Transcription

1 Thi aicle wa oiginally publihed in a jounal publihed by levie, and he aached copy i povided by levie fo he auho benefi and fo he benefi of he auho iniuion, fo non-commecial eeach and educaional ue including wihou limiaion ue in inucion a you iniuion, ending i o pecific colleague ha you know, and poviding a copy o you iniuion adminiao. All ohe ue, epoducion and diibuion, including wihou limiaion commecial epin, elling o licening copie o acce, o poing on open inene ie, you peonal o iniuion webie o epoioy, ae pohibied. Fo excepion, pemiion may be ough fo uch ue hough levie pemiion ie a: hp://

2 Sochaic Pocee and hei Applicaion Backwad ochaic diffeenial euaion wih ingula eminal condiion Abac A. Popie L.A.T.P., 39 ue F. Jolio Cuie, 3453 Maeille cedex 3, Fance Received 8 June 24; eceived in evied fom 6 Febuay 26; acceped 24 May 26 Available online 23 June 26 In hi pape, we ae concened wih backwad ochaic diffeenial euaion BSD fo ho of he following ype: Y = ξ Y Y d Z db, whee i a poiive conan and ξ i a andom vaiable uch ha Pξ = + >. We udy he link beween hee BSD and he aociaed Cauchy poblem wih eminal daa g, whee g = + on a e of poiive Lebegue meaue. c 26 levie B.V. All igh eeved. Keywod: Backwad ochaic diffeenial euaion; Non-inegable daa; Vicoiy oluion of paial diffeenial euaion Conen. Inoducion and main eul Appoximaion and conucion of a oluion Poof of i and ii fom Theoem Poof of iii The cae ξ bounded away fom zeo The cae of ξ non-negaive Auho' peonal copy Coeponding adde: cole Polyechniue, CMAP, Roue de Palaieau, 928 Palaieau, Fance. Tel.: ; fax: mail adde: popie@cmap.polyechniue.f /$ - ee fon mae c 26 levie B.V. All igh eeved. doi:.6/j.pa

3 A. Popie / Sochaic Pocee and hei Applicaion Poof of iv Coninuiy of Y a T Lowe bound and aympoic behaviou in a neighbouhood of T Coninuiy a ime T : The fi ep Coninuiy when > When aumpion H3, B, D and ae aified A peliminay eul Coninuiy wih H3, B, and D Minimal oluion Paabolic PD, vicoiy oluion Minimal oluion Regulaiy of he minimal oluion Sign aumpion on he eminal condiion ξ Acknowledgemen Refeence Inoducion and main eul Backwad ochaic diffeenial euaion BSD fo ho in he emainde ae euaion of he following ype: Y = ξ + f, Y, Z d Z db, T, whee B T i a andad d-dimenional Bownian moion on a pobabiliy pace Ω, F, F T, P, wih F T he andad Bownian filaion. The funcion f : [, T ] R n R n d R n i called he geneao, T he eminal ime, and he R n -valued F T -adaped andom vaiable ξ a eminal condiion. The unknown ae he pocee {Y } [,T ] and {Z } [,T ], which ae euied o be adaped wih epec o he filaion of he Bownian moion: hi i a cucial poin. Such euaion, in he non-linea cae, wee inoduced by Padoux and Peng in 99 in []. They gave he fi exience and uniuene eul fo n-dimenional BSD unde he following aumpion: f i Lipchiz coninuou in boh vaiable y and z and he daa, ξ, and he poce, { f,, } [,T ], ae uae inegable. Since hen, BSD have been udied wih gea inee. In paicula, many effo have been made o elax he aumpion on he geneao and he eminal condiion. Fo inance Biand e al. in [2] poved an exience and uniuene eul unde he following aumpion: f i Lipchiz in z, coninuou and monoone in y, he daa, ξ, and he poce, { f,, } [,T ], ae in L p fo p >. The eul i ill ue fo p = wih anohe echnical condiion. The eul of [2] ae he aing poin of hi wok, whee we conide a one-dimenional BSD wih a non-linea geneao: Y = ξ Auho' peonal copy Y Y d Z db wih R +. The geneao f y = y y aifie all he aumpion of he Theoem 4.2 and of [2]: f i coninuou on R, doe no depend on z, and i monoone: y, y R 2, y y f y f y. 2

4 26 A. Popie / Sochaic Pocee and hei Applicaion Theefoe hee exi a uniue oluion Y, Z fo ξ L p Ω fo p we do no make pecie he cla of Y, Z in which uniuene hold. The oluion of he elaed odinay diffeenial euaion, namely y = y y, y T = x, i given by he fomula: ignx T + x whee ignx = if x < and ignx = if x >. We emak ha, even if x i eual o + o, y i finie on [, T [. Numeou heoem ee fo inance [3,4] and [5] how he connecion beween BSD aociaed wih ome fowad claical ochaic diffeenial euaion SD fo ho o fowad backwad yem and oluion of a lage cla of emi-linea and uailinea paabolic and ellipic paial diffeenial euaion. Thoe eul may be een a a non-linea genealizaion of he celebaed Feynman Kac fomula. The BSD i conneced wih he following ype of PD ee [5]: { u, x + Lu, x u, x u, x =,, x [, T [ R m ; 3 ut, x = gx, x R m. whee L i he infinieimal geneao: L = σ σ 2 i j + 2 x i, j i x j i b i = x i 2 Tace σ σ D 2 + b ; 4 whee in he e of he pape, and D 2 will alway denoe epecively he gadien and he Heian maix w... he pace vaiable. Indeed Baa and Piee [6], Macu and Veon [7] have given exience and uniuene eul fo hi PD. In [7] i i hown ha evey poiive oluion of 3 poee a uniuely deemined final ace g which can be epeened by a couple S, µ whee S i a cloed ube of R m and µ a non-negaive Radon meaue on R = R m \ S. The final ace can alo be epeened by a poiive, oue egula Boel meaue ν, and ν i no neceay locally bounded. The wo epeenaion ae elaed by: { A R m νa = if A S, A Boel, νa = µa if A R. The e S i he e of ingula final poin of u and i coepond o a blow-up e of u. Fom he pobabiliic poin of view Dynkin and Kuzneov [8] and Le Gall [9] have poved imila eul fo he PD 3 in he cae < : hey ue he heoy of upepocee. In hi pape we ae concened wih a eal F T -meauable andom vaiable uch ha: Pξ = + o ξ = >. 5 Auho' peonal copy Thu ξ i no in L Ω. We give a new definiion of a oluion of he BSD. Definiion. Le u have > and ξ an F T -meauable andom vaiable. We ay ha he poce Y, Z i a oluion of he BSD Y = ξ Y Y d Z db

5 A. Popie / Sochaic Pocee and hei Applicaion if Y, Z veifie: D fo all < T : Y = Y Y Y d Z db ; D2 fo all [, T [, up Y 2 + Z 2 d < + ; D3 P-a.. lim T Y = ξ. The ouline of he pape i a follow. xcep in Secion 5, ξ i uppoed o be non-negaive. In he fi ecion, wihou any fuhe aumpion on ξ, we conuc a poce Y, Z which aifie all condiion fo being a oluion in he ene of he peviou definiion, excep he la one. Moe peciely we eablih in Secion he Theoem 2. Le ξ a.. Thee exi a pogeively meauable poce Y, Z, wih value in R + R d, uch ha: D and D2 ae aified: a fo all [, T [, and all : Y = Y Y + d Z db, b fo all [, T [, Y, and Z T 2 d ; ii T 2 2 Y i coninuou on [, T [, he limi of Y, when goe o T, exi and: lim Y ξ, P a..; iii T 3 Z aifie alo: 2 T 2/ Z 2 d 8. iv Noe ha hi eul doe no pecify whehe Y aifie lim T Y = ξ. The exience of hi poce Y, Z i obained by appoximaion. Fo evey inege n, le Y n, Z n be he oluion of he BSD wih eminal condiion ξ n L Ω. Y, Z i he limi of hi euence Y n, Z n n N. In Secion 2, we udy ou poce Y in he neighbouhood of T. In a fi pa we make pecie he aympoic behaviou of Y on he blow-up e. Popoiion 3. On he e {ξ = + } / lim T / Y = a.. 6 In he econd pa we will pove he coninuiy of Y unde onge condiion on ξ. So fa we only have he ineualiy: Auho' peonal copy lim Y ξ = Y T. T Wihou addiional aumpion, we wee unable o pove he convee ineualiy. The fi hypohei on ξ i he following: i ξ = gx T, H

6 28 A. Popie / Sochaic Pocee and hei Applicaion whee g i a meauable funcion defined on R m wih value in R + uch ha he e F = {g = + } i cloed; and whee X T i he value a = T of a diffuion poce o moe peciely he oluion of a ochaic diffeenial euaion fo ho SD: X = x + b, X d + σ, X db, fo [, T ]. 7 We will alway aume ha b and σ ae defined on [, T ] R m, wih value epecively in R m and R m d, ae meauable w... he Boelian σ -algeba, and ha hee exi a conan K >.. fo all [, T ] and fo all x, y R m R m : Lipchiz condiion: b, x b, y + σ, x σ, y K x y ; 2 Gowh condiion: b, x + σ, x K + x. The econd hypohei on ξ i: fo all compac e K R m \ F gx T K X T L Ω, F T, P; R. Moeove in he cae 2 we will add he following condiion: σ and b ae bounded: hee exi a conan K.. Auho' peonal copy L G H2, x [, T ] R m, b, x + σ, x K ; B 2 he econd deivaive of σ σ belong o L : 2 σ σ L [, T ] R m. x i x j 3 σ σ i unifomly ellipic, i.e. hee exi λ >.. fo all, x [, T ] R m : y R m, σ σ, xy.y λ y 2. 4 g i coninuou fom R m o R + and: M, g i a Lipchiz funcion on he e O M = { g M}. H3 Theoem 4 Coninuiy of Y a T. Unde he aumpion H, H2, L and G, and wih eihe > 2 o H3, B, D and, Y i coninuou a ime T lim Y = ξ, T P a.. In Secion 3, we pove ha ou oluion i he minimal oluion. Theoem 5 Minimal Soluion. The oluion Y, Z obained in Theoem 2 and 4 i minimal: if Ȳ, Z i anohe non-negaive oluion in he ene of Definiion, hen fo all [, T ], P-a..: Ȳ Y. Moeove we pove ha: / Ȳ. T D

7 A. Popie / Sochaic Pocee and hei Applicaion The fouh ecion povide connecion beween hi conuced oluion of he BSD and vicoiy oluion of elaed emi-linea PD 3. Fo all, x [, T ] R m, we denoe by X,x he oluion of he SD: X,x = x + b, X,x d + σ, X,x db, fo [, T ], 8 and X,x = x fo [, ]. The coefficien b and σ veify alway he aumpion of he econd ecion. A a final condiion of he BSD, we ake gx,x T, whee g i a funcion defined fom R m o R + uch ha he e F = {g = + } i cloed and uch ha he condiion H2 i veified. Moeove g i uppoed o be coninuou fom R m o R +. Theoem 6 Vicoiy Soluion. The minimal oluion of he BSD wih ξ = gx,x T i denoed by Y,x. Then Y,x i a deeminiic numbe and if we e u, x = Y,x, hen u i lowe emi-coninuou fom [, T ] R m o R + wih u,. < + wheneve < T and u i a diconinuou vicoiy oluion of he PD 3. We pove ha he peviou oluion i minimal among all non-negaive vicoiy oluion. Theoem 7 Minimal Vicoiy Soluion. If v i a non-negaive vicoiy oluion of he PD 3, hen fo all, x [, T ] R m : u, x v, x. We give ufficien condiion o have u coninuou on [, T ] R m. In Secion 5, we exend ou eul when hee i no ign aumpion on ξ. Theoem 8. Le ξ be an F T -meauable andom vaiable, poibly negaive, uch ha: Pξ = + o ξ = >. 5 Moeove ξ aifie ξ = gx T, wih g : R m R {, + } uch ha: he wo e {g = + } and {g = } ae cloed; 2 he condiion H2 i veified: fo all compac e K R m \ {g = ± } gx T K X T L Ω, F T, P; R; 3 if 2, H3 hold. The coefficien of he SD 7 aify L, G and in he cae < 2, hey alo veify B, D and. Thee exi a poce Y, Z which i a oluion of he BSD Y = ξ Auho' peonal copy Y Y d in he ene of Definiion. Z db In he coninuaion, unimpoan conan will be denoed by C. H

8 22 A. Popie / Sochaic Pocee and hei Applicaion Appoximaion and conucion of a oluion Fom now and in Secion 2 4, ξ aifie: Pξ = and Pξ = + >. In hi ecion we pove Theoem 2. Fo >, le u conide he funcion f : R R, defined by f y = y y. f i coninuou and monoone ineualiy 2. By Theoem 2.2 and xample 3.9 in [5], fo ζ L 2 F T, he BSD wih ζ a eminal condiion ha a uniue oluion Y, Z wih value in R R d, uch ha Y i coninuou on [, T ] and ha: up T Y 2 + Z 2 d <. 9 Remak ha a aighfowad applicaion of he Tanaka fomula ee [] and of he compaion Theoem 2.4 in [5] how ha: / [, T ], Y. T Fo evey n N, we inoduce ξ n = ξ n. ξ n belong o L 2 Ω, F T, P; R. We apply he peviou eul wih ξ n a he final daa, and we build a euence of andom pocee Y n, Z n which aify and 9. Fom he compaion Theoem 2.4 in [5], if n m, ξ n ξ m m, which implie fo all in [, T ], a.., Y n Y m T + m. T We define he pogeively meauable R-valued poce Y, a he inceaing limi of he euence Y n n : [, T ], Y = lim n + Y n. 2 Then we obain fo all T Y. 3 T In paicula Y i finie on he ineval [, T [ and bounded on [, T δ] fo all δ >... Poof of i and ii fom Theoem 2 Auho' peonal copy Hee we will pove he popeie i and ii. Le δ > and [, T δ]. Fo all, Iô fomula lead o he eualiy: Y n Y m Z n Z m 2 d = Y n Y m 2 Y n Y m Z n Z m db + 2 Y n Y m f Y n f Y m d

9 A. Popie / Sochaic Pocee and hei Applicaion Y n Y m 2 2 Y n Y m Z n Z m db, fom he monooniciy of f ineualiy 2. Thank o 9: Y n Y m Z n Z m db =. Fom he Bukholde Davi Gundy ineualiy, we deduce he exience of a univeal conan C wih: up Y n Y m 2 + Z n Z m 2 d C Y n Y m 2. 4 Fom he eimae 3, fo T δ, Y n and Y δ /. Since Y n δ / convege o Y a.., he dominaed convegence heoem and he peviou ineualiy 4 imply: fo all δ >, Z n n i a Cauchy euence in L 2 Ω [, T δ]; R d, and convege o Z L 2 Ω [, T δ]; R d, 2 Y n n convege o Y unifomly in mean uae on he ineval [, T δ]; in paicula Y i coninuou on [, T [, 3 Y, Z aifie fo evey < T, fo all : Y = Y Y + d Z db. The elaion i i poved. Since Y i malle han /T / by 3, and ince Z L 2 Ω [, T δ]; R d, applying he Iô fomula o Y 2, wih < T and, we obain: Y 2 + Z 2 d = Y 2 2 T 2 Y Z db Auho' peonal copy Y Z db, Y f Y d again hank o he monooniciy of f ineualiy 2. Fom 3, ince Z L 2 [, ] Ω, we have: Y Z db =. Theefoe, we deduce: [, T [,.2. Poof of iii Z 2 d. ii T 2 Fom now, he poce Y i coninuou on [, T [ and we define Y T = ξ. The main difficuly will be o pove he coninuiy a ime T. I i eay o how ha: i ξ lim inf T Y. Indeed, fo all n and all [, T ], Y n Y, heefoe: ξ n = lim inf Y n lim inf Y. 6 T T 5

10 222 A. Popie / Sochaic Pocee and hei Applicaion Thu, Y i lowe emi-coninuou on [, T ] hi i clea ince Y i he upemum of coninuou funcion. Wihou ohe aumpion on ξ, we ae unable o pove he coninuiy of Y a = T. Bu now we will how ha Y ha a limi on he lef a ime T. We will diinguih he cae when ξ i geae han a poiive conan fom he cae ξ non-negaive..2.. The cae ξ bounded away fom zeo We can how ha Y ha a limi on he lef a T, by uing Iô fomula applied o he poce /Y n. We pove he following eul: Popoiion 9. Suppoe hee exi a eal α > uch ha ξ α >, P-a.. Then Y = T + F Φ, T, 7 ξ whee Φ i a non-negaive upemaingale. Poof. Fom he compaion eul 2.4 of [5], fo evey n N and evey T : n Y n / / T + /α T + /α >. By he Iô fomula Y n = + Z n + T ξ n 2 2 Y n +2 d + = F + T ξ n + T F 2 The poce: Y n, Z n Y n + Auho' peonal copy Z n 2 Y n +2 d. Z n Y n + db i he maximal oluion in L [, T ] Ω L 2 [, T ] Ω; R d of he BSD: T Y = ξ n + + Z 2 2 Y /n d Z db. Indeed, fo hi BSD hee exi a maximal oluion U, V ee [], Theoem 2.3 and ince U /Y n, we can apply Iô fomula o he poce U /. We find ha U /, /V/U +/ aifie he BSD wih eminal value ξ n. Thu U / = Y n and he concluion follow. Le n m. Since ξ n ξ m, we obain fo all T : Y m Y n = F ξ m ξ n + T F Z m 2 T 2 Y m +2 d F Z n 2 Y n +2 d.

11 A. Popie / Sochaic Pocee and hei Applicaion Now: + Z m 2 F 2 T Y m +2 d F Z n 2 Y n +2 d [ ] [ F ξ m ξ n Y m ] Y n. Fo a fixed [, T ], he euence F ξ n and n Y n convege a.. and in L n dominaed convegence heoem. Then F T Z n 2 d convege a.. and in L and Y n +2 n we denoe by Φ he limi: Φ = lim n F Z n 2 Y n +2 d. We can alo emak ha: + T F Z n 2 2 Y n +2 d = F ξ n + T Y n, wih Y n /T /, o + T F Z n 2 2 Y n +2 d F ξ n α. Theefoe, Fo, Φ F F Z n 2 ξ Y n +2 d. Z n 2 Φ F Φ. Z n 2 Y n +2 d Y n +2 d F F Z n 2 Y n +2 d We deduce ha Φ <T i a non-negaive bounded upemaingale. Now fo all n N, Y n = T + + T F ξ n F Z n 2 2 Y n +2 d. Auho' peonal copy Fix < T. Taking he limi a n +, we deduce: Y = T + F ξ Φ. Fom he above expeion, Φ <T i igh-coninuou.

12 224 A. Popie / Sochaic Pocee and hei Applicaion Φ being a igh-coninuou non-negaive upemaingale, he limi of Φ a goe o T exi P-a.. and hi limi Φ T i finie P-a.., ince i i bounded by /α. The L -bounded maingale F ξ convege a.. o /ξ, a goe o T ; hen he limi of Y a T exi and i eual o: lim Y = T ξ Φ T. / If we wee able o pove ha Φ T i zeo a.., we would have hown ha Y T = ξ The cae of ξ non-negaive Now we ju aume ha ξ. We canno apply he Iô fomula o /Y n becaue we have no poiive lowe bound fo Y n. We will appoach Y n in he following way. We define fo n and m, ξ n,m by: ξ n,m = ξ n m. Thi andom vaiable i in L 2 and i geae han o eual o /m a.. The BSD, wih ξ n,m a eminal condiion, ha a uniue oluion Ỹ n,m, Z n,m. I i immediae ha if m m and n n hen: Ỹ n,m Ỹ n,m. A fo he euence Y n, we can define Ỹ m a he limi when n gow o + of Ỹ n,m. Tha limi Ỹ m i geae han Y = lim n + Y n. Bu fo m m, fo [, T ]: [ Ỹ Ỹ n,m = ξ n,m ] ξ n,m n,m + n,m Ỹ + d Ỹ n,m [ Z n,m ξ n,m ξ n,m ] Z n,m db [ Z n,m and aking he condiional expecaion given F : Ỹ n,m Ỹ n,m F ] Z n,m db, Auho' peonal copy ξ n,m ξ n,m m. 8 Recall ha Y n, Z n i he oluion of he BSD wih ξ n = ξ n a eminal daa. Thu we alo have: Ỹ n,m Y n F ξ n,m ξ n m. Leing m + in he la eimae lead o lim m + Ỹ n,m ineualiy 8: Ỹ n,m Y n m. = Y n a.. and uing he

13 Theefoe P-a..: Ỹ m up [,T ] A. Popie / Sochaic Pocee and hei Applicaion Y m. Since Ỹ m ha a limi on he lef a T, o doe Y..3. Poof of iv In ode o complee he poof of Theoem 2, we need o eablih he aemen iv. Popoiion. Suppoe hee exi a conan α > uch ha P-a.. ξ α. In hi cae: 2/ T 2/ Z 2 d 8. iv Poof. If ξ α, by compaion, fo all inege n and all [, T ]: Y n / T + /α >. Le δ > and θ : R R, θ : R R defined by: { { θx = x on [δ, + [, and θ x = x 2 on [δ, + [, θx = on ], ], θ x = on ], ], and uch ha θ and θ ae non-negaive, non-deceaing and in epecively C 2 R and C R. We apply he Iô fomula on [, T δ] o he poce θ T θy n, wih δ < T +/α / : o: θ δθy n T δ θ T θy n = 2 8 δ and ince Y n δ δ δ T /2 Y n /2 Z n db T 2 Y n 2 Y n T /2 Z n 2 δ Y n d 3/2 d T T /2 Z n 2 Y n d T /2 θy n 3/2 + T /2 2 Y n Z n /2 db + δ T /2 Y Y n 2 /2 n d T δ Auho' peonal copy /T / and T / Y n /, aking he expecaion we obain: ha i fo all n and all δ > : δ T /2 Z n 2 Y n 3/2 d θ T θy n //2, T /2 Z n 2 Y n 3/2 d 8//2.

14 226 A. Popie / Sochaic Pocee and hei Applicaion Now, ince /Y n T /, δ T 2/ Z n 2 d 8/ 2/, and leing δ and wih he Faou lemma, we deduce ha T 2/ Z 2 d 8/ 2/. Thi achieve he poof of he popoiion. Now we come back o he cae ξ. We canno apply he Iô fomula becaue we do no have any poiive lowe bound fo Y n. Bu we can appoximae he poce Y n, Z n by a euence of pocee Ỹ n,m, Z n,m a in he poof of he exience of a limi fo Y a ime T. Le u ecall ha we olve he BSD wih ξ n,m = ξ n m a eminal condiion. Fo all δ >, he Iô fomula, applied o he poce T. 2/ Ỹ n,m Y n 2, lead o he ineualiy: δ T 2/ Z n,m Z n 2 d δ 2/ Ỹ n,m T δ Y T n δ Auho' peonal copy δ T 2/ Ỹ n,m Y n 2 d. Le δ go o in he peviou ineualiy. We can do ha becaue fo all [, T ], Ỹ n,m Y n, which implie Ỹ n,m Y n 2 n 2, and becaue T. 2/ i inegable on he ineval [, T ]. Finally, uing he ineualiy: Ỹ n,m Y n m, we have: T 2/ Z n,m Z n 2 d 2 Theefoe, fo all ε > : T 2/ Z n 2 d m 2 T 2/ d = T 2/ m 2. T 2/ Z n,m 2 d + ε T 2/ Z n,m Z n 2 d T 2/ Z n,m Z n,m Z n d T 2/ Z n,m 2 d + + T 2/ Z n,m Z n 2 d ε 8 + ε/ 2/ + T 2/ +. ε We have applied he peviou eul o Z n,m. Now we le fi m go o + and hen ε go o, we have: T 2/ Z n 2 d 8/ 2/. 9 The eul follow by leing finally n go o. m 2

15 2. Coninuiy of Y a T A. Popie / Sochaic Pocee and hei Applicaion In hi ecion, ξ i ill uppoed non-negaive. We make pecie he behaviou of Y in a neighbouhood of T Popoiion 3 and we how he coninuiy of Y a T unde onge aumpion Theoem Lowe bound and aympoic behaviou in a neighbouhood of T Now we conuc an adaped poce which i malle han Y. Lemma. Fo < T, P-a.. / Y F T +. ξ Remak 2. The igh hand ide i obained hough he following opeaion: fi, we olve he odinay diffeenial euaion y = y + wih ξ a eminal condiion; hen we pojec hi oluion on he σ -algeba F. Poof. Le n N and conide fo T : / Γ n = F T +. ξ n Γ n i well defined becaue he em in he condiional expecaion i bounded by n. We have: + / T T + ξ n = ξ n d. T + So Γ n veifie: Γ n = F T ξ n d + T + ξ n = F ξ n Γ n + d U n d, wih U n he adaped and bounded by n + poce: U n = F ξ n Auho' peonal copy + T + ξ n Γ n + ; he Jenen ineualiy + > howing ha U n, fo T. Then, he compaion Theoem 2.4 in [5] allow u o conclude ha, fo all [, T ], a.. Γ n Y n Y.

16 228 A. Popie / Sochaic Pocee and hei Applicaion We hen deduce fom he monoone convegence heoem: / lim Γ n n + = F T + ξ = Γ. We now eablih he: Popoiion 3. On he e {ξ = + }, limt / Y = T /, a.. 6 Poof. Indeed, [ Γ = F ξ ] / ξ< + T ξ + T / F ξ= [ F ξ ] / ξ< + T ξ + T / F ξ=. Then, [ T / Γ T / F ξ ] / ξ< / + T ξ + F ξ=. The fi em, in he igh hand ide, convege o on he e {ξ = + } and he econd convege o / /. Indeed, we have: ξ + T ξ T and we can apply he maingale convegence heoem. Since Y i bounded fom above by /T /, hi achieve he poof Coninuiy a ime T : The fi ep We now wan o pove Theoem 4, i.e. ξ lim up T Y. Recall ha we aleady know ha he limi of Y a T exi a.. Fom he ineualiy 5, we ju have o how ha on he e {ξ < + }, Auho' peonal copy ξ lim Y = lim inf Y. T T The main difficuly hee i o find a good uppe bound of Y. We hall ue a mehod widely inpied by he aicle of Macu and Véon [7] and moe peciely by he poof of Lemma 2.2 page 45. We y o adap hi mehod o ou cae. We make onge aumpion on ξ. Fom now and fo he e of hi pape, we uppoe ha he condiion H, H2, L and G hold.

17 A. Popie / Sochaic Pocee and hei Applicaion Le ϕ be a funcion in he cla C 2 R m wih a compac uppo. Le Y, Z be he oluion of he BSD wih he final condiion ζ L 2 Ω. Fo any [, T ]: Y ϕx = Y ϕx + + = Y ϕx + + ϕx [ Y Y ] d + Z. db + Y dϕx Z. ϕx σ, X d Y LϕX d + ϕx Y Y d + whee L i he opeao defined by 4. Taking he expecaion: Y ϕx = Y ϕx + + Auho' peonal copy Z. ϕx σ, X d Y ϕx σ, X + ϕx Z. db ϕx Y Y d Z. ϕx σ, X d + Y LϕX d. 2 The idea i o ue he elaion 2 wih a uiable funcion ϕ. The e F c = {g < + } i open in R m. Le U be a bounded open e wih a egula bounday and uch ha he compac e U i included in F c. We denoe by Φ = Φ U a funcion which i uppoed o belong o C 2 R m ; R + and uch ha Φ i eual o zeo on R m \U, i poiive on U. Le α be a eal numbe uch ha α > 2 + /. Fo n N, le Y n, Z n be he oluion of he BSD wih he final condiion g nx T. The eualiy 2 become fo T : YT n Φα X T = Y n Φα X + + Φ α X Y n + d + Z n. Φα X σ, X d Y n LΦα X d. 2 Uing 2, we will fi pove ha fo evey eal α > 2 + / and fo evey n: Y n + Φ α X d C <, whee he conan C i independen of n. In paicula, we will have o find a bound fo he em conaining Z in he igh hand ide of he euaion 2. We know how o conol hi em in he wo cae of Theoem 4: we uppoe ha eihe > 2 o H3, B, D and ae aified. Thank o he Faou lemma, Y + Φ α X will belong o L Ω [, T ]. Then, we will deduce ha he limi a goe o T of Y i le han o eual o ξ a.. on he e {ξ < + }.

18 23 A. Popie / Sochaic Pocee and hei Applicaion Coninuiy when > 2 In hi ecion, we will uppoe ha > 2. In ha cae, we can eaily conol he em Z n. Φα X σ, X d. Uing he Cauchy Schwaz ineualiy, we obain: Z n. Φα X σ, X d Z n 2 T 2/ 2 d Fom he ineualiy 9 of he fi ecion: Z n 2 T 2/ d 8 And he econd em Φ α X σ, X 2 T 2/ d 2. Φ α X σ, X 2 T 2/ d i finie if > 2. Indeed, ecall ha Φ i compacly uppoed and α > 2. Hence he numeao i bounded fo all, x [, T ] R m : Φ α xσ, x 2 α 2 Φ 2α x Φx 2 K + x 2 C. Theefoe, hee exi a conan C = C, Φ, α, σ uch ha fo all [, T ] and n N: Z n. Φα X σ, X d C. 22 The em Y n LΦα X d can be bounded uing he Hölde ineualiy. Le p be uch ha /p + / + = o p = + / and p/ + = p. We will pove ha Φ αp LΦ α p L [, T ] R m. Uing he gowh condiion G on σ, we have: Φ αp TaceD 2 Φ α σ σ,. p CΦ αp D 2 Φ α p + x 2 p, Auho' peonal copy 2. and D 2 Φ α = αφ α D 2 Φ + αα Φ α 2 Φ Φ Φ αp D 2 Φ α p 2 p α p Φ α p D 2 Φ p + αα p Φ α 2p Φ 2p.

19 A. Popie / Sochaic Pocee and hei Applicaion Hence: Φ αp TaceD 2 Φ α σ σ,. p CΦ α 2p Ψ whee Ψ i he coninuou funcion on R m : [ Ψ = α p Φ p D 2 Φ p + αα p Φ 2p] [ + x 2p]. Since α > 2 + / = 2 p and ince Φ ha a compac uppo, Φ αp TaceD 2 Φ α σ σ,. p L [, T ] R m. 23 Fo he econd em Φ αp Φ α.b,. p, we have: Φ α = αφ α Φ Φ αp Φ α p = α p Φ α p Φ p. Since α > 2p and b, x K + x, Theefoe, Φ αp Φ α.b,. p 2 p K p Φ αp Φ α p + x p = 2 p K α p Φ α p Φ p + x p. Φ αp Φ α.b,. p L [, T ] R m. 24 Since LΦ α p 2 p Φ α.b,. p + 2 p TaceD 2 Φ α σ σ,. p, we apply he Hölde ineualiy: Y n X [ Φ α X Y n [ C Φ α X Y n +] [ + Φ αp X LΦ α X p] /p +] +, and he conan C depend only on, b, σ, Φ and α, no on n, o on. Finally, we have: Y n LΦα X [ d C Φ α X Y n + d [ C We come back o he. 2 fo = : Φ α X Y n +] ] + + d. 25 Auho' peonal copy Y n T Φα X T = Y n Φα X + + Φ α X Y n + d + Z n. Φα X σ, X d Y n LΦα X d. Recall ha YT nφα X T gx T Φ α X T ; ince Φ i eual o zeo ouide a compac e included in F c = {g < + }, uing he condiion H2 he lef hand ide of he peviou eualiy i bounded by a conan independen of n.

20 232 A. Popie / Sochaic Pocee and hei Applicaion In he igh hand ide, uing he ineualiy 22, we deduce ha he fi wo em ae alo bounded. Thu, we have: Φ α X Y n + d + which implie wih he ineualiy 25: [ Φ α X Y n + d C Y n LΦα X d C, Φ α X Y n ] + + d } The e {x R +, x Cx + C i bounded. Theefoe, we deduce ha: Φ α X Y n + d C. Hence, we have poved: C. 26 Lemma 4. The euence Φ α XY n + i a bounded euence in L Ω [, T ] and wih he Faou lemma, Y + Φ α X belong o L Ω [, T ]. Thi ineualiy allow u o how ha lim inf T Y ξ. Indeed, le θ be a funcion of cla C 2 R m ; R + wih a compac uppo icly included in F c = {g < + }. Thee exi an open e U.. he uppo of θ i included in U and U F c. Le Φ = Φ U be he peviouly ued funcion. Le u ecall ha α i icly geae han 2 + / > 2. Thank o a eul in he poof of he lemma 2.2 of [7], hee exi a conan C = Cθ, α uch ha: θ CΦ α, θ CΦ α and D 2 θ CΦ α 2. We wie again he. 2 fo θ, n N and T : Y n T θx T = Y n θx + + Y n θx Y n LθX d + + d Z n θx σ, X d. 27 In he lef hand ide, we ue he aumpion H2 o pa o he limi a n end o. We ju have o conol he igh hand ide a n end o infiniy. Fo he fi em, hee i no poblem: we ue he dominaed convegence heoem. Fo he econd, we apply he monoone convegence heoem. Fo he hid one, we can do he ame calculaion uing he peviouly given eimaion on θ, θ and D 2 θ in em of powe of Φ α and Hölde ineualiy: Auho' peonal copy Φ αp Lθ p L [, T ] R m. 28 Now we can wie: Y n LθX = Y n Φα/+ Φ α/+ LθX.

21 A. Popie / Sochaic Pocee and hei Applicaion The euence Y n Φ α/+ = Y n Φ α /p i a bounded euence in L + Ω [, T ] ee Lemma 4. Wih 28, uing a weak convegence eul and exacing a ubeuence if neceay, we can pa o he limi in he em Y nlθx d. Moeove: n N, Y n LθX d C. 29 Fo he emaining em Z n. θx σ, X d ecall ha fom Secion.3, hee exi a conan C = C fo all n N: Z n 2 T 2 d C. Hence, hee exi a ubeuence, which we ill denoe a Z n T /, and which convege weakly in he pace L 2 Ω, T, dp d; R d o a limi, and he limi i ZT /, becaue we aleady know ha Z n convege o Z in L 2 Ω, T δ fo all δ >. θxσ., XT. / i L 2 Ω, T, becaue θ i compacly uppoed and > 2. Theefoe, and lim n + n N, Z n. θx σ, X d = Paing o he limi a n + in 27: ξθx T = Y θx + + Z. θx σ, X d Z n. θx σ, X d d C. 3 θx Y + d + Z. θx σ, X d. Auho' peonal copy Y LθX d We le go o T and we apply Lemma 4, ineualiie 29 and 3, and Faou lemma: [ ] [ξθx T ] = lim [Y θx ] lim inf Y θx T T T [ ] = lim Y θx T. 3 T Bu ecall ha we aleady know iii: lim T Y gx T. Hence, he ineualiy in 3 i in fac a eualiy, i.e. [ ] [gx T θx T ] = θx T lim Y. +

22 234 A. Popie / Sochaic Pocee and hei Applicaion And uing again iii, we conclude ha: lim Y = gx T, P-a.. on {gx T < }. + Theefoe, we have poved he coninuiy of Y on [, T ] in he cae > 2. Remak 5. A lile modificaion in he peviou agumen how ha he euence Y n LΦ α X and Y n LθX ae bounded in L Ω [, T ] fo < +. Theefoe he euence Y n LθX i unifomly inegable and he paage o he limi and he eimae 29 follow When aumpion H3, B, D and ae aified If we ju aume >, ou peviou conol on he em conaining Z in 2 fail. Bu wih he aumpion H3, B, D and, we ae able o pove ha hee exi a funcion ψ uch ha fo < T : Z n. θx σ, X d = Y n ψ, X d, and hen, we apply again he Hölde ineualiy in ode o conol Y n ψ, X d by Y n + Φ α X d. We need he exience of a egula deniy fo he poce X oluion of he SD 7. Accoding o he aicle of Aonon [2], Theoem 7 and, hee exi a deniy Geen funcion fo X, px;.,. L 2 δ, T ; H 2 fo all δ >. Moeove, fom he Theoem 7 of [2] and he Theoem II.3.8 of [3], he deniy i Hölde coninuou in x and aifie he following ineualiy fo ], T ]: exp C y x 2 C exp y x 2 C C m/2 px;, y m/2, 32 C depending only on T, on he bound K and λ in B and, and C i independen of he egulaiy of hee funcion. Fom now on, we omi he vaiable x in px;., A peliminay eul We now pove he following eul fo he oluion Y, Z of he BSD: Y = hx T + Auho' peonal copy f Y d Z db, whee h : R m R i a bounded and Lipchiz funcion. Popoiion 6. Unde he aumpion B, D and, fo each funcion ϕ in he cla C 2 R m wih a compac uppo, hee exi a eal Boel funcion ψ defined on ], T ] R m.. fo all >, Y ψ, X < + and [Z. ϕx σ, X ] = [Y ψ, X ].

23 A. Popie / Sochaic Pocee and hei Applicaion The funcion ψ i given by he following fomula: ψ, x = d i= + ϕσ i x divpσ i, x p, x + Tace D 2 ϕxσ σ, x d ϕx.[ σ i σ i ], x; 33 i= whee σ i i he i-h column of he maix σ and p i he deniy of he poce X. Poof. To find hi funcion ψ, we ue he following eul: Popoiion 7. Fo all i d, { D i Y, T } i a veion of Z i. Z i = {Z i, T } denoe he i-h componen of Z. Thi eul come fom he Popoiion 5.3 in he aicle of l Kaoui e al. [4]. Hee, D i Y ha he following ene: D i Y = lim D i Y. < Fom he condiion L and G and he Theoem 2.2. of [5], we know ha X T belong o D,, and ince h i Lipchiz, wih he Popoiion.2.3 of [5], ξ = hx T D,2. Moeove, ince h i bounded by M, Y i alo bounded by M and f i a C -funcion. Hence, he concluion of he Popoiion 5.3 of [4] hold. We mu calculae: d [ ] [Z. ϕx σ, X ] = [D Y. ϕσ X ] = D i Y. ϕσ i X, whee ϕσ X = ϕx σ, X and ϕσ i X denoe he i-h componen of ϕσ X. Le ν i j, j N, be he following funcion: ν i j = j [ /j,]e i, wih [, T ] and wih e,..., e d he canonical bai of R d. Hee, we need >. We define D ν i j Y = DY, ν i j H, H being he Hilbe pace L 2 [, T ]; R d. The inegaion by pa fomula fo he Malliavin deivaive i he following: [ ] [ ] D ν i Y ϕσ i X = Y ϕσ i X ν i j j. db Auho' peonal copy i= [ ] Y D ν i ϕσ i X. 34 j Now we calculae he fi em of he igh hand ide. [ ] ] Y ϕσ i X ν i j. db = j [Y ϕσ i X B i Bi /j [ = j Y ϕσ i X /j divpσ i u, X u pu, X u whee p i he deniy of X and σ i i he i-h column of he maix σ. ] du, 35

24 236 A. Popie / Sochaic Pocee and hei Applicaion The la eualiy i juified by he Lemma 3. and 4. of he aicle of Padoux [6]. We mu how ha he aumpion of hee lemma ae aified. Fi, Y i a funcion of X: hee exi a coninuou funcion u : [, T ] R m R uch ha Y = u, X. Hence, we can wie: ] ] [Y ϕσ i X B i Bi /m = [v, X B i Bi /m, wih v, x = u, x ϕσ i x. Since ϕ ha a compac uppo, o doe v. And v i meauable and bounded. So he condiion of he Lemma 4. ae veified, he ime dependence of g playing no ole in he demonaion. L, B, ae no exacly he aumpion of he Lemma 3. and 4., bu wih hee condiion, he concluion ae he ame. In fac, he main poblem i o give a ene o he facion divpσ i /p. Fom he lowe bound in 32, he e {, y ], T ] R m ; p, y = } i empy. Thu he concluion of he Lemma 3. hold. Moeove he popey p L 2 δ, T ; H 2, δ >, implie ha divpσ i belong o L 2 [δ, T ] R m fo all δ > Lemma 2. in [6] wih m = and δ inead of. Since > i fixed hee, he poof of he Theoem 2.2 in [6] ee page 53 give u he eualiy 35. We have an addiional egulaiy popey on div p. Thi popey i given by he Theoem 2., Secion 3, page 223 of [7]. Since p.,. i oluion of he PD: p = L p, and ince he coefficien of L ae bounded and Lipchiz in x, he fi deivaive of σ σ belong o L [, T ] R m. Theefoe L i unifomly ellipic and can be wien in divegence fom. Fom B and L, b and b ae bounded. Fom B, L and D, all coefficien of L ae bounded. Hence he concluion of he Theoem III.2. ae valid: p/ x i aifie a Hölde condiion in x. Fom he lowe bound in 32, /p i a coninuou funcion. Moeove, divpσ i i alo coninuou. Le j goe o + in he ideniy 34: [ ] D i Y ϕσ i X = [ Y ϕσ i X divpσ i ], X p, X [ ] Y D i ϕσ ix. To find D i ϕσ ix, ecall he following eul ee Popoiion.2.3 and Theoem 2.2. in [5]: ince he poce X ha a deniy, if µ i a Lipchiz funcion, hen: D i µx = µx D i X = µx σ i X. Applying hi eul wih µ = ϕσ i, we obain: Finally, D i ϕσ ix = σ D 2 ϕσ ii X + ϕ.[ σ i.σ i ]X. [Z. ϕx σ, X ] = d [ ] D i Y. ϕσ i X = [Y ψ, X ] Auho' peonal copy i= [ ] d = Y ϕσ i X divpσ i, X p, X i= ] d [Y TaceD 2 ϕσ σ X + ϕ.[ σ i σ i ]X. i=

25 A. Popie / Sochaic Pocee and hei Applicaion The hypohei H3 implie ha g n i a Lipchiz funcion on R m. Indeed if we define { } gx gy K n = up ; gx gy n, x y hen he aumpion H3 implie ha K n i a non-deceaing euence of eal poiive numbe. Moeove, if x and y aify gx gy n o gx gy n, hen g nx g ny K n x y. And if gy < n gx, hen he coninuiy of g lead o: g nx g ny = n gy K n+ diy, { z R m ; gz n } K n+ y x. Finally g n ha a Lipchiz nom malle han K n+. We can apply Popoiion 6 wih Y n, Z n, ϕ. Coming back o he. 2 fo > : YT n ϕx T = Y n ϕx + Y n + 2 ϕx Y n ψ, X d + + d Y n whee ψ i given by he fomula 33 in Popoiion 6. ϕx b, X d Y n TaceD2 ϕx σ σ, X d Coninuiy wih H3, B, and D Recall ha U i a bounded open e uch ha U F c = {g < }, ha Φ = Φ U i a funcion which i uppoed o belong o C 2 R m ; R + and uch ha Φ i eual o zeo on R m \U, i icly poiive on U. α i a eal uch ha α > 2 + /. Fo n N, le Y n, Z n be he oluion of he BSD wih he final condiion g nx T. Fo < T, he elaion 36 i: YT n Φα X T = Y n Φα X + + Φ α X Y n wih Ψ α he following funcion: fo ], T ] and x R m + d Y n Ψ α, X d, 37 Ψ α, x = Φ α x.b, x 2 TaceD2 Φ α xσ σ, x d Φ α divp, xσ i, x xσ, x i p, x Auho' peonal copy i= d i= Φ α x.[ σ i, xσ i, x]. Ou goal now i o pove ha fo a fixed ε > and p = + /: Φ αp Ψ α p L [ε, T ] R m.

26 238 A. Popie / Sochaic Pocee and hei Applicaion If i i ue, hen he la em in 37 aifie: Y n Ψ α, X d C Φ α X Y n + + d and he end of he poof will be he ame a in he cae > 2. Fom he cae > 2 ee 23 and 24, we aleady know ha Φ αp Φ α.b,. p and Φ αp 2 Tace D 2 Φ α σ σ p,. ae in L [, T ] R m. The nex em i: d Φ αp Φ α.[ σ i,.σ i,.] i= p σ aifie he condiion L and B. We ue again he calculaion done fo he gadien of Φ α ee he poof of 24 o deduce ha if α > 2p hi em i in L [, T ] R m. Now, we come o he la em which involve he deniy of X: d Φ αp Φ α divpσ i,. p σ,. i p,. i= d = α p Φ α p divpσ i,. p Φσ,. i. p,. i= We pli hi em ino wo pa: d Φ Φσ α p,. i divσ i i p + σ,. p i= 2d p d i= Φ α p Φσ,.i div σ i,. + 2d p d Φ α p Φσ,.i σ i,. p p,. p p,. p. i= Fo he fi pa, hee i no poblem becaue α p >, o Φ α p i coninuou and compacly uppoed and Φσ i div σ i i bounded becaue of condiion L and B. Fo he econd pa, we ue he ineualiy 32 and he fac ha he uppo of Φ α p i a compac e K. So he minimum of p.,. exi on he e [ε, T ] K and i poiive. Theefoe, we conol he denominao. Fo he numeao, we aleady know ha p/ x i aifie a Hölde condiion in x. We can now conclude ha he econd pa i bounded by a conan K independen of n and. Finally, we have:. Auho' peonal copy p p Φ αp Ψ α p L [ε, T ] R m, and hu, fo ε: [ Y n Ψ α, X ] [ C Φ α X Y n +] +, 38

27 A. Popie / Sochaic Pocee and hei Applicaion whee C i a conan independen of and n. The. 37 i: Y n T Φα X T Y n ε Φα X ε = + ε ε [ Φ α X Y n +] d [ Y n Ψ α, X ] d. We have: YT nφα X T gx T Φ α X T ; ince Φ i eual o zeo ouide a compac e included in F c = {g < + }, uing he condiion H2 he lef hand ide of he peviou eualiy i bounded by a conan independen of n. Theefoe, we obain: ε [ [ Φ α X Y n +] d C ε Φ α X Y n ] + + d C. Since he e {x R +, x Cx + C} i bounded, we immediaely deduce ha Φ α XY n + i a bounded euence in L Ω [ε, T ]. Wih he Faou lemma, fo < ε T, Y + Φ α X belong o L Ω [ε, T ]. A Y i bounded on he ineval [, ε], ε < T, by /T ε /, we have poved: Y + Φ α X d < A in he cae > 2, hi ineualiy allow u o how ha: lim inf T Y ξ. Indeed, le θ be a funcion of cla C 2 R d ; R + wih a compac uppo icly included in F c = {g < + }. We wie again he. 36 fo θ, n N, ε > and ε: wih ξ nθx T = YT n θx T = Y n θx + Θ, x = θx.b, x θx Y n + d + Auho' peonal copy d divp, xσ i, x θxσ, x i p, x i= 2 TaceD2 θxσ σ, x Wih he ame agumen a in he cae > 2, we can pove ha ξ = lim Y, P-a.. on {ξ < + }. T 3. Minimal oluion Y n Θ, X d, 4 d θx.[ σ i, x.σ i, x]. i= In hi ecion we pove Theoem 5: he oluion conuced in Secion and 2 i he minimal one. Befoe we obain he following eimae: Popoiion 8. Wih he aumpion of Theoem 5, we pove:

28 24 A. Popie / Sochaic Pocee and hei Applicaion [, T ], Ȳ. T Poof. Fo evey < h < T, we define on [, T h] Λ h =. T h Λ h i he oluion of he odinay diffeenial euaion: Λ h = Λ h +, wih final condiion Λ h T h = +. Bu on he ineval [, T h], Ȳ, Z i a oluion of he BSD wih final condiion Ȳ T h. Fom he aumpion Ȳ T h i in L 2 Ω, o i finie a.. Now we ake he diffeence beween Ȳ and Λ h fo all < T h: Λ h Ȳ = Λ h Ȳ Λ h + + Ȳ d Z db wih = Λ h Ȳ α Λh Ȳ d Λ h + Ȳ + fo Ȳ α = Λ h Λ h Ȳ if Ȳ = Λ h. Auho' peonal copy Z db So α i a non-negaive pogeively meauable poce. Then we deduce: [ Λ h Ȳ = F Λh ] Ȳ exp α d. Moeove we know ha: Ȳ F Ȳ T h. Theefoe [ Λ h Ȳ F Λ h F ] Ȳ T h exp α d [ = F Λh ] Ȳ T h exp α d. Now Faou lemma lead, a goe o T h, o: Λ h Ȳ. Thi ineualiy i ue fo all [, T h] and fo all < h < T. So i i clea ha fo evey [, T ]: Ȳ. T Thi achieve he poof of he popoiion. In he cae whee ξ = + a.. hi ineualiy and Lemma give he uniuene of he oluion. If ξ = +, hee i a uniue oluion, namely Y = and Z =. T

29 A. Popie / Sochaic Pocee and hei Applicaion Poof of Theoem 5. We will pove ha Ȳ i geae han Y n fo all n N, which implie ha Y i he minimal oluion. Le Y n, Z n be he oluion of he BSD wih ξ n a eminal condiion. By compaion wih he oluion of he ame BSD wih he deeminiic eminal daa n: Y n T + /n / n. Beween he inan < T : Ȳ Y n = Ȳ Y n Ȳ + Y n + d Z Z n db = Ȳ Y n Ȳ + Y n + Ȳ Ȳ Y n Y n d Z Z n db wih = Ȳ Y n Ȳ + Y α n n+ = Ȳ Y n fo Ȳ Y n if Ȳ = Y n α n Ȳ Y n d Z Z n db 4 The poce α n i well defined, pogeively meauable and veifie: α n + Ȳ Y n. We deduce ha: [ Ȳ Y n = F Ȳ Y n exp. α n d ] uing he lineaiy of he BSD 4 and he fac ha he geneao of hi BSD i monoone. Then wih Faou lemma: { [ Ȳ Y n = lim inf F Ȳ Y n ]} exp α n d T { [ F lim inf Ȳ Y n ]} exp α n d. T I i legal o apply Faou lemma becaue wha i inide he condiional expecaion ha a lowe bound eual o n: Ȳ hi belong o he hypohei and Y n n and exp α n d. Auho' peonal copy Finally Ȳ Y n. A i i ue fo evey n N and evey [, T ], we have Ȳ Y. 4. Paabolic PD, vicoiy oluion In he inoducion, we have aid ha hee i a connecion beween BSD whoe eminal daa i a funcion of he value a ime T of a oluion of a SD o fowad backwad yem, and oluion of a lage cla of emi-linea paabolic PD. Le u make hi connecion pecie in ou cae.

30 242 A. Popie / Sochaic Pocee and hei Applicaion To begin wih, we modify he. 7. Fo all, x [, T ] R m, we denoe by X,x he oluion of he following SD: X,x = x + b, X,x d + σ, X,x db, fo [, T ], 8 and X,x = x fo [, ]. b and σ aify he aumpion L and G, and we add ha b and σ ae joinly coninuou in, x. We conide he following BSD fo T : Y,x = hx,x T Y,x Y,x d Z,x db, 42 whee h i a funcion defined on R m wih value in R + uch ha h i coninuou and bounded. The wo. 8 and 42 ae called a fowad backwad yem. Thi yem i conneced wih he PD 3 wih eminal condiion h. Thi eul i poved in he Theoem 3.2 of he aicle [5]: Theoem 9 Theoem 3.2 of [5]. If we olve he. 8 and 42 and if we define u, x fo, x [, T ] R m by u, x = Y,x, hen u i a coninuou funcion and i i a vicoiy oluion of he PD 3. Le u ecall he definiion of a vicoiy oluion ee [8,9] page 8 and 99 o [2] fo v coninuou. Fo v : [, T ] R m R, we define he uppe and lowe emi-coninuou envelope of v, namely: v, x = lim up,x,x v, x and v, x = lim inf,x,x v, x. Definiion 2. In hi definiion, h i coninuou and bounded on R m.. We ay ha v i a uboluion of 3 on [, T ] R m if v < +, if v T, x hx, and if, fo evey funcion ϕ C,2 [, T ] R m and local maximum, x of v ϕ, ϕ, x Lϕ, x + v, x v, x. 2. We ay ha v i a upeoluion of 3 on [, T ] R m if v >, if v T, x hx, and if, fo evey funcion ϕ C,2 [, T ] R m and local minimum, x of v ϕ, ϕ, x Lϕ, x + v, x v, x. 3. A funcion v i a vicoiy oluion if i i boh a vicoiy uboluion and upeoluion. Now, in ou cae, he funcion h i eplaced by he funcion g : R m R + which i uppoed o be coninuou fom R m o R + and uch ha he e F = {g = + } i cloed and non-empy. We canno apply he Theoem 3.2 in [5], o he peviou definiion, becaue g i unbounded on R m. Auho' peonal copy Definiion 2 Vicoiy Soluion wih Unbounded Daa. We ay ha v i a vicoiy oluion of he PD 3 wih eminal daa g if v i a vicoiy oluion on [, T [ R m and aifie: lim v, x = gx.,x T,x

31 A. Popie / Sochaic Pocee and hei Applicaion We ake he noaion of he conucion of he minimal oluion. Fo all n N and, x [, T ] R m, we obain a euence of andom vaiable Y,x,n, Z,x,n aifying : Y,x,n = gx,x T n Y,x,n Y,x,n d Auho' peonal copy Z,x,n db, and 9. We know now ha hi euence convege o Y,x, Z,x, which i he minimal oluion veifying he concluion of Theoem 2 and 4. In paicula, hi mean ha eihe > 2 o ele H3, B, D and ae aified by σ and b. We wan o pove Theoem 6. If we define he funcion u n by u n, x = Y,x,n, hen fom Theoem 3.2 in [5], we know ha u n i joinly coninuou in, x and i a vicoiy oluion of he paabolic PD 3 wih eminal value g n. The fac ha g i uppoed o be coninuou implie ha g n i bounded and coninuou on R m. By he compaion heoem fo BSD, Y,x,n = u n, x n N i a non-deceaing euence, and hence i convege o Y,x = u, x when n end o infiniy. Some emak abou he funcion u. I i a non-negaive funcion aifying he following bound:, x [, T ] R m, u, x. 43 T Moeove, ut, x = gx fo all x R m. A lea, u i lowe emi-coninuou on [, T ] R m a he upemum of coninuou funcion he euence u n i a non-deceaing euence, and fo all x R m : lim inf u, x gx.,x T,x Poof of Theoem 6. The main ool i he half-elaxed uppe and lowe limi of he euence of funcion {u n }, i.e. u, x = lim up n +,x,x u n, x and u, x = lim inf n +,x,x u n, x. In ou cae, u = u u = u becaue he euence {u n } i non-deceaing and u n i coninuou fo all n N. Fi, u i a upeoluion of he PD 3 on [, T [ R m. u = u = u i lowe emi-coninuou on [, T [ R m. Fom he eimae 43, fo all δ >, n N and all, x [, T δ] R m, / u n, x u, x. δ Since u n i a upeoluion of he PD 3, paing o he limi wih he Lemma 6., page 33, of [2], we obain ha u i a upeoluion of 3 on [, T [ R m. The ame agumen how ha u i a uboluion on [, T [ R m. A in he cae of he BSD, he main difficuly i in howing ha lim up u, x gx = ut, x.,x T,x We will pove ha u i locally bounded on a neighbouhood of T on he e {g < + }. Then, we deduce u i a uboluion and we apply hi o demonae ha u T, x gx if x {g < + }, which how he waned ineualiy on u.

32 244 A. Popie / Sochaic Pocee and hei Applicaion We make he ame calculaion a in he poof of he coninuiy of Y a T. Le θ be a funcion of cla C 2 R m ; R + wih a compac uppo included in {g < + }. We will pove ha u n θ i unifomly bounded on [, T ] R m. On [, T δ] R m he bound 43 give immediaely he eul. I emain o ea he poblem on a neighbouhood of T. Fi cae: > 2: We wie he eualiy 2 beween and T, fo x R m ; u n, xθx = Y,x,n T θx,x T Y,x,n [ LθX,x d θx,x Y,x,n Z,x,n +] d. θx,x σ, X,x d. The la em i conolled by: Z,x,n. θx,x σ, X,x T d Z,x,n 2 T 2/ d T θx,x σ, X,x /2 2 T 2/ d C = C, θ, σ. Hee, we ue he fac ha > 2, θ i compacly uppoed, and he condiion G. Thu, we have: θx,x Y,x,n Y,x,n T θx,x T + d + Y,x,n Z,x,n LθX,x d. θx,x σ, X,x d. The igh hand ide i bounded by he upemum of gθ and C. In he lef hand ide, he econd em i conolled by he fi one aied o a powe icly malle han uing Hölde ineualiy ee 25 and 28. Theefoe, hee exi a conan C independen of n,, x: θx,x Y,x,n + d C. We deduce ha: u n, xθx C = CT, g, θ,. Second cae: he aumpion H3, B, D and ae aified: Fo n N,, x [, T ] R m, >, he. 4 become: u n, xθx = Y,x,n T θx,x T Y,x,n Auho' peonal copy θx,x Y,x,n + d /2 Θ, X,x d. 44 Fom Popoiion 6 o fomula 36, he funcion Θ i defined by: d divp, xσ i, x Θ, x = θx.b, x θxσ, x i p, x i= 2 TaceD2 θxσ σ, x d i= θx.[ σ i, xσ i, x].

33 A. Popie / Sochaic Pocee and hei Applicaion A u n and θ ae wo non-negaive funcion, θx,x Y,x,n + d + Y,x,n Θ, X,x d Y,x,n T θx,x T. The igh hand ide in he peviou ineualiy i bounded by he upemum of gθ; hi upemum i well defined becaue gθ i coninuou wih compac uppo. And fom he calculaion made on he BSD, we know ha he abolue value of he econd em in he lef hand ide i conolled by he fi em which i non-negaive aied o a powe icly malle han. Thu, we deduce: θb,x Y,x,n + d C = CT, g, θ,. I i impoan o noe ha hi conan i independen of n,, x. If we come back o he ineualiy 44, we deduce ha fo all, x [, T ] R m : u n, xθx C = CT, g, θ,. Le U be an open ube.. U F c = {g < + }. Thu, u n i unifomly bounded on [, T ] U w... o n. Theefoe, u i bounded on [, T ] U. We know ha u n i a uboluion of he PD 3 eiced o [, T ] U, i.e. { u n, x Lu n, x + u n, x u n, x =,, x [, T [ U; u n T, x = g nx, x U. We can apply Theoem 4. in [9] ee alo Secion in [9]. Since g i coninuou, gx = gx = lim upg nx. n + x x Thu u i a uboluion of he PD: u Lu + u u =, in [, T [ U; ] min [ u Lu + u u ; u g, in {T } U. Now he Theoem 4.7 wih aighfowad modificaion how ha u g in {T } U. Thi achieve he poof of Theoem 6. The nex popoiion make pecie he behaviou of he oluion u on a neighbouhood of T. Popoiion 22. The peviouly defined oluion u aifie fo all x in he ineio of {g = + }: Auho' peonal copy lim [T T ]/ u, x =. Poof. We ake he noaion of Lemma. Fo all, x [, T [ R m, Y,x,n F / T + /ξ n.

34 246 A. Popie / Sochaic Pocee and hei Applicaion Thu, fo all inege n: Theefoe, [T ] / u, x [T ] / u n, x T / T + /n F {g= }X,x T [ + F T ξ n ] / + T ξ n {ξ< }. [T ] u, x F {g= } X,x T + F [ T ξ n + T ξ n {ξ< } The la em i bounded by [ F T ξ n ] / [ ] / + T ξ n {ξ< } [T ] /. 45 T Hence: lim inf [T T ]/ u, x lim F {g= } X,x T T ; hi limi i eual o fo x in he ineio of {g = + }. We conclude uing he bound Minimal oluion The goal of hi paagaph i o demonae ha he vicoiy oluion obained wih he BSD i minimal among all non-negaive vicoiy oluion Theoem 7. We compae a vicoiy oluion v in he ene of Definiion 2 wih u n, fo all inege n: fo all, x [, T ] R m, u n, x v, x. We deduce ha u v v. Remak ha he only ued aumpion in he poof will be L and G. Recall ha g : R m R + i coninuou, which implie ha g n : R m R + i coninuou. Popoiion 23. u n v, whee v i a non-negaive vicoiy oluion of he PD 3. Poof. Thi eul eem o be a diec coneuence of a well-known maximum pinciple fo vicoiy oluion ee [2,9] o [2]. Bu o he be of ou knowledge, hi pinciple wa no poved fo oluion which can ake he value +. Thu, following he poof of he Theoem 8.2 in [2], we ju give hee he main poin. Recall ha u n i he bounded by n and coninuou vicoiy oluion aociaed wih he eminal condiion f = g n. Fo ε >, we define u n,ε, x = u n, x ε. u n,ε i bounded by u n and i a uboluion of he PD 3 ee [2], poof of Theoem 8.2: u n,ε Auho' peonal copy Lu n,ε + u n,ε u n,ε ε T Moeove, a T, u n,ε T, x = u n T, x ε/t g nx and a, u n,ε end unifomly in x o. We will pove ha u n,ε v fo evey ε; hence we deduce u n v. Fom now on, n and ε ae fixed. We uppoe ha hee exi, z [, T ] R m uch ha u n,ε, z v, z δ > and we will find a conadicion. Fi of all, i i clea ha i no eual o o T, becaue v T, z = gz. ].

35 A. Popie / Sochaic Pocee and hei Applicaion u n,ε and v ae bounded fom above on [, T ] R m epecively by n and. Thu, fo α, β R 2, if we define: m, x, y = u n,ε, x v, y α 2 x y 2 β x 2 + y 2, m ha a upemum on [, T ] R m R m. Moeove, he penalizaion em aue ha he upemum i aained a a poin ˆ, ˆx, ŷ = α,β, x α,β, y α,β. Denoe by µ α,β hi maximum. Since δ 2β z 2 u n,ε, z v, z 2β z 2 = m, z, z µ α,β, chooing β ufficienly mall in ode o have δ/2 δ 2β z 2, we obain δ/2 µ α,β. Fom hi ineualiy and ince u n,ε n and v, we have: µ α,β = mˆ, ˆx, ŷ = u n,ε ˆ, ˆx v ˆ, ŷ α 2 ˆx ŷ 2 β ˆx 2 + ŷ 2 n α 2 ˆx ŷ 2 β ˆx 2 + ŷ 2 and hence ˆx 2 + ŷ 2 n β and ˆx ŷ 2 2n α. Now, we will pove ha ˆ canno be eual o zeo, o o T. Recall ha u n,ε,. =, hu ˆ canno be eual o zeo. Aume ha ˆ i eual o T. We have: µ α,β = g n ˆx ε T gŷ α 2 ˆx ŷ 2 β ˆx 2 + ŷ 2 g n ˆx g nŷ. Le γ β be a modulu of coninuiy of g n defined by: η, γ β η = up{ g nx g ny ; x y η, x, y B β 2 }, whee B β i he cloed ball wih adiu eual o n/β. Theefoe, we obain: δ/2 µ α,β g n ˆx g nŷ γ β ˆx ŷ γ β 2n α Since we have uppoed ha g n i coninuou, g n i unifomly coninuou on B β and heeby he limi of γ β η i eual o zeo a η goe o. Hence, he peviou ineualiy i fale when α i ufficienly lage. We deduce ha ˆ < T. We now ue he Theoem 8.3 of [2] wih he u n,ε uboluion and v upeoluion. Fo all ν > hee exi wo ymmeic maice X and Y of ize m m and wo eal a and b uch ha X A + ν A 2, Y a b = and aifying: I I wih A = α + 2β I I Auho' peonal copy a Fˆ, ˆx, u n,ε, ˆx, α ˆx ŷ + 2β ˆx, X ε T 2 b Fˆ, ŷ, v, ŷ, α ˆx ŷ 2β ŷ, Y. I I. 47, 48

36 248 A. Popie / Sochaic Pocee and hei Applicaion We ubac he wo peviou ineualiie: ε T 2 Fˆ, ˆx, u n,ε, ˆx, α ˆx ŷ + 2β ˆx, X Fˆ, ŷ, v, ŷ, α ˆx ŷ 2β ŷ, Y = 2 Tace σ σ ˆ, ˆxX 2 Tace σ σ ˆ, ŷy + bˆ, ˆx bˆ, ŷ.α ˆx ŷ + 2β bˆ, ˆx. ˆx + bˆ, ŷ.ŷ u n,ε ˆ, ˆx + + v ˆ, ŷ +. Since b i Lipchiz and gow a mo linealy, hee exi a conan K uch ha bˆ, ˆx bˆ, ŷ.α ˆx ŷ + 2β bˆ, ˆx. ˆx + bˆ, ŷ.ŷ αk ˆx ŷ 2 + 2β K + ˆx 2 + ŷ 2. Uing again he lowe bound 47 of µ α,β we have: δ/2 µ α,β u n,ε ˆ, ˆx v ˆ, ŷ v ˆ, ŷ u n,ε ˆ, ˆx and hu u n,ε ˆ, ˆx + + v ˆ, ŷ +. One em emain o be conolled: Tace σ σ ˆ, ˆxX Tace σ σ ˆ, ŷy. Fom he uppe bound 48, we deduce ha hee exi a conan κ = + 2αν + 2βν uch ha X I I I α + 2β. κ Y I I I If we chooe ν = /α, hen he conan κ i bounded: 3 κ 3 + 2β/α. We muliply hi ineualiy by he following non-negaive maix: σ σ ˆ, ˆx σ ˆ, ˆxσ ˆ, ŷ σ ˆ, ŷσ ˆ, ˆx σ σ, ˆ, ŷ and we ake he ace: Tace σ σ ˆ, ˆxX Tace σ σ ˆ, ŷy 2κβ [ σ σ ˆ, ˆx + σ σ ˆ, ŷ ] + κα [ σ ˆ, ˆx σ ˆ, ŷ ] [ σ ˆ, ˆx σ ˆ, ŷ ]. Uing he fac ha σ aifie L and G, we obain he exience of a conan K uch ha Tace σ σ ˆ, ˆxX Tace σ σ ˆ, ŷy K α ˆx ŷ 2 + Kβ + ˆx 2 + ŷ 2. Finally we have: ε T 2 C α ˆx ŷ 2 + β + ˆx 2 + ŷ 2, 49 whee C i a conan independen of α and β. Thi ineualiy i no exacly he ame a in he poof of he Theoem 8.2 in [2], becaue he oluion ae defined on R m and no on ome Auho' peonal copy bounded open e. Thu hee i hi addiional em wih β. Since ˆx 2 + ŷ 2 =, lim lim α α + β 2 ˆx ŷ 2 + β he ineualiy 49 lead o a conadicion aking β ufficienly mall and α ufficienly lage. Hence u n,ε v and i i ue fo evey ε >, o he eul i poved.

37 A. Popie / Sochaic Pocee and hei Applicaion Regulaiy of he minimal oluion The funcion u i he minimal non-negaive vicoiy oluion of he PD 3. We know ha u i finie on [, T [ R m ee 43. Fo δ >, u i bounded on [, T δ] R m by a conan which depend only on δ. Popoiion 24. If he coefficien of he opeao L aify L, B and, and ae Hölde coninuou in ime, hen u i coninuou on [, T ] R m, and fo all δ > : u C,2 [, T δ] R m ; R +. 5 Poof. The poof of Popoiion 23 how ha hee i a uniue bounded and coninuou vicoiy oluion of he Cauchy poblem: { v + Lv v v =, on [, T δ] R m, vt δ, x = φx on R m 5 whee φ i uppoed bounded and coninuou on R m. Moeove, he Cauchy poblem 5 ha a claical oluion fo evey bounded and coninuou funcion φ ee Lemma 25 below. Recall ha u n i joinly coninuou in, x and on [, T δ] R m, u n i bounded by: / u n, x. δ Thu, he poblem 5 wih condiion φ = u n T δ,. ha a bounded claical oluion. Since evey claical oluion i a vicoiy oluion and ince u n i he uniue bounded and coninuou vicoiy oluion of 5, we deduce ha: δ >, u n C,2 [, T δ[ R m ; R +. Fom he conucion of he claical oluion u n, we alo know ha he euence {u n } i locally bounded in C α,+α [, T δ/2] R m. The bound i given by he L nom of u n which i malle han T δ/4 /. Theefoe u i coninuou on [T δ/2] R m and if we conide he poblem 5 wih coninuou eminal daa ut δ,., wih he ame agumen a fo u n, we obain ha u i a claical oluion, i.e. u C,2 [, T δ] R m ; R +. In paicula, u i coninuou on [, T [ R m. And he eminal condiion in Theoem 6 how ha u i coninuou a ime T. Lemma 25. Fo evey bounded and coninuou funcion φ, he Cauchy poblem 5 ha a claical oluion v. Poof. If L can be wien in divegence fom, he concluion i given by he Theoem 8. and Remak 8., Secion V, in [7]. Moe peciely, hee exi a uniue coninuou and bounded oluion v uch ha, fo all δ > δ, v belong o H +β/2,2+β [, T δ ] R m. H +β/2,2+β [, T δ ] R m i he e of funcion which ae + β/2-hölde coninuou in ime and 2 + β-hölde coninuou in pace. Remak ha if he aumpion D hold, hen L can be wien in divegence fom. In geneal, we ue a booap mehod. If v denoe he uniue bounded and coninuou vicoiy oluion of 5, if f = v v L, fom he heoem 3. of [22], he linea Cauchy poblem: Auho' peonal copy