, 2, α R, β R + and q 1 uch ha A 3 f, h : [, T ] R d R k R k and g : R d R k ae coninuou funcion and, fo all x, x R d, y, y R k,, [, T ], i y y, f, x, y f, x, y α y y 2, ii h, x, y h, x, y β + x x + y y, iii f, x, y + h, x, y C y, iv gx C x q. 2 Le u conide he following yem of emi-linea PDE conideed on he whole pace: u n i, x + Lun i, x + f i, x, u n, x u n i, x, nδx lx, nδx h i, x, u n, x = u n i T, x = g ix,, x [, T ] R d, i = 1, k, 3 and he nex emi-linea PDE conideed wih Neumann bounday condiion: u i, x + Lu i, x + f i, x, u, x =,, x [, T ] G, u i n, x + h i, x, u, x =,, x [, T ] G, u i T, x = g i x, x G, i = 1, k, 4 whee L i he econd ode paial diffeenial opeao L = 1 σ σ 2 ij + 2 x i,j i x j i b i x i, 4

5 and, fo any x G u i n, x = l x, u i, x i he exeio nomal deivaive in x G. Ou goal i o eablih a connecion beween he vicoiy oluion fo 3 and 4 epecively. The poof will be given uing a pobabiliic appoach. Theefoe we a by udying an SDE wih eflecing bounday condiion and hen we aociae a coeponding backwad SDE. Since he coefficien of he fowad equaion ae meely coninuou, ou eing i ha of weak fomulaion of oluion. Fo, x [, T ] Ḡ we conide he following ochaic diffeenial equaion wih eflecing bounday condiion: i ii iii X,x K,x = + K,x = x + K,x [,] = bx,x d + lx,x d K,x [,], σx,x dw, 1 {X,x G} d K,x [,], [, T ], 5 whee K,x [,] i he he oal vaiaion of K,x on he ineval [, ]. We denoe by k,x follow ha he coninuou inceaing poce defined by k,x k,x = lx,x := K,x [,]. I, dk,x. 6 Uing he penalizaion em δ we can define he appoximaion pocedue fo he efleced diffuion X. Unde aumpion A 1 we know ha ee, e.g., [6, Theoem ], fo each n N, hee exi a weak oluion of he following penalized SDE Le X,x,n = x + [ bx,x,n nδx,x,n ] d + K,x,n := k,x,n := nδx,x,n d, lx,x,n σx,x,n dw, [, T ]. 7 8, dk,x,n, [, T ]. We menion ha ee, e.g., [6] he oluion poce X,x,n [,T ] i unique in law unde he upplemenay aumpion A 2. Fo < T, he oal vaiaion of Y on [, ] i given by Y [,] ω = up whee : = < 1 < < n = i a paiion of he ineval [, ]. { n 1 i= } Y i+1 ω Y i ω, 5

6 Hee and ubequenly, we hall denoe by V and V n : V,x := x + V,x,n := x + bx,x,n d + Hence 5 and 7 become epecively bx,x,n d + σx,x,n dw, σx,x,n dw, [, T ]. 9 X,x + K,x = V,x and X,x,n + K,x,n = V,x,n, [, T ]. Definiion 2 We ay ha Ω, F, P, {F }, W, X, K i a weak oluion of 5 if Ω, F, P, {F } i a ochaic bai, W i a d -dimenional Bownian moion wih epec o hi bai, X i a coninuou adaped poce and K i a coninuou bounded vaiaion poce uch ha X Ḡ, [, T ] and yem 5 i aified. The main eul ae he following wo heoem. The fi one coni in eablihing he weak uniquene in law of he oluion fo 5 and he coninuou dependence in law wih epec o he iniial daa. Theoem 3 Unde he aumpion A 1 A 2, hee exi a unique weak oluion X,x, K,x of SDE 5. Moeove, X,x,n, K,x,n X,x, K,x u and he applicaion i coninuou in law. [, T ] Ḡ, x X,x, K,x [,T ] Once hi eul fo he fowad pa i eablihed we hen aociae a BSDE involving Sielje inegal wih epec o he inceaing poce k,x in ode o obain he pobabiliic epeenaion fo he vicoiy oluion of PDE 3. The nex eul povide he appoximaion of a vicoiy oluion fo yem 4 by he oluion equence of 3. Theoem 4 Unde he aumpion A 1 A 3, hee exi coninuou funcion u n : [, T ] R d R d and u : [, T ] Ḡ Rd uch ha u n i a vicoiy oluion fo yem 3, u i a vicoiy oluion fo yem 4 wih Neumann bounday condiion and, in addiion, 3 Poof of Theoem 3 lim n un, x = u, x,, x [, T ] Ḡ. We hall divide he poof of hi Theoem ino eveal lemma. Fi of all we ecall ha he exience of a weak oluion i given, unde aumpion A 1, by [11, Theoem 3.2]. Fo he impliciy of peenaion we uppe fom now on he explici dependence on, x in he noaion of he oluion of 5 and 7. We fi pove an eimaion eul fo he oluion of he penalized SDE 7. 6

7 Lemma 5 Unde aumpion A 1, fo any q 1, hee exi a conan C >, depending only on d, T and q, uch ha E up [,T ] X n 2q + E up [,T ] K n 2q + E K n q [,T ] C, n N. 1 Poof. Wihou lo of genealiy we can aume ha G. Fom Iô fomula applied fo X n 2 i can be deduced ha X n X n, dk n = x X n, b X n d + 2 Wie τ m := inf { [, T ] : X n m} T, m N and by he above, + X n, σ X n dw σx n 2 d, [, T ]. τ X τ n m 2 m τ + 2 X n, dk n C + x 2 m τ m + C X n d + 2 X n, σ X n dw, [, T ]. Hee and in wha follow C > will denoe a geneic conan which i allowed o vay fom line o line. and Theefoe τ X τ n m 2 m q + X n, dk n C 1 + x 2q τ m + C τ m +C q X n 2 d q X n, σ X n dw, τ E up [,] X τ n m 2 m + +C E τ m q Xu n, dku n C 1 + x 2q τ up u [,] Xu n 2q m q d + C E up [,] Xu n, σ Xu n dw u. 11 By Bukholde-Davi-Gundy inequaliy we deduce τ m q E up [,] Xu n, σ Xu n dw u C E τ m C E Xu n 2 q/2 du C 1 + E τ m τ m Xu n 2 σ Xu n 2 du q/2 up u [,] Xu n 2q d, and 11 yield E up [, τ m] X n 2q C 1 + x 2q + E up u [, τ m] Xu n 2q d, [, T ], 7

8 ince fom 1 applied fo z = G, we have Fom he Gonwall lemma, Taking m i follow ha X n, dk n = n E up [, τ m] X n 2q C X n, δx n d. 1 + x 2q, n N. E up [,T ] X n 2q C, n N. 12 Once again fom 11 and 12 we obain q E X n, dk n C 1 + x 2q. We have ha hee exi ε > uch ha he ball B, ε G, and, fo z = ε Kn K n K n Kn G, inequaliy 1 become ε K n v K n u v u X n, dk n, u v T and by he definiion of oal vaiaion of K n, i follow ha ε q E K n q T q [,T ] E X n, dk n C. Lemma 6 Unde aumpion A 1 he equence X n, K n, k n [,T ] i igh wih epec o he S- opology. Poof. In ode o obain he S-ighne of a equence of inegable càdlàg pocee U n, n 1, we hall ue he ufficien condiion given e.g. in [9, Appendix A] which coni in poving he unifom boundedne fo: whee CV T U n + E up [,T ] U n, m 1 E[U CV T U n := up E[ n U i+1 n /F i i ] ] 13 π i= define he condiional vaiaion of U n, wih he upemum aken ove all finie paiion π : = < 1 < < m = T. Uing Lemma 5, we deduce ha hee exi a conan C > uch ha fo evey n N CV T K n + E up [,T ] K n E K n [,T ] + E 8 up [,T ] K n C.

9 Since k n i inceaing and l Cb 2 R d, hen hee exi conan M, C > uch ha fo evey n N CV T k n + E up [,T ] k n 2E kt n T = 2E 2E up [,T ] 2M E K n [,T ] 2M C. lx n, dk n lx n K n [,T ] By he definiion of V n, aumpion A 1 and he fac he condiional vaiaion of a maingale i, we obain fo each n N, CV T V n CV T = CV T T M C bx,x,n d + CV T bx,x,n d E σx,x,n dw bx,x,n d Theefoe ee alo Lemma 5, hee exi C > uch ha fo evey n N CV T X n + E up [,T ] X n CV T V n + CV T K n + E up [,T ] X n C. Lemma 7 Unde he aumpion A 1 A 2, he uniquene in law of he ochaic poce X [,T ] hold. Poof. Le Ω, F, P, {F }, W, X, K be a weak oluion of 5 and f C 1,2 [, T ] Ḡ. We apply Iô fomula o f, X : f, X = f, x + + f + Lf, X d x f, X, σ X dw. x f, X, l X dk Since σσ i uppoed o be inveible, we deduce, uing Kylov inequaliy fo he eflecing diffuion ee [17, Theoem 5.1], ha fo any [, T ], E f + Lf, X 1 {X G}d C de σσ 1 f 1 d+1 + Lf d+1 1 G ddx =. G 14 9

10 and equaliy 14 become f, X = f, x + Theefoe + f + Lf, X 1 {X G}d x f, X, σ X dw, P-a.. f, X f, x f + Lf, X 1 {X G}d i a P-upemaingale wheneve f C 1,2 [, T ] Ḡ aifie x f, x, l x, x G. x f, X, l X dk Fom [19, Theoem 5.7] applied wih φ = l, γ := φ and ρ := we have ha he oluion o he upemaingale poblem i unique fo each aing poin, x, heefoe ou oluion poce X [,T ] i unique in law. Remak 8 Following he emak of El Kaoui [4, Theoem 6] we obain he uniquene in law of he couple X, K, ince he inceaing poce k depend only on he oluion X and no on he Bownian moion. The uniquene i eenial in ode o fomulae he iue of he coninuiy wih epec o he iniial daa. Lemma 9 We uppoe ha he aumpion A 1 A 2 ae aified. Then i X n, K n ii k n u k. Poof. i Fi we will pove he convegence: X n, K n X, K, u S X, K. 15 We hall apply [8, Theoem 4.3 iii]. We ecall ha we have he uniquene of he weak oluion. Fo any n N, [, T ], le H n := x Ḡ and he pocee Zn :=, W. Ou equaion can be wien a X n = H n + b, σ X,x,n, dz n K n, [, T ]. The pocee Z n aifie he UT condiion inoduced in [18], ince fo any dicee pedicable pocee U n, Ū n of he fom U n := U n + k i= U i n, epecively Ū n := Ū n + k i= Ū i n wih Ui n, Ū i n 1, q q 2 E U n d + Ū n q dw 2E U n 2 q 2 d + 2E Ū n dw q 2q 2 + 2E Ū n 2 d 2q q

11 Theefoe he aumpion of [8, Theoem 4.3] ae aified and hu we obain ha X n S X. Uing once again [8, Theoem 4.3 ii] and definiion 9 we deduce ha X n 1, X n 2,..., X n m, V n X 1, X 2,..., X m, V, fo any paiion = < 1 < < m = T. The above convegence i conideed in law, on he pace R d m D[, T ], d R endowed wih he poduc beween he uual opology on R d m and he Skoohod opology J1. Hence X n, V n X, V, S ince X n, V n n i igh. I i known ha he pace D[, T ], R d of càdlàg funcion endowed wih S-opology i no a linea opological pace, bu he equenial coninuiy of he addiion, wih epec o he S-opology, i fulfilled ee Jakubowki [5, Remak 3.12]. Theefoe K n = V n X n S V X = K. In ode o obain he unifom convegence of he equence X n, K n n we emak ha, ince V n V and V n, V ae coninuou, hi convegence i unifom in diibuion: J1 V n u V. Uing he Skoohod heoem, hee exi a new pobabiliy pace ˆΩ, ˆF, ˆP on which we can define andom vaiable ˆV, ˆV n uch ha and Le ˆX n be he oluion of he equaion X n be he oluion of and denoe ˆV ˆX n + ˆK n := X n + ==== law n law V, ˆV ==== V n, n N up ˆV n ˆV [,T ] a... nδ ˆX n d = ˆV n, [, T ], nδ X n d = ˆV, [, T ], nδ ˆX n d, Kn := nδ X n d. We ae hankful o pofeo L. Słomińki fo hi ueful uggeion in he poof of hi pa. 11

12 I i eay o pove ee, e.g., [7, Lemma 2.2] o [2, Lemma 2.2] ha heefoe up [,T ] ˆXn X n 2 n ˆV ˆXn X n 2 up [,T ] ˆV ˆV n ˆV n ˆV ˆV n ˆV, d ˆK n K n, ˆV up n ˆV ˆV ˆKn [,T ] + K n [,T ]. 16 [,T ] Since ˆX n, ˆK n law ==== X n, K n and K n [,T ] i bounded in pobabiliy by inequaliy 1, ˆK n [,T ] i bounded in pobabiliy. Applying [7, Theoem 2.7], i follow ha K n [,T ] i alo bounded in pobabiliy. Bu heefoe, fom 16, up n ˆV ˆV pob, [,T ] up ˆXn X n [,T ] 2 pob On he ohe hand, le ˆX be he oluion of he Skoohod poblem ˆX + ˆK = ˆV, [, T ].. 17 I can be hown ee he poof of [11, Theoem 2.1] o he poof of [16, Theoem 4.17] ha up Xn ˆX 2 pob, [,T ] heefoe, fom 17, up ˆXn ˆX 2 pob. [,T ] Since ˆK = ˆV ˆX, ˆKn = ˆV n ˆX n, [, T ], and he concluion follow. ˆX n, ˆK n pob ˆX, ˆK, u ˆX n, ˆK n law ==== X n, K n, ii In ode o pa o he limi in he inegal lx, dk, we apply he ochaic veion of Helly-Bay heoem given by [21, Popoiion 3.4]. Fo he convenience of he eade we give he aemen of ha eul: Lemma 1 Le X n, K n : Ω n, F n, P n C [, T ], R d be a equence of andom vaiable and X, K uch ha X n, K n X, K. u 12

13 If K n n ha bounded vaiaion a.. and up P K n [,T ] > a, a a, n N hen K ha a.. bounded vaiaion and X n, dk n X, dk, a n. u Reuning o he poof of Lemma 9, he concluion ii follow now eaily, ince k and k n ae defined by 6 and 8 epecively. Remak 11 Le he aumpion A 1 A 2 be aified. Then he weak oluion X,x [,T ] i a ong Makov poce. Indeed, aking ino accoun he equivalence beween he exience fo he ub-maingale poblem and he exience of a weak oluion fo efleced SDE 5 ee [4, Theoem 7], we obain ha he weak oluion X,x [,T ] i a ong Makov poce ince he uniquene hold ee [4, Theoem 1]. In ou iuaion, hi equivalence can be obained by uing Kylov inequaliy fo eflecing diffuion. The following eul will finalize he poof of Theoem 3. We exend he oluion poce o [, T ] by denoing X,x := x, K,x :=, [,. 18 Lemma 12 We uppoe ha he aumpion A 1 A 2 ae aified and le X,x, K,x [,T ] be he weak oluion of 5. Then i he family X,x, K,x [,T ] i igh wih epec o he iniial daa, x, a family of C[, T ],R d R d -valued andom vaiable and ii he weak oluion X,x, K,x [,T ] i coninuou in law wih epec o he iniial daa, x. Poof. i Fi le, x [, T ] Ḡ be fixed. Denoe a befoe X, K := X,x, K,x. Applying Iô fomula fo he poce X X, whee i fixed and, we deduce X X 2 = ince X u, X Ḡ and X u X, b X u du 2 X u X, σ X u dw u X u X, b X u du + z X u, dk u = X u X, dk u + σ X u 2 du + 2 σ X u 2 du X u X, σ X u dw u, z X u, l X u dk u,, z R d. 13

14 Theefoe, uing ha b, σ ae bounded funcion and Ḡ i a bounded domain, v E X X 8 C 4 + CE up v [,] 4 X u X, σ X u dw u C 4 + CE X u X 2 σ X u 2 du C 4 + C 2 C max 2 4, Concening K, we emak fi ha K K = b X u du + σ X u dw u X X. Hence E K K 8 CE X X CE b X u du v 8 +CE up v [,] σ X u dw u C max 4, 2 + C 8 + CE C max 8, 2. 4 σ X u 2 du 2 Obeve ha he conan in he igh hand of he inequaliie 19 and 2 do no depend on, x. Theefoe, applying a ighne cieion ee, e.g. [16, Cap. I] he deied concluion follow. ii Taking ino accoun he concluion i and he Pokhoov heoem, we have ha if n, x n, x, a n, hen hee exi a ubequence, ill denoed by n, x n, uch ha X n := X n,xn X, K n := K n,xn K, a n. u u I emain o idenify he limi, i.e. X ==== law X,x and K ==== law K,x. Since X n, K n, W n n i a C[, T ], R d R d R d igh equence, by he Skoohod Theoem, we can chooe a pobabiliy pace ˆΩ, ˆF, ˆP which can be aken in fac a [, 1], B[,1], µ whee µ i he Lebegue meaue, and ˆX n, ˆK n, Ŵ n, ˆX, ˆK, Ŵ defined on hi pobabiliy pace, uch ha ˆX n, ˆK n, Ŵ n ==== law X n, K n, W n law, ˆX, ˆK, Ŵ ==== X, K, W and ˆX n, ˆK n, Ŵ n a.. ˆX, ˆK, Ŵ, a n. 14

15 Then, uing [16, Popoiion 2.15], we deduce ha Ŵ n i an F Ŵ n, ˆX n -Bownian moion, Ŵ i an F Ŵ, ˆX -Bownian moion and ogehe wih he Lebegue heoem we infe ha, fo all q 1, E up n ˆV ˆV q, a n, [,T ] whee If V n i defined by ˆV n := x + ˆV := x + V n := x + b ˆX n d + b ˆX d + bx n d + σ ˆX n dŵ n and σ ˆX dŵ, [, T ]. σx n dw n hen X n + K n = V n, P-a..; uing [16, Coollay 2.14], we ee ha and X n, K n, W n, V n law ==== ˆX n, ˆK n, Ŵ n, ˆV n on C[, T ], R d R d R d R d which yield, paing o he limi, ha ˆX n + ˆK n = ˆV n, a.. ˆX + ˆK = ˆV, a.. Then he coupled poce ˆX, ˆK i a oluion of 5 coeponding o he iniial daa, x. Taking ino accoun he uniquene in law of he oluion X,x, K,x [,T ] ee Remak 8 we deduce ha he whole equence X n, K n [,T ] convege o he poce X,x, K,x [,T ], and heefoe he coninuiy wih epec o, x follow. 4 BSDE and nonlinea Neumann bounday poblem Le u now conide he pocee X,x,n, k,x,n T and X,x, k,x T given by elaion 5-8, fo, x [, T ] Ḡ. In ode o give he poof of Theoem 4 we aociae he following genealized backwad ochaic diffeenial equaion BSDE fo ho on [, T ]: Y,x = gx,x T + f, X,x, Y,x d U,x dm X,x and epecively he BSDE coeponding o he oluion of 7 h, X,x, Y,x dk,x, 21 Y,x,n = gx,x,n T + f, X,x,n, Y,x,n d h, X,x,n U,x,n dm X,x,n, Y,x,n dk,x,n, 22 15

16 whee M X,x := σx,x dw, M X,x,n := σx,x,n dw 23 ae he maingale pa of he efleced diffuion poce X,x and X,x,n epecively. We aume fo impliciy ha he pocee X,x,n, K,x,n [,T ] and X,x, K,x [,T ] ae conideed on he canonical pace. We ecall ha he coefficien f, g and h aify aumpion A 3. Then, given he pocee X,x,n, k,x,n [,T ] and X,x, k,x [,T ], hi aumpion enue ee [15] he exience and he uniquene fo he couple Y,x,n, U,x,n [,T ] and Y,x, U,x [,T ] epecively. Aguing a in [2], one can eablih he following eul. Popoiion 13 Le he aumpion A 1 A 3 be aified. Le Y,x,n, U,x,n [,T ] and Y,x, U,x [,T ] be he oluion of he BSDE 22 and 21, epecively. Then Y,x,n, M,x,n, H,x,n Y,x, M,x, H,x, S S S whee M,x,n := M,x := U,x,n dm X,x,n, H,x,n := and M X,x,n and M X,x ae defined by 23. U,x dm X,x, H,x := Moeove, we have ha lim Y,x,n n = Y,x. h, X,x,n h, X,x, Y,x,n, Y,x dk,x dk,x,n, 24 Remak 14 The oluion poce Y,x [,T ] i unique in law. Indeed, following [3, Theoem 3.4], i can be poven ha, ince he coefficien b and σ aify he aumpion A 1 A 2 and he oluion poce ha he Makov popey, hee exi a deeminiic meauable funcion u uch ha he oluion Y,x = u, X,x, [, T ] dp d a.. The concluion follow by Popoiion 7 and he uniquene a a ong oluion of Y. In he following, we exend X,x, K,x o [, T ] a in 18 and Y,x, U,x by denoing Y,x := Y,x, U,x := and M X,x :=, [,. Popoiion 15 Le n, x n, x, a n. Then hee exi a ubequence nk, x nk k N uch ha Y n k,xn k Y,x. S Poof. The poof will follow he echnique ued in [2, Theoem 3.1] ee alo [14, Theoem 6.1]. I i clea ha Y n,xn = gx n,xn T + 1 [n,t ] f, X n,xn, Y n,xn d h, X n,xn 16 U n,xn dm Xn,xn, Y n,xn dk n,xn, [, T ]. 25

17 Fo he poof we will adap he ep fom he poof of [2, Theoem 3.1]. Sep 1. The oluion aify he boundedne condiion fo he poof ee, e.g., [15, Popoiion 1.1]: E up [,T ] Y n,xn 2 +E E up [,T ] Y,x 2 +E U n,xn σx n,xn 2 d C, [, T ], n N U,x σx,x 2 d C, [, T ], whee C > i a conan no depending on n. Sep 2. To obain he ighne popey wih epec o he S-opology i i ufficien o compue he condiional vaiaion CV T ee definiion 13 of he pocee Y n,xn, M n,xn and H n,xn epecively; we ecall he noaion 24 fo he quaniie M n,xn and H n,xn. A in [2, Theoem 3.1], afe ome eay compuaion we deduce ha hee exi a conan C > independen of n, uch ha CV T Y n,x n + E up [,T ] Y n,xn +E up [,T ] M n,xn +CV T H n,x n +E up [,T ] H n,xn C, n N. Sep 3. The above condiion enue ee [9, Appendix A] o [2, Theoem 3.5] he ighne of he equence Y n,xn, M n,xn, H n,xn wih epec o he S-opology. Theefoe hee exi a ubequence, ill denoed by Y n,xn, M n,xn, H n,xn, and a poce Ȳ, M, H D [, T ], R k 3 uch ha X n,x n, K n,xn, Y n,xn, M n,xn, H n,xn X,x, K,x, Ȳ, M, H, 26 U U S S S weakly on C[, T ], R d 2 D[, T ], R k 3. In ode o pa o he limi in 25 we ue he coninuiy of f, [5, Coollay 2.11], he Lipchizianiy of h, k n,xn k,x, u and we apply [2, Lemma 3.3]; we pecie ha he concluion of hi lemma i ill ue in he poin T, hence hee exi a counable e Q [, T uch ha, fo any [, T ] \ Q, Ȳ = gx,x T + 1 [,T ] f, X,x, Ȳd M T M h, X,x, Ȳdk. Since he pocee Ȳ, M and H ae càdlàg, he above equaliy hold ue fo any [, T ]. We menion ha M X,x and M ae maingale wih epec o he ame filaion. Indeed, M i F X,x,Ȳ, M -adaped and, moeove, M i an F X,x,Ȳ, M-maingale fo he poof ee [9, Lemma A.1]. Le now ψ be a bounded coninuou mapping fom C[, ], R d D[, ], R k 2, ϕ C R d and L = 1 σ σ 2 ij + 2 x i,j i x j i 17 b i x i

18 be he infinieimal geneao of he diffuion poce X n,xn. Fom Iô fomula we obain ha ϕx n,xn ϕx n i a maingale. Theefoe, fo any 1 < 2 T, [ E LϕX n,xn d + n ψ 1 X n,x n, Y n,xn, M n,xn ϕx n,xn 2 I can be poved, uing 26, ha [ lim n E ϕx n,xn n 2 n ϕx n,xn 1 2 n + 1 n ψ 1 X n,x n, Y n,xn, M n,xn ϕx n,xn 2 [ = E ψ 1 X,x, Ȳ, M ϕx,x 2 ϕx,x 1 On he ohe hand, 2 1 lim n E [ψ 1 X n,x n, Y n,xn, M n,xn 2 n 1 n [ψ 1 X,x, Ȳ, M 2 ] = E by [21, Popoiion 3.4]. Theefoe 1 ϕx,x ϕx n,xn 1 n dk n,xn dk n,xn 2 n ϕx n,xn 1 ] LϕX,x d. dk,x [ E ψ 1 X,x, Ȳ, M ϕx,x 2 ϕx,x , ϕx n,xn 2 1 ϕx,x 1 n LϕX n,xn d ] =, n. ] LϕX n,xn d ] dk n,xn ] LϕX,x d dk,x ] =. Uing Iô fomula we ee ha E [ψ 1 X,x, Ȳ, M 2 1 ] ϕx,x dm X,x i alo F X,x,Ȳ, M- and heefoe M X,x i a F X,x,Ȳ, M-maingale. Now ince Y,x and U,x ae F X,x -adaped, M,x := maingale. = U,x dm X,x 18

19 Le u ake 1 2 T. Iô fomula yield Y,x 1 ince Y,x 2 Ȳ [M,x M] 2 [M,x M] 1 = Y,x 2 Ȳ Y,x Ȳ, f, X,x, Y,x f, X,x, Ȳ d Y,x Y,x Ȳ, h, X,x Ȳ, dm,x M Ȳ α β Y,x Ȳ, dm,x 2 1, Y,x h, X,x, Ȳ da,x Y,x 2 Ȳ 2 d + A,x 2 Y,x Ȳ, dm,x M. 1 M i a F X,x,Ȳ, M-maingale. Hence, fom a genealized Gonwall lemma ee, e.g., [12, Lemma 12], by aking 2 = T, we deduce he idenificaion Y,x = Ȳ and M,x = M. 4.1 Poof of Theoem 4 Le u denoe u n, x := Y,x,n and u, x := Y,x. 27 Hence, u n and u ae deeminiic funcion ince Y,x,n i adaped wih epec o he filaion geneaed by X,x,n and Y,x i adaped wih epec o he filaion geneaed by X,x. Fi we pove ha he funcion u n : [, T ] R d R d and u : [, T ] Ḡ Rd defined by 27 ae coninuou. We will how only ha he funcion u i coninuou. Le n, x n, x [, T ] Ḡ, a n. Fom he poof of Popoiion 15, we can exac a ubequence ill denoed n, x n, uch ha X n,x n, K n,xn, Y n,xn, M n,xn X,x, K,x, Y,x, M,x. U U S S We know fom [2, Lemma 3.3] applied fo = T, ha h, X n,xn, Y n,xn dk n,xn Uing [5, Remak 2.4], we ee ha M n,xn T Hence we can pa o he limi in u n, x n = Y n,xn n = Y n,xn = gx n,xn T + h, X,x, Y,x dk,x in law, a n. M,x T in law, ince M n,xn M,x. S 19 1 [n,t ] f, X n,xn h, X n,xn, Y n,xn d M n,xn T, Y n,xn dk n,xn

20 and, a in he he poof of Popoiion 15, we deduce ha he limi of u n, x n i gx,x T + 1 [,T ] f, X,x, Y,x d M,x T h, X,x, Y,x dk,x = Y,x = Y,x = u, x. I i eay o how ha, even if b and σ ae only coninuou funcion, he poof fom [15, Theoem 4.3] ee alo [14, Theoem 3.2] fo noneflecing cae ill wok in ode o how ha he funcion u n and u defined by 27 ae vicoiy oluion of he PDE 3 and 4 epecively. Finally, a a conequence of Popoiion 13 we deduce he oluion u of he deeminiic yem 4 i appoximaed by he funcion u n, i.e. lim n un, x = u, x,, x [, T ] Ḡ. Acknowledgemen. The auho would like o hank he efeee fo he emak and commen which have led o a ignifican impovemen of he pape. The auho L. Maiciuc and A. Zălinecu hank he IMATH laboaoy of Univeié du Sud Toulon Va fo i hopialiy. Refeence [1] K. Bahlali, A. Elouaflin, E. Padoux, Homogenizaion of emilinea PDE wih diconinuou aveaged coefficien, Eleconic Jounal of Pobabiliy 14 29, [2] B. Boufoui, J. van Caeen, An appoximaion eul fo a nonlinea Neumann bounday value poblem via BSDE, Sochaic Poce. Appl , [3] N. El Kaoui, Backwad ochaic diffeenial equaion: a geneal inoducion, Backwad ochaic diffeenial equaion ed. N. El Kaoui, L. Mazliak, 7-26, Piman Reeach Noe in Mahemaic Seie, 364, Longman [4] N. El Kaoui, Poceu de éflexion dan R n, Séminaie de Pobabilié IX, Lecue Noe in Mah. no , , Spinge, Belin. [5] A. Jakubowki, A non-skoohod Topology on he Skoohod pace, Eleconic Jounal of Pobabiliy , [6] I. Kaaza, S.E. Sheve, Bownian moion and ochaic calculu, Spinge-Velag, New- Yok, [7] W. Łaukajy, L. Słomińki, Penalizaion mehod fo eflecing ochaic diffeenial equaion wih jump, Soch. Soch. Rep , [8] W. Łaukajy, L. Słomińki, Penalizaion mehod fo he Skookhod poblem and eflecing SDE wih jump, Benoulli, fohcoming pape, 213. [9] A. Lejay, BSDE diven by Diichle poce and emi-linea Paabolic PDE. Applicaion o Homogenizaion, Sochaic Poce. Appl , [1] P. L. Lion, J. L. Menaldi, A. S. Szniman, Conucion de poceu de diffuion éfléchi pa pénaliaion du domaine, C.R. Acad. Sci. Pai Sé. I Mah ,

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