Présentée pour obtenir le grade de. Docteur en Science **************TITRE**************

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1 UNIVRSITÉ MOHAMD KHIDR FACULTÉ DS SCINCS XACTS T SCINC D LA NATUR T D LA VI BISKRA *************************** THÈS Préeée pour obeir le grade de Doceur e Sciece Spécialié: Probabilié **************TITR************** Sur ceraie propriéé de équaio différeielle ochaique doubleme rérograde. *************************** par Badreddie MANSOURI Soueue le deva la commiio d exame:

2 Ackowledgeme I i a pleaure o ackowledge Profeor Brahim Mezerdi, my upervior, who ha maerly guided me io he world of reearch wih icomparable dipoabiliy ad ice my fir ep. Thak you Profeor for your edle care, help, ecourageme. I am deeply graeful o Profeor Khaled Bahlali for heir help, co-operaio, ecourageme i variou apec ad for warm hopialiy, he cieific ehuiam he ha ramied me durig my ay i Toulo, hak you Profeor for hi. I would like o expre my icere graiude o Profeor : Lamie Melkemi, Khaled Bahlali, Abdelhafide Moukrae for accepig o repor my hei. Pleae do fid here he expreio of my coideraio. I am alo graeful o Profeor Seid Bahlali for accepig o be chairma. May hak are alo due o Dr. Boubakeur Labed for hi co-operaio, help, ecourageme ad fried-hip ad for guidig me io he area of reearch. 1

3 Coe Ackowledgeme 1 Iroducio 6 I BACKGROUND ON BACKWARD STOCHASTIC DIF- FRNTIAL QUATIONS xiece reul of backward ochaic differeial equaio Iroducio BSD wih Lipchiz coefficie Compario heorem BSD wih coiuou coefficie Refleced Backward Sochaic Differeial quaio Iroducio Refleced BSD wih Lipchiz coefficie Compario heorem for RBSD Refleced BSD wih coiuou coefficie II BACKWARD DOUBLY STOCHASTIC DIFFRNTIAL QUATIONS Backgroud o backward doubly ochaic differeial equaio Iroducio BDSD wih Lipchiz coefficie Noaio ad aumpio xiece ad uiquee heorem Compario Theorem of BDSD

4 3 3.4 BDSD wih coiuou coefficie Refleced Backward doubly ochaic differeial equaio Iroducio Aumpio ad Defiiio xiece of a oluio of he RBDSD wih Lipchiz codiio RBDSD wih coiuou coefficie xiece reul of Double barrier Refleced backward doubly ochaic differeial equaio Iroducio Aumpio ad Defiiio xiece of a oluio of he DRBDSD wih Lipchiz codiio Double barrier BDSD wih coiuou coefficie

5 Abrac Thi hei i compoed of wo par. The fir par i devoed he udy of backward ochaic differeial equaio (BSD) uder Lipchiz codiio i boh mulidimeioal cae, we prove a exiece ad uiquee reul for BSSD i hi cae. Afer, we give a exiece reul for a BSD where he coefficie are aumed o be oly coiuou. I The ecod par, we prove he exiece of he oluio where i forced o ay above a give ochaic proce, called he obacle. We prove he exiece ad uiquee uder Lipichiz coefficie. I he cae where he coefficie i coiuou we prove he exiece oly. I he ecod par, we pree ome ew reul i he heory of backward doubly ochaic differeial equaio (BDSD), I fir, we give he reul he exiece ad uiquee uder ome Liphiz aumpio o he coefficie, fially we eablih exiece ad uiquee reul for a refleced BDSD wih oe barrier ad wih wo barrier, udy he cae Lipchiz ad Coiuou. 4

6 5 Réumé Cee hèe e compoe de deux parie. La première parie e coacré à l éude de équaio différeielle ochaique rérograde (DSR), ou le codiio de Lipichiz e prouve l exiece e l uicié, mai avec coefficie coiu, ou doo u réula d exiece pour ue DSR. Nou éudio da chapire 2, ue clae de équaio différeielle ochaique rérograde. Da la deuxième parie, ou préeo de ouveaux réula da la héorie de équaio différeielle doubleme ochaique rérograde (DDSR), o commece par le réula claique ur cee clae de équaio. efi e défiie le équaio différeielle doubleme ochaique rérograde réfléchie (DDSRR). D abord, ou la codiio de Liphiz ur le coefficie ou éablio l exiece e l uicié pour ue DDSR réfléchie. uie, u réula d exiece de la oluio da le ca où le coefficie o coiu à croiace liéaire.

7 Iroducio I wa maily durig he la decade ha he heory of backward ochaic differeial equaio ook hape a a diic mahemaical diciplie. Thi heory ha foud a wide field of applicaio a i mahemaical fiace, he heory of hedgig ad oliear pricig heory for imperfec marke (ee l Karoui, Peg ad Queez [14]) ad a he ame ime, i ochaic opimal corol ad ochaic game (ee Hamadèe ad Lepelier [23]), ad hey provide probabiliic formulae for oluio o parial differeial equaio (ee Pardoux ad Peg [34]). The Liear backward ochaic differeial equaio : Y = ξ + (Y β + Z γ + φ )d Z dw. (1) have bee iroduced by Bimu [8] ad [9] whe he wa udyig he adjoi equaio aociaed wih he ochaic maximum priciple i opimal corol. However, he fir publihed paper o oliear BSD (ee Pardoux ad Peg [34]) Y = ξ + f(, Y, Z )d Z dw. (2) I [34], Pardoux ad Peg have eablihed he exiece ad uiquee of he oluio of equaio 2 uder he uiform Lipchiz codiio, i.e. here exi a coa K > uch ha f(, y, z) f(, y, z ) K( y y + z z ). for all y, y R d, z, z R d. Noe ha, ice he boudary codiio i give a he ermial ime T, i i o really aural for he oluio Y be o adaped a each ime o he pa of he Browia moio W before ime. The preece of he proce Z eem uperfluou. However, we poi ou ha i i he preece of hi proce ha make i poible o fid adaped proce Y o aify equaio (2). Hece, a oluio of BSD (2) o he probabiliy pace of Browia moio, a meioed above, i a pair (Y, Z) of adaped procee ha aifie (2) almo urely. 6

8 7 I [27], Lepelier ad Sa Mari have prove he exiece of a oluio for oe-dimeioal BSD where he coefficie i coiuou, i ha a liear growh, i.e aume he for fixed, ω, f(,.,.) o coiuou, ad here exi a coa K > uch ha for all, y, z we have f(, y, z) K(1 + y + z ). Kobylaki i [26] prove exiece ad uiquee reul for BSD (2) i oe dimeioal whe he geeraor (f(, y, z) ha a quadraic growh i Z. f(, y, z) C(1 + y + z 2 ). Thee reul are ipired by he aalogou oe for quailiear parial differeial equaio ad hold for procee (Y, Z ) T uch ha (Y ) T i bouded. K.Bahlali i [2] we deal wih mulidimeioal BSD wih locally Lipchiz coefficie ad a quare iegrable ermial daa. We udy he exiece ad uiquee, a well a he abiliy of oluio. We how ha if he coefficie f i locally Lipchiz i boh variable y, z ad he Lipchiz coa L N i he ball B(, N) i uch ha L N = o( log N), he he BSD 2 ha a uique oluio. I he cae where he oluio i forced o remai above a obacle, l Karoui e al. [15] have derived a exiece ad uiquee reul for Refleced BSD wih Lipchiz coefficie by Picard ieraio mehod a well a a pealizaio argume. I hi cae, he oluio i a riple (Y, Z, K), where K i a icreaig proce, aifyig Y = ξ + f(, Y, Z )d + K T K Z db. (3) Noe ha he udy of Refleced BSD o oe barrier ha bee primarily moivaed by he evaluaio of a America opio i a marke coraied, which may be a marke where iere rae are o he ame if we wa o borrow or ive moey. Ideed, i ha bee proved ha he price of a America coige acio i a oluio of a refleced BSD ha he barrier i give by he payoff ad he opimal ime of exercie i he fir ime whe he price reache he payoff (ee [16]). Oher applicaio i i mixed ochaic corol ee [2]. I he paper of Maoui [29] he will be ipired by he work of l Karoui [15] ad Lepelier [27] o eablih exiece of Refleced oluio of oedimeioal BSD wih coiuou ad liear growh coefficie. Recely, a ew cla of BSD, called doubly ochaic, ha bee coidered by Pardoux ad Peg (1994) (ee [35]). Thi ew kid of backward SD eem o be uiable o give a probabiliic repreeaio for a yem of parabolic ochaic parial differeial equaio (SPD ). We refer o Pardoux ad Peg (1994) [35] for he lik bewee

9 8 SPD ad BDSD i he paricular cae where oluio of SPD are regular. I [35] Pardoux-Peg udy exiece ad uiquee of he oluio o a backward doubly ochaic differeial equaio a follow Y = ξ + f(, Y, Z )d + g(, Y, Z )d B Z dw. (4) where he dw iegral i a forward Iô iegral ad he d B iegral i a backward Iô iegral, we prove uder he codiio f ad g are Lipchiz, he BDSD 4 ha a uique oluio. I [41]. Shi e al., we hall prove he compario heorem of BDSD (4). The we udy he cae where he geeraor are coiuou wih liear growh. We how he exiece of he miimal oluio of (4). Thi mehod i due o [27]. I hi hei, we pree ome ew reul i he heory of Backward Doubly Sochaic Differeial quaio. Fir, we udy he cae where he oluio i forced o ay above a give ochaic proce, called he obacle. We obai he real valued refleced backward doubly ochaic differeial equaio : Y = ξ + f(, Y, Z )d + g(, Y, Z )d B + K T K Z dw. (5) We eablih he exiece ad uiquee of oluio for equaio (5) uder uiformly Lipchiz codiio o he coefficie [4]. I cora o claical refleced BSD, he ecio heorem ca o be eaily ued for RBDSD. Ideed, i i o poible o prove ha he oluio ay above he obacle for all ime, by oly uig he claical BSD echic. Thi i due o he fac ha he oluio hould be adaped o a family (F ) which i o a filraio. We give here a mehod which allow u o overcome hi difficuly i he Lipchiz cae. The idea coi o ar from he pealized baic RBDSD wih f ad g idepede from (y, z). We raform i o a RBDSD wih f = g =, for which we prove he exiece ad uiquee of oluio by pealizaio mehod. The ecio heorem i he oly ued i he imple coex where f = g = o prove ha he oluio of he RBDSD (wih f = g = ) ay above he obacle for each ime. A ew ype of compario heorem i alo eablihed ad ued i hi coex. The (geeral) cae, where he coefficie f, g deped o (y, z), i reaed by a Picard ype approximaio. I he cae where he coefficie f i oly coiuou, we eablih he exiece of a maximal ad a miimal oluio. I hi cae, we approximae f by a equece of Lipchiz fucio (f ) ad ue a compario heorem which i eablihed here for RBDSD.

10 9 Oher ew reul, we geeralize he above reul o he cae of wo reflecig barrier procee, we obai he real valued double refleced backward doubly ochaic differeial equaio (i hor DRBDSD): Y = ξ+ f (, Y, Z ) d+ g (, Y, Z ) d B + dk + dk Z dw, T. (6) We eablih he exiece ad uiquee of oluio for equaio (6) uder uiformly Lipchiz codiio o he coefficie. I he cae where he coefficie f i oly coiuou, we eablih he exiece of a oluio. The hei i orgaized a follow. The fir par of hi hei i o he Backward Sochaic Differeial quaio Y = ξ + f(, Y, Z )d Z dw. (7) I Chaper 1, we pree, a exiece ad uiquee heorem for oluio of BSD (7) where he coefficie i Lipuchiz, ad we give he compario reul i hi cae ad we prove he exiece i he cae where he geeraor are coiuou wih liear growh. I Chaper 2, we prove exiece ad uiquee reul of oluio of refleced backward ochaic differeial equaio Y = ξ + f(, Y, Z )d + K T K Z db. (8) where he coefficie i Lipichiz ad exiece oly i he cae where he geeraor are coiuou wih liear growh. The ecod par of hi hei i o he Backward Doubly Sochaic Differeial quaio Y = ξ + f(, Y, Z )d + g(, Y, Z )d B Z dw. (9) I Chaper 3, we give a backgroud o BDSD, we prove exiece ad uiquee reul of oluio of BDSD (9) where f ad g are Lipichiz, ad he geeralize hi cae where f i coiuou wih liear growh ad we prove alo he reul compario for BDSD. I Chaper 4, i hi chaper we eablih a ew reul of BDSD, i whe he oluio i forced o ay above a give ochaic proce, called he obacle. We obai he real

11 1 valued refleced BDSD Y = ξ + f(, Y, Z )d + g(, Y, Z )d B + K T K Z dw. (1) We prove he exiece ad uiquee of oluio for equaio (1) uder uiformly Lipchiz codiio o he coefficie. I he cae where he coefficie f i oly coiuou, we eablih he exiece of a maximal ad a miimal oluio. I Chaper 5, we give oher ew reul i for double refleced backward doubly ochaic differeial equaio (i hor DRBDSD): Y = ξ+ f (, Y, Z ) d+ g (, Y, Z ) db + dk + dk Z dw, T. (11) We eablih he exiece ad uiquee of oluio for equaio (11) uder uiformly Lipchiz codiio o he coefficie. I he cae where he coefficie f i oly coiuou, we eablih he exiece of a oluio.

12 Par I BACKGROUND ON BACKWARD STOCHASTIC DIFFRNTIAL QUATIONS. 11

13 Chaper 1 xiece reul of backward ochaic differeial equaio We prove he exiece ad uiquee of oluio for backward ochaic differeial equaio wih lipchiz geeraor ad quared iegrable ermial codiio, we prove moreover he exiece of a oluio whe he geeraor i merely coiuou. 1.1 Iroducio Backward ochaic differeial equaio (BSD) have bee fir iroduced by. Pardoux ad S. Peg [39] who proved exiece ad uiquee of adaped oluio for hee equaio uder uiable Lipchiz ad liear growh codiio o he coefficie. The pricipal iere of BSD i ha hey provide a ueful framewark for formulaig may problem a i fiace heory, ochaic corol ad i he game heory. Followig he idea of J.P. Lepelier ad J. Sa Mari [27], we ue a approximaio argume o prove he exiece of a oluio of oe dimeioal BSD wih a coiuou coefficie. 1.2 BSD wih Lipchiz coefficie Le coider a filered pace (Ω, F, P, F, W, [, 1]) be a complee Wieer pace i R,i.e. (Ω, F, P ) i a complee probabiliy pace, (F, [, 1]) i a righ coiuou 12

14 Secio 1.2. BSD wih Lipchiz coefficie 13 icreaig family of complee ub σ algebra of F, (W, [, 1]) i a adard Wieer proce i R wih repec o (F, [, 1]). We aume ha F = σ[w, ] N, where N deoe he oaliy of P -ull e. Now, we defie he followig wo objec : (H1.1) A ermial value ξ L 2 (Ω, F 1, P ). (H1.2) A fucio proce f defied o Ω [, 1] R k R k wih value i R k ad aifie he followig aumpio : (i) for all (y, z) R k R k : (ω, ) f(ω,, y, z) i F -progreively meaurable. (ii) 1 f(,, ) 2 d < (iii) for ome K > ad all y, y L k, z, z R k, ad (ω, ) Ω [, 1] f(ω,, y, z) f(ω,, y, z ) K( y y + z z ). Le u coider he followig kid of pace of variable or procee. (1) L 2 T (Rk ) i he pace of all F T -meaurable radom variable X : Ω R k aifyig X 2 = ( X 2 ) <. (2) H 2 T (Rk ) i he pace of all he predicable proce φ : Ω [, T ] R k uch ha φ 2 = ( φ 2 d) <. Such procee are aid o be quare iegrable. Le u ow iroduce our BSD : Give a daa (f, ξ) we wa o olve he followig backward ochaic differeial equaio: Y = ξ + f(, Y, Z )d Z dw, T. (1.1) Defiiio A oluio of equaio (1.1) i a pair of procee (Y, Z) progreively meaurable ad aifyig : ( f(, Y, Z ) + Z 2 )d <, ad equaio (1.1). We ow make more precie he depedece of he orm of he oluio (Y, Z) upo he daa (ξ, f). Propoiio Le aumpio (H1.1) ad (H1.2) hold. The here exi a coa C, which deped oly o K, uch ha (up T Y 2 ) + ( Y 2 (e α(1 ) ξ 2 + where α = 1 + 2K + 2K 2. Z 2 d) C( ξ 2 + e α( ) f(,, ) 2 d/f ), f(,, ) 2 d)

15 Secio 1.2. BSD wih Lipchiz coefficie 14 Before provig propoiio (1.2.2), le u fir prove he iequaliy up T Y 2 + ( Defie for each N, he oppig ime τ = if{ T ; Y }, Z 2 d) <. (1.2) ad he procee By oig Y = Y τ. Z = 1 [,τ]()z, we have Y = ξ + 1 [,τ]()f(, Y, Z )d Z dw, T. If we apply Iô formula o he proce Y 2, he Y 2 + which implie ( Y 2 + If we ake K 2ε 2 Z 2 d = ξ [,τ]() ) f(, Y, Z )d Z 2 d) ξ 2 + 1, we ge 2 + K 2ε 2 Z 2 d. Y Now i follow from Growall lemma ha O he oher had, up N From Faou lemma, we ca ee ha Y, Z dw, ( f(,, ) 2 d + (1 + 2K + 2ε 2 ) Y 2 )d Z 2 d C(1 + up T Y 2 C. up Z 2 d. N up Y 2. T Y 2 )d.

16 Secio 1.2. BSD wih Lipchiz coefficie 15 Burkholder-Davi-Gudy iequaliy implie ha up T Y 2. I follow ha τ T a.. Uig agai Faou lemma, we obai Z 2 d. Proof. (of propoiio 1.2.2) Sice (Y, Z) aifie equaio (1.1) ad (2.12), Y, Z dw =, becaue he local marigale, { Y, Z dw, T } i uiformly iegrable from he Burkholder-Davi-Gudy iequaliy for ochaic iegral ee (barlow ad proer) ad he fac ha up T ad from Doob iequaliy, we obai up T (Y ) Z dw C( up Y 2 ) 1/2 ( up T T (Y ) Z dw C( up Y 2 ) 1/2 ( up T T From Iô formula, (H1.2)(iii) ad chwarz iequaliy, Y 2 + Z 2 d ξ ξ (Y ) f(, Y, Z )d 2 Z dw 2 ) 1/2, Z 2 d) 1/2 <, Y, Z dw ( f(,, ) 2 + (1 + 2K + 2K 2 ) Y Z 2 )d Y, Z dw. Takig expecaio ad uig Growall lemma we ge up Y 2 + Z 2 d C( ξ 2 + f(,, ) 2 d) <. T The he reul follow from he Burkholder-Davi-Gudy iequaliy. The ecod reul follow by akig he codiioal expecaio i he followig iequaliy e a Y e a Z 2 d e at ξ e a f(,, ) 2 d 2 e a Y, Z dw. 2 We hall ow prove exiece ad uiquee for BSD (1.1) uder codiio (H1.1) ad (H1.2).

17 Secio 1.2. BSD wih Lipchiz coefficie 16 Theorem (Pardoux-Peg) Uder codiio (H1.1), (H1.2), here exi a uique oluio for equaio (1.1). Proof. xiece.fir, le u prove ha he BSD ha oe oluio. Le Y = ξ + Y = (ξ + f()d Z dw, f()d/f ), ad {Z, T } i give by Iô marigale repreeaio heorem applied o he quare iegrable radom variable ξ + f()d, ha i ξ + f()d = (ξ + f()d) + Z dw, T, Takig he codiioal expecaio wih repec o F, we deduce ha Y = ξ + f()d Z dw, T, i.e. (Y, Z) i a oluio of our BSD. Le u defie he followig equece, Z ) N uch ha Y = Z = ad +1, Z +1 ) i he uique oluio of he BSD (1) Z +1 i a predicable proce ad( Z +1 2 d) <, (2) Y +1 = ξ + f(, Y, Z )d Z +1 dw, T. We hall prove ha he equece, Z ) i Cauchy. Uig Iô formula, we obai for every > m ad he, e α Y +1 Y m = e α Y +1 Y m e α Z +1 Z m+1 2K 2 d + α e α Y +1 Y m+1 2 d e α +1 Y m+1 ) [f(, Y, Z ) f(, Y m, Z m )]d e α +1 Y m+1 ) (Z +1 Z m+1 )dw, e α Z +1 Z m+1 2 d + α e α Y +1 Y m+1 [ Y e α Y +1 Y m+1 2 d Y m + Z Z m ]d

18 Secio 1.2. BSD wih Lipchiz coefficie 17 which implie e α Y +1 Y m (K 2 ε 2 α) Chooig α ad ε uch ha 2 ε 2 e α Z +1 Z m+1 2 d + 2 ε e α Z 2 Z m 2 ]d. e α Y +1 Y m ( I follow immediaely ha e α Y e α Y e α Y +1 Y m+1 2 d + 2 ε 2 = 1 2 ad α 4K2 = 1, he Y m 2 d + Coequely,, Z ) N i a Cauchy equece. Le e α Z +1 Z m+1 2 d e α Z Z m 2 ]d). Y m 2 d + e α Z Z m 2 ]d C 2. Y = lim Y, ad Z = lim Z. I i eay o ee ha (Y, Z) i a oluio of our BSD. e α Y Y m 2 d Uiquee.Le {(Y, Z ); T } ad {(Y, Z ); T } deoe wo oluio of our BSD, ad defie {( Y, Z ), T } = {(Y Y, Z Z ), T }. I follow from Iô formula ha Hece [ Y 2 + [ Y 2 + Z 2 d] = 2 Z 2 d] C he reul follow from Growall lemma. Y, f(, Y, Z ) f(, Y, Z ) d. Y 2 d + frac12 Z 2 d. The followig propoiio how, i paricular, he exiece ad uiquee reul for liear backward ochaic differeial equaio. Such a way i well-kow i mahemaical fiace, where he oluio of a liear BSD i i fac he pricig ad hedgig raegy of he coige claim ξ (ee [14]).

19 Secio 1.3. Compario heorem 18 Propoiio Le (β, γ) be a bouded (R, R )-valued progreively meaurable proce, Φ be a eleme of H 2 T (R), ad ξ be a eleme of L2 T (Rk ). The he Liear BSD Y = ξ + (Φ + Y β + Z γ )d ha a uique oluio (Y, Z) i H 2 T (R) H2 T (R ) give explicily by : Λ Y = [ξλ T + Z dw (1.3) Λ Φ d/f ], (1.4) where Λ i he adjoi proce defied by he forward liear ochaic differeial equaio, Λ = 1 + Λ β d + Λ γ dw, Proof. By heorem1.2.3, here exi a uique oluio o he BSD (1.3). Uig Iô formula we obai Λ Y + Λ Φ d = Y + Λ Y γ dw + Λ Y Z dw. Sice up T Y ad up T Λ are quare iegrable, herefore he local marigale {Λ Y + Λ Φ d, [, T ]} i a uiformly iegrable marigale, whoe -ime value i he F -codiioal expecaio of i ermial value. 1.3 Compario heorem We ae i he oe dimeioal cae a compario heorem (fir obaied by Peg,[38]). Theorem (Compario heorem)le (f 1, ξ 1 ) ad (f 2, ξ 2 ) be wo daa of BSD, ad le (Y 1, Z 1 ) ad (Y 2, Z 2 ) be he aociaed oluio. We uppoe ha ξ 1 ξ 2 P a.., ad f 1 (, y, z) f 2 (, y, z) d P a.. The we have Y 1 Y 2 Proof. We ue he follow oaio We obai he follow BSD δ(y ) = δ(ξ ) + we ca wrie P a.. δ(y ) = Y 1 Y 2, δ(z ) = Z 1 Z 2, δ(ξ ) = ξ 1 ξ 2. δ(y ) = δ(ξ ) + (f 1 (, Y 1, Z 1 ) f 2 (, Y 2, Z 2 ))d δ(z )dw, T, (α δ(y ) + β δ(z ) + f 1 (, Y 2, Z 2 ) f 2 (, Y 2, Z 2 ))d δ(z )dw, T,

20 Secio 1.4. BSD wih coiuou coefficie 19 where {α ; T } i defied by α = { (f 1 (, y 1, z 1 ) f 1 (, y 2, z 2 ))(δ(y )) 1 if Y 1 Y 2,, if Y 1 = Y 2. ad he R valued proce {β, T } a follow. For 1 i, le Z (i) deoe he -dimeioal vecor whoe compoe are equal o hoe of Z 2, ad whoe i la compoe are equal o hoe of Z 1. Wih hi oaio, we defie for each 1 i, { (f 1 (, Y β i 2, Z (i) ) f 1 (, Y 2, Z i 1 )(δ(z)) i 1 if Z 1i =, if Z 1i = Z 2i. Z 2i, Sice f i a Lipchiz fucio, α ad β are bouded procee, for T, le We have for T, δ(y ) = Γ, δ(y ) + Hece Γ, = exp[ δ(y ) = (Γ, δ(y ) + β r, dw r + (α r β r 2 Γ,r (f 1 (r, Y 2 r, Z 2 r ) f 2 (r, Y 2 r, Z 2 r ))dr 2 )dr]. Γ,r (δ(z r ) + β r δ(y r ))dw r. Γ,r (f 1 (r, Y 2 r, Z 2 r ) f 2 (r, Y 2 r, Z 2 r ))dr/f ) The reul follow from hi formula ad he egaiviy of δ(ξ) ad (f 1 (r, Y 2 r, Z 2 r ) f 2 (r, Y 2 r, Z 2 r )). 1.4 BSD wih coiuou coefficie Now we prove he exiece of a oluio for oe dimeioal backward ochaic differeial equaio where he coefficie i coiuou, i ha a liear growh, ad he ermial codiio i quared iegrable. we alo obai he exiece of a miimal oluio. Fir le u coider he followig aumpio : (H1.3) (i) The fucio f i R-valued. (ii) here exi a coa C uch ha P a.., f(, y, z) C(1 + y + z ) for ay (, y, z) [, T ] R 1+d (iii) P a.. for ay [, T ], he fucio which wih (y, z) (, y, z) i coiuou. The we have he followig reul:

21 Secio 1.4. BSD wih coiuou coefficie 2 Theorem (Lepelier-Sa Mari [27]).The BSD aociaed wih (f, ξ) ha a maximal oluio (Y, Z), i.e., we have Y S 2, Z H 2 (R d ), Y = ξ + f(, Y, Z )d Z dw. ad if (Y, Z ) i aoher oluio for (1.5) he P a.., Y Y. (1.5) The idea of he proof i o give a approximaio of he coefficie f by a Lipchiz equece of fucio f, ad eablih ha he limi of he equece, Z ) of correpodig oluio for BSD (ξ, f ) i a oluio of BSD wih parameer (ξ, f). Lemma Le g : R R be a coiuou fucio wih liear growh, ha i here exi a coa K < uch ha x R p g(x) K(1 + x ). The he equece of fucio i well defied for K ad i aifie : {g ) ( 1) (x) = if {g(y) + x y } y Q (i) liear growh : x R, g (x) K(1 + x ); (ii) moooiciy i : x R, g (x) ; (iii) Lipchiz codiio : x, y R, g (x) g (y) x y, (iv) Srog covergece : if x which coverge o x R we have, lim g (x ) = g(x), d a.e, Proof. By he liear growh hypohei o g, g i well defied for K. Alo i follow a oce ha g g. Agai, from he liear growh codiio o g, we obai g (x) if K K y + K x y = K(1 + x ) y Q from which (i) hold. Propery (ii) i evide from he defiiio of he equece (g ). Take ε > ad coider y ε Q uch ha g (x) g(y ε ) + x y ε ε g(y ε ) + y y ε + x y ε y y ε ε g(y ε ) + y y ε x y ε ε g(y ε ) x y ε.

22 Secio 1.4. BSD wih coiuou coefficie 21 Therefore, ierchagig he role of x ad y, ad ice ε i arbirary we deduce ha g (x) g (y) x y. I order o prove (iv), coider lim x = x. Take for every, y Q uch ha g(x ) g (x ) g(y )+ x y 1. Sice (g(x )) i bouded ad g ha liear growh, we deduce ha (y ) i bouded, ad o i (g(y )). Coequely, lim up y x <, ad i paricular y x. Moreover from which he reul follow. g(x ) g (x ) g(y ) 1. Proof. (of heorem 1.4.1) For, le, Z ) be he oluio of he BSD aociaed wih (f, ξ). O he oher had le (Ỹ, Z) be he oluio of he BSD aociaed wih ( C(1 + y + z ), ξ). The compario heorem (1.3.1), implie ha for ay, Y Y +1 Ỹ. I follow ha P a.. for ay T, Y Y ad he equece ) coverge i H 2 (R) o a proce Y which alo upper emi-coiuou. Now by Iô formula wih ) 2, uig he iequaliy ab ε a ε b 2 for ay ε > ad a, b R ad ice f (, y, z) C(1 + y + z ) we deduce ha he equece (Z ) i uiformly bouded i H 2 (R). Nex le, m. The uig Iô formula yield: Y m ) 2 + Z Z m 2 d = 2 2 Y m )(f (, Y, Z ) f m (, Y m Y m )(Z Z m )dw, T., Z m ))d The he equece (Z ) i of Cauchy ype i H 2 (R d ) ad coverge o a proce (Z ) which belog o H 2 (R d ). Now he pair of procee aifie, Y = ξ + = Y f (, Y, Z )d f (, Y, Z ) + Z dw Z dw, T. Bu for ay oppig ime ν we have lim ( Yν Y ν ) =, lim ( ν (Z Z )dw ) = hrough Bulkhoder-Davi-Gudy iequaliy. Fially ( ν (f (, Y, Z ) f(, Y, Z ))d ) ( + ( f (, Y, Z ) f(, Y, Z ) 1 [ Y + Z k] d) f (, Y, Z ) f(, Y, Z ) 1 [ Y + Z >k] d)

23 Secio 1.4. BSD wih coiuou coefficie 22 Bu here exi a ubequece which we ill deoe by uch ha he fir erm i he righ coverge o a ice P a.. ad for ay T, (f (, y, z)) coverge uiformly o f(, y, z) o compac ube of R 1+d. The ecod erm i majoriie by a coa δ k which coverge o a k. Heceforh he lef coverge o a. I follow ha for ay ν a oppig ime we have : Y ν = Y ν f(, Y, Z )d + ν Z dw. Fially he opioal ecio heorem (ee [1], page22) implie ha (Y, Z) i a oluio for (1.5). Now if (Y, Z ) i aoher oluio for (1.5), he compario heorem implie ha Y Y ice f f. Therefore akig he limi we obai Y Y ad he Y i maximal. Remark I he ame way a previouly ha we ued a icreaig approximaio of f we would have coruced he miimal oluio of (1.5).

24 Chaper 2 Refleced Backward Sochaic Differeial quaio We udy refleced oluio of oe-dimeioal backward ochaic differeial equaio. We prove uiquee ad exiece by approximaio via pealizaio, we how ha whe he coefficie ha Lipchiz, ad we prove he exiece of a oluio of RBSD wih coiuou ad liear growh coefficie. 2.1 Iroducio I hi ecio, we udy he cae where he oluio i forced o ay above a give ochaic proce, called he obacle. A icreaig proce i iroduced which puhe he oluio upward, o ha i may remai above he obacle. The exiece i eablihed via approximaio i coruced by pealizaio of he corai, we prove alo a compario heorem, imilar o ha i [14], for o-refleced BSD. Fially we prove he exiece of a refleced oluio of oe-dimeioal backward ochaic differeial equaio wih coiuou ad liear growh coefficie. 2.2 Refleced BSD wih Lipchiz coefficie Alog wih hi ecio he dimeio i equal o 1. So we are goig o deal wih oluio of BSD whoe compoe Y are forced o ay above a give barrier. Le {B, T } be a d-dimeioal browia moio defied o a probabiliy pace 23

25 Secio 2.2. Refleced BSD wih Lipchiz coefficie 24 (Ω, F, P ). Le {F, T } be he aural filraio of {B }, where F coai all P ull e of F ad le P be he σ algebra of predicable ube of Ω [, T ]. Le u iroduce ome oaio. L 2 = {ξ i a F T meaurable radom variable.. ( ξ 2 ) < } H 2 = {{ϕ, T } i a predicable proce.. ( ϕ 2 d) < } S 2 = {{ϕ, T } i a predicable proce.. ( up ϕ 2 ) < } T ad we defie : (I) a ermial value ξ L 2. (II) a coefficie f, which i a map : f : Ω [, T ] R R d R uch ha (y, z) R R d, f(., y, z) H 2, (III) for ome K > ad all y, y R, z, z R d, a.. f(, y, z) f(, y, z ) K( y y + z z ) (IV) A obacle {S, T }, which i a coiuou progreively meaurable realvalued proce aifyig : (up T (S + ) 2 ) <. We hall alway aume ha S T ξ a... Defiiio The oluio of RBSD i a riple {(Y, Z, K ), T } of F progreively meaurable procee akig value i R, R d ad R +, repecively, ad aifyig: (V) Z H 2, i paricular Z 2 d < (V ) Y S 2 ad K T L 2 (VI) Y = ξ + f(, Y, Z )d + K T K Z db, T (VII) Y S, T (VIII) {K } i coiuou ad icreaig, K = ad (Y S )dk =. Our mai reul i hi ecio i Theorem (L karoui e al.[15]) Uder he above aumpio, i paricular (I), (II), (III) ad (IV), he RBSD wih (V), (VI), (VII), (VIII) ha a uique oluio (Y, Z, K). Our prove baed o approximaio via pealizaio. I he followig c will deoe a coa whoe value ca vary from lie o lie.

26 Secio 2.2. Refleced BSD wih Lipchiz coefficie 25 Proof. For each N, le {, Z ); T } deoe he uique pair of F progreively meaurable procee wih value i R R d aifyig Z 2 d < ad Y = ξ + f(, Y, Z )d + where ξ ad f aify he above aumpio. We defie K = S ) d S ) d, T. Z db (2.1) We ow eablih a priori eimae, uiform i, o he equece, Z, K ). Ideed uig Iô formula wih ) 2 ad akig expecaio yield : Y 2 + Z 2 d = ξ f(, Y, Z )Y d + 2 Y dk bu for ay T we have, ξ c(1 + (f(,, ) + K Y + K Z ) Y d + 2 Y 2 d) Z 2 d + 1 ( up (S + ) 2 ) + α(kt K ) 2, α T S dk Hece K T K = Y ξ f(, Y, Z )d + Z db. (K T K ) 2 c{( Y 2 ) + ξ ( chooig α = (1/3c), we have 2 3 ( Y 2 ) From Growall lemma i follow ha up ( Y 2 ) + T Z 2 d c(1 + ( Y 2 + Z 2 )d)}, Y 2 d). Z 2 d + (K T ) 2 c, N. Uig agai equaio (2.1) ad he Burkholder-Davi-Gudy iequaliy, we deduce ha ( up Y 2 + T Z 2 d + (KT ) 2 ) c, N. (2.2)

27 Secio 2.2. Refleced BSD wih Lipchiz coefficie 26 We defie f (, y, z) = f(, y, z) + (y S ), f (, y, z) f +1 (, y, z), ad i follow from he compario heorem (for orefleced BSD ee ([34])) ha Y Y +1, T, a.. Hece Y Y, T, a.. ad from (2.2) ad Faou lemma, ( up Y 2 ) c. T I he follow by domiaed covergece ha (Y Y ) 2 d a. (2.3) Now we prove lim (up T (Y S ) 2 =, hi propery i he key poi i he proof of our reul. Le (Ỹ, Z ) be he oluio of he followig BSD : Ỹ = ξ + (f(, Y, Z ) (Ỹ S ))d Z db By compario we have P a.. T, Y F oppig ime uch ha ν T. The, Ỹ, for ay. Now le ν be a Ỹ ν = [ξ exp( (T ν)) + ν {f(, Y, Z ) + S } exp( ( ν))d/f ν }] Sice S i coiuou he ξ exp( (T ν)) + ν S exp( ( ν))d ξ1 {ν=t } + S ν 1 {ν<t } P a.. ad i mea quare. O he oher had Therefore, exp( ( ν))f(, Y, Z )d 1 { ν 2 Ỹ ν ξ1 {ν=t } + S ν 1 {ν<t } L 2 a (f(, Y, Z )) 2 d} 1/2. ad he Y ν S ν a.. From ha ad he ecio heorem (ee [1]), i follow ha P a.., [, T ], Y S ad he heorem implie ha up T S ), T, P a.. Now Dii S ) a. Fially he cocluio em from he domiaed covergece heorem ice for ay, Y 1 S + Y S ad

28 Secio 2.2. Refleced BSD wih Lipchiz coefficie 27 he S ) Y 1 + S +. Now i follow from Iô formula ha for ay m ( Y Y m 2 ) + Sice m, he Z Z m 2 d = K + 2 [f(, Y, Z m ) f(, Y m, Z m )](Y Y m )d(k K m ) ( Y Y m 2 + Y S ) dk m + 2 Y m )(dk dk m ) up T (Y eimae (2.2) ad he fac lim (up T Z Z m 2 d c Y Y m S ) 2 = yield Y m )d Y m Z Z m )d (Y m S ) dk (2.4) Y m ) KT m, ad uig he 2 d + 2 up S ) KT m a heceforh here exi a proce (Z ) T which belog of H 2 (R d ) ad which i he H 2 (R d ) limi of (Z ). Nex goig back o (2.4), akig he upermum ad uig Burkholder-Davi-Gudy iequaliy o obai up (Y T Y m ) 2 + T Z Z m 2 d 2[ up (Y S ) KT m ] + ε up T T + 1 ε Z Z m 2 d + c Y Y Y m 2 Y m 2 d where ε >. We choe ε < 1 implie ha up 2 T (Y Y m ) a, m ad he up T (Y Y ) a, moreover Y = (Y ) T i a coiuou proce. Now ice for ay ad T K = Y Y f(, Y, Z )d + Z db he we have alo, up T K K m 2 a, m. Hece here exi a F adaped o-decreaig ad coiuou proce (K ) T, K = uch ha up T K K 2 a. Fially we prove he limiig proce (Y, Z, K) = (Y, Z, K ) T refleced BSD aociaed wih (f, ξ, S). Obviouly he procee (Y, Z, K ) T aify : Y = ξ + f(, Y, Z )d + K T K Z db, T. i he oluio of he

29 Secio 2.3. Compario heorem for RBSD 28 O he oher had ice lim [up T ((Y S ) ) 2 ] = he P a.., T, Y S. We have alo (Y S )dk = ice he equece ) ad (K ) coverge uiformly repecively o Y ad K ad hak o S )dk = ( S ) ) 2 d. 2.3 Compario heorem for RBSD We prove a compario heorem, imilar o ha of [34] for o-refleced BSD. Theorem Le (ξ, f, S) ad (ξ, f, S ) be wo Refleced BSD, each oe aifyig all he aumpio (I), (II), (III) ad (IV), ad uppoe i addiio he followig : 1) ξ ξ a.. 2) f(, y, z) f (, y, z) dp da.e., (y, z) R R d, 3) S S, T, a.. Le (Y, Z, K) be a oluio of he RBSD (ξ, f, S) ad (Y, Z, K ) a oluio of he RB- SD (ξ, f, S ). The Y Y, T a.. Proof. Applyig Iô formula o (Y Y ) + 2, ad akig expecaio, we ge (Y Y ) {Y>Y } Z Z 2 d 2 (Y Y ) + [f(, Y, Z ) f (, Y, Z )]d Sice o {Y > Y }, Y > S S, we have + 2 (Y Y ) + (dk dk ) = Aume ow ha he Lipchiz codiio o f. The (Y Y ) K 1 {Y>Y } Z Z 2 d (Y Y ) + (dk dk ). (Y Y ) + dk. (Y Y ) + [f(, Y, Z ) f(, Y, Z )]d (Y Y ) + ( (Y Y 1 {Y>Y } Z Z 2 d + c ) + Z Z )d (Y Y ) + 2 d.

30 Secio 2.4. Refleced BSD wih coiuou coefficie 29 Hece (Y Y ) + 2 c (Y Y ) + 2 d. ad from Growall lemma, (Y Y ) + =, T. 2.4 Refleced BSD wih coiuou coefficie We are give hree objec : 1) a ermial value ξ L 2 (Ω, F T, P ). 2) a coefficie f which i a map uch ha : f : [, T ] Ω R R d R i) (y, z) R R d, (, ω) f(, ω, y, z) i P meaurable. ii) P a.. [, T ], f(, ω, y, z) i coiuou i (y, z) o R R d. Moreover here exi a coa C > uch ha for ay (, y, z) [, T ] R R d, f(, ω, y, z) C(1+ y + z ) 3) A obacle {S, T }, which i coiuou ad F progreively meaurable proce aifyig : We hall alway aume ha S T ξ a.. ( up (S ) 2 ) <. T Theorem (A.Maoui)[29]. Le (ξ, f, S) be a riple aifyig he above aumpio 1)-3). The here exi a F progreively meaurable riple{(y, Z, K ), T } oluio of he refleced BSD : Y = ξ + uch ha : i ) (Y 2 + Z 2 )d <, ii ) Y S, T, f(, y, z)d + K T K Z db ; T. (2.5) iii ) {K, T } i a coiuou ad icreaig proce, K =, ad (Y S )dk =. To prove heorem (2.4.1), we eed a impora reul which give a approximaio of coiuou fucio by Lipchiz fucio (ee [27]) f (x) = if y Q {f(y) + x y }.

31 Secio 2.4. Refleced BSD wih coiuou coefficie 3 Proof. Coider, f defie below. The f i a meaurable fucio a well a a Lipchiz fucio. Moreover, ice ξ L 2 ad {S, T } aify 3), we ge from l karoui e al. [15] ha here i a uique oluio, Z, K ), T for RBSD aifyig equaio 2.6 ad: Y = ξ + f (, Y, Z )d + KT K Z db (2.6) ( Y 2 + Z 2 )d <, Y S, T, {K, T } i a coiuou ad icreaig proce, K = ad S )dk =. Uig he compario heorem of RBSD i l karoui e al. (1996)[15], we obai m K, Y Y m, d dp a.. (2.7) The idea of he proof of heorem (2.4.1) i o eablih ha he limi of he equece, Z, K ) i a oluio of he RBSD (2.6) wih parameer (f, ξ, S). From ow o he proof will be divided io four ep. Sep 1: There exi a coa C, uch ha K, (up T Y 2 + Z 2 d + (KT )2 ) C. C > deoe a coa, whoe value may vary from lie o lie. From Iô formula applied o ) 2, i follow ha ) 2 + Z 2 = ξ f (, Y, Z )Y d + 2 Y dk 2 Y Z db, Takig expecaio, ad uig he fac ha Y Z db i uiformly iegrable(ee [15]) ad ued he ideiy S )dk =, we deduce Y 2 + Z 2 = ξ f (, Y, Z )Y d + 2 S dk. uig he uiform liear growh of f ad he iequaliy 2ab a2 + ε εb2, ε >, Y 2 + Z 2 C(1 + Y 2 d) Z 2 d + 1 ( up (S + ) 2 ) + ε((kt K ) 2 ). ε T

32 Secio 2.4. Refleced BSD wih coiuou coefficie 31 Now from (2.6), we ge ((K T K ) 2 ) C( ) 2 + ξ Chooig α = 1, we obai 3C 2 3 ) Z 2 C(1+ ( ) 2 + Z 2 )d). ) 2 d), ) 2 C(1+ ) 2 d). (2.8) I he follow from Growall lemma ha : up T Y 2 C, ad from (2.8) ad he la iequaliy we ge Z 2 d C, (K T ) 2 C. The reul of ep 1 he from equaio (2.6), he above eimae ad he Burkholder- Davi-Gudy iequaliy. Sep 2.We hould prove ha he equece of procee Z coverge i H 2 (R). We have from (2.7) ad he reul of ep 1. Y Y +1, T, P a.. ad up T ( Y 2 ) C. Hece Y Y, T, P a.., ad from Faou lemma, we have (up T Y 2 ) C. I he follow by he domiaed covergece heorem ha Y Y 2 d a. (2.9) For all p K, from Iô formula for =, ad uig he fac Y obai Y Y p 2 + Z Z p 2 d S, we Y p )(f (, Y, Z ) f p (, Y p, Z p ))d S )dk + 2 (Y p S )dk p, From he ideiy S )dk =, ad uig he Hölder iequaliy, we have Z Z p 2 d 2( Y Y p 2 d) 1/2 ( f (, Y, Z ) f p (, Y p, Z p ) 2 d) 1/2, Now, uig he uiform liear growh o he equece (f ) ad he fac, Z ) i bouded, we obai he exiece of a coa C depedig oly o K, T, ξ 2 ad (up T (S + ) 2 ) uch ha, p, Z Z p C Y Y p 1/2.

33 Secio 2.4. Refleced BSD wih coiuou coefficie 32 The from (2.9), (Z ) i a Cauchy equece i H 2 (R d ), ad here exi a F progreively meaurable proce Z uch ha Z Z i H 2 (R d ), a. Hece, ( Y Sep 3.We prove ha (up T Y From Iô formula, we have Y Y p 2 + Z Z p 2 d = 2 Y p 2 + Z Z p 2 )d a, p. (2.1) + 2 Y p 2 ) a, p. Y p )(f (, Y, Z ) f p (, Y p, Z p ))d Y p )(dk dk p ) 2 From he above proof, we have p, Y p )(dk dk p ). The Y Y p 2 2 from which we deduce ( up Y Y p 2 ) 2( T Y p )(f (, Y, Z ) f p (, Y p, Z p ))d 2 Y + 2( up T Y p 2 d) 1/2 ( Y p )(Z Z p )db ). Y p )(Z Z p )db. Y p )(Z Z p )db, f (, Y, Z ) f p (, Y p, Z p )) 2 d) 1/2 Uig agai he uiform liear growh o he equece (f ) ad he fac ha, Z ) i bouded, we deduce ( f (, Y, Z ) f p (, Y p, Z p )) 2 d) 1/2 C. (2.11) Aferward, from he Burkholder-Davi-Gudy iequaliy, we obai 2( up T Y p )(Z Z p )db ) 1 ( up Y Y p 2 ) + C Z Z p 2 d. 2 T Hece, from (2.11) ad he above iequaliy ( ) ( up Y Y p 2 ) C ( Y Y p 2 d) 1/2 + Z Z p 2 d. T The from (2.1), we have ( up Y Y p 2 ), a, p, (2.12) T

34 Secio 2.4. Refleced BSD wih coiuou coefficie 33 from which we deduce ha P a.., Y coverge uiformly i o Y ad ha Y i a coiuou proce. Sep 4. Now accordig o (2.6), we have for all, p K, ( up K K p 2 ) Y Y p 2 + ( up T + C + ( up T T Y Y p 2 ) f (, Y, Z ) f p (, Y p, Z p ) 2 d (Z Z p )db 2 ). (2.13) We eed o how ha he equece of procee (f (., Y, Z )) coverge o f(., Y, Z) i H 2 (R). Thi i deduce from he followig fac : a) Y Y i H 2 (R) ad d dp a.. b) Sice Z Z i H 2 (R) he here exi a proce Z i H 2 (R d ) ad a ubequece uch ha, Z Z, Z Z, d dp a.. Therefore, from he lemma (1.4.2) we ge f (, Y, Z ) f(, Y, Z ), d a.. a ad f (, Y, Z ) K(1 + Y + Y + Z ). Thu, i follow by he domiaed covergece heorem ha f (, Y, Z ) f(, Y, Z ) 2 d a. (2.14) From Burkholder-Davi-Gudy iequaliy ad (2.12)-(2.14) we obai ( up K K p 2 ) a, p. T Coequely, here exi a progreively meaurable proce K wih value i R uch ha ( up K K 2 ) a, (2.15) T ad he {K, T } i clearly a icreaig ( wih K = ) ad a coiuou proce. Takig limi i he RBSD (2.6) we obai ha he riple {(Y, Z, K ), T } i a oluio of he RBSD (2.5). Now from Sep 1., we have ( Y 2 + Z 2 )d C, akig limi i hi iequaliy, we obai i ) ( Y 2 + Z 2 )d C. O he oher had, we have K, Y S, [, T ], akig limi we have clearly ii ). From (2.12) ad (2.15) we have S )dk (Y S )dk, P a.. a, uig he ideiy S )dk =, we obai (Y S )dk =.

35 Par II BACKWARD DOUBLY STOCHASTIC DIFFRNTIAL QUATIONS. 34

36 Chaper 3 Backgroud o backward doubly ochaic differeial equaio I hi chaper, a ew cla of backward ochaic differeial equaio i iveigaed. which allow u o produce a probabiliic repreeaio of cerai quai-liear ochaic parial differeial equaio, we prove he exiece ad uiquee of a oluio where he coefficie i Lipchiz, afer we obai a compario heorem of hee Backward Doubly SD. A oe of i applicaio, we prove he exiece of a oluio for BDSD wih coiuou coefficie. 3.1 Iroducio Thi ew kid of backward SD eem o be uiable o give a probabiliic repreeaio for a yem of parabolic ochaic parial differeial equaio (SPD). We refer o Pardoux ad Peg (1994)[35] for he lik bewee SPD ad BDSD i he paricular cae where oluio of SPD are regular. I ecio 1, we udy exiece ad uiquee of he oluio where he coefficie i Lipchiz, i ecio 2 ad 3, we hall prove he compario heorem of BDSD. The we udy BDSD wih coiuou coefficie a a applicaio of he compario heorem. 35

37 Secio 3.2. BDSD wih Lipchiz coefficie Backward doubly ochaic differeial equaio wih Lipchiz coefficie Noaio ad aumpio Le T be a fixed fial ime. Throughou hi par {W, T } ad {B, T } will deoe wo idepede d-dimeioal Browia moio (d 1), defied o he complee probabiliy pace (Ω, F, P). Le N deoe he cla of P-ull e of F. For each [, T ], we defie F F W F B,T N, I oher word he σ-field F, T, are P-complee. We oice ha he family of σ-algebra F = {F } T i eiher icreaig or decreaig; i paricular, i i o a filraio. For ay 1, we coider he followig pace of procee: 1. The Baach pace M 2 (F, [, T ]; R ) of all equivalece clae (wih repec o he meaure dp d) where each equivalece cla coai a -dimeioal joily meaurable radom proce {ϕ, [, T ]} which aifie: (i) ϕ 2 d < ; (ii) ϕ i F -meaurable, for d-almo all [, T ]. Uually a equivalece cla will be ideified wih (oe of) i member. 2. S 2 (F, [, T ]; R ) i he e of coiuou -dimeioal radom procee which aify: (i) up ϕ 2 < ; T (ii) ϕ i F -meaurable, for a.e [, T ]. We coider coefficie (f, g) wih he followig properie: f : Ω [, T ] R R d R, g : Ω [, T ] R R d R d, uch ha here exi F -adaped procee {f, g : T } wih value i [1, + ) ad wih he propery ha for ay (, y, z) [, T ] R R d, he followig hypohee are aified for ome ricly poiive fiie coa L ad < α < 1 uch ha for ay

38 Secio 3.2. BDSD wih Lipchiz coefficie 37 (y 1, z 1 ), (y 2, z 2 ) R R d, : f (, y, z), g (, y, z) are F -meaurable procee, (H3.1) (i) f (, y 1, z 1 ) f (, y 2, z 2 ) 2 c ( y 1 y z 1 z 2 2), (ii) g (, y 1, z 1 ) g (, y 2, z 2 ) 2 c y 1 y α z 1 z 2 2. We poi ou ha by C we alway deoe a fiie coa whoe value may chage from oe lie o he ex, ad which uually i (ricly) poiive xiece ad uiquee heorem Suppoe ha we are give a ermial codiio ξ L 2 (Ω, F T, P) Defiiio By defiiio a oluio o a BDSD (ξ, f, g, ) i a pair (Y, Z) S 2 (F, [, T ]; R ) M 2 ( F, [, T ]; R d), uch ha for ay T Y = ξ + f (, Y, Z ) d + g (, Y, Z ) db Z dw. (3.1) Here db deoe he claical backward Iô iegral wih repec o he Browia moio B. Our mai goal i hi ecio i o prove he followig heorem. Theorem Uder he above hypohei (H3.1) here exi a uique oluio for he BDSD (3.1). Le u fir eablih he reul i Theorem for BDSD, where he coefficie f, g do o deped o (y, z). More preciely, le f, ad g : Ω [, T ] R d aify (H3.1), ad le ξ be a before. Coider he equaio: Y = ξ + The we have he followig reul. f () d + g () db Z dw. (3.2) Theorem Uder he hypohei (H3.1), here exi a uique oluio o equaio (3.2). Proof. xiece. To how he exiece, we coider he filraio G = F W F B T ad he marigale [ M = ξ + f () d + g () ] db /G, (3.3)

39 Secio 3.2. BDSD wih Lipchiz coefficie 38 which i clearly a quare iegrable marigale by (H3.1). A exeio of Iô marigale repreeaio heorem yield he exiece of a G -progreively meaurable proce (Z ) wih value i R d uch ha Z 2 d < ad M T = M + Z dw. (3.4) We ubrac he quaiy f () d + g () db from boh ide of he marigale i (3.3) ad we employ he marigale repreeaio i (3.4) o obai where Y = ξ + f () d + h () dk + g () db Z dw, [ Y = ξ + f () d + g () ] db /G. I remai o how ha (Y ) ad (Z ) are i fac F -adaped. For Y, hi i obviou ice for each, Y = (Θ/F F B ) Where Θ i F F B meaurable. Hece F B i idepede of F σ(θ), ad Now Z dw = ξ + Y = (Θ/F ). f () d + g () db Y, ad he righ ide i FT W F,T B meaurable. Hece, from Iô marigale repreeaio heorem, Z, < < T i F W F,T B adaped. Coequely Z i F W F,T B meaurable, for ay <, o i i F W F,T B meaurable. Uiquee. I immediae, ice if (Y, Z) i he differece of wo oluio, Hece by orhogoaliy Y + Z dw =, T. ( Y 2 ) + ad Y P a.., Z = ddp a.e. We will alo eed he followig Iô-formula. T r[z Z ]d =,

40 Secio 3.2. BDSD wih Lipchiz coefficie 39 Lemma Le α S 2 (F, [, T ]; R ), β M 2 (F, [, T ]; R ), γ M 2 ( F, [, T ]; R d), ad δ M 2 ( F, [, T ]; R d) be uch ha: α = α + The, for ay fucio φ C 2 (R, R) φ (α ) = φ (α ) + I paricular, + Proof. See [35]. α 2 = α β d + φ (α ), β d + φ (α ), δ dw 1 2 γ 2 d + α, β d + 2 δ 2 d. γ db + δ dw. φ (α ), γ db Tr[φ (α ) γ γ ] d α, γ db + 2 Tr[φ (α ) δ δ ] d. α, δ dw Nex, we eablih a a priori eimae for he oluio of he BSD i (3.1). for ha ake, we eed a addiioal aumpio o g. (H3.2) { here exi c uch ha for all(, y, z) [, T ] R k R k d, gg (, y, z) zz + c( g(,, ) 2 + y 2 )I. Propoiio Aume, i addiio o he codiio of Theorem(3.2.3), ha (H3.2) hold ad for ome p > 2, ξ (Ω, F, P, R k ) ad The ( f(,, ) p + g(,, ) p )d <. ( up Y p + ( Z 2 ) p/2 ) <. T Proof. By lemma applied o ϕ(x) = x p, we obai ha Y p + p 2 = ξ p + p Y p 2 Z 2 d + p 2 (p 2) Y p 4 (Z Z Y, Y )d Y p 2 f(, Y, Z ), Y d + p Y p 2 Y, g(, Y, Z )db + p Y p 2 g(, Y, Z ) 2 d 2 + p 2 (p 2) Y p 4 gg (, Y, Z )Y, Y d p Y p 2 Y, Z dw.

41 Secio 3.2. BDSD wih Lipchiz coefficie 4 Takig he expecaio, we ge ( Y p ) + p 2 Y p 2 Z 2 d + p 2 (p 2) Y p 4 Z Z Y, Y d ( ξ p ) + p Y p 2 f (, Y, Z ), Y d + p 2 Y p 2 g (, Y, Z ) 2 d + p 2 (p 2) Y p 4 gg (, Y, Z )Y, Y )d. We ca coclude from (H3.1) ha for ay α < α < 1, here exi c(α ) uch ha g(, y, z) c(α )( y 2 + g(,, ) 2 ) + α z 2. Bu from (H3.1), (H3.2) ad he fac ha 2 ab 1 α 2c a2 + 2c 1 α b2, c >, i follow ha here exi a coa θ > ad c uch ha ( Y p ) + θ ( ξ p ) + c Y p 2 Z 2 d ( Y p + f (,, ) p + g (,, ) p )d The, from Growall Lemma we obai ( ) up Y p + Y p 2 Z 2 d < T Applyig he ame iequaliie we have already ued o he fir ideiy of he proof, we deduce ha Y p ξ p + c ( Y p + f(,, ) p + g(,, ) p )d + p Y p 2 Y, g(, Y, Z )db p Y p 2 Y, Z dw from he Burkholder-Davi-Gudy iequaliy, we ge (up T Y p ) ξ p + c ( Y p + f(,, ) p + g(,, ) p )d + c Y 2p 4 gg (, Y, Z )Y, Y d + c Y 2p 4 Z Z Y, Y d

42 Secio 3.2. BDSD wih Lipchiz coefficie 41 We eimae he la erm a follow : T Y 2p 4 Z Z Y, Y d (Y p/2 T Y p 2 Z 2 d we deduce ha Now we have + Z 2 d = ξ 2 Y Hece for ay δ >, g(, Y, Z ) 2 d (up T Y p ) Y p 2 Z 2 d (up T Y p ) < f(, Y, Z ), Y d + 2 Y, Z dw ( Z 2 d) p/2 (1 + δ)( g(, Y, Z ) 2 d) p/2 + c(δ, p)[ ξ p + Y p + + Paig o expecaio Y, g(, Y, Z )db p/2 + Y, Z dw p/2 ] ( Z 2 d) p/2 (1 + δ) 2 α( Z 2 d) p/2 + c (δ, p) + c(δ, p)[ Z Y d p/2 ] + c(δ, p)[( Y, g(, Y, Z )db Y 2 Z 2 d) p/4 ] (1 + δ) 2 α( Z 2 d) p/2 + c (δ, p) + c(δ, p){(up T Y p/2 )[( Z d) p/2 + ( Z 2 d) p/4 ]} [(1 + δ) 2 α + (1 + δ)][( Z 2 d) p/2 ] + c (δ, p). The ecod par of he reul ow follow, if we chooe δ > mall eough uch ha (1 + δ) 2 α + (1 + δ) < 1 f(, Y, Z ), Y d p/2 We ca ow ur o he proof of heorem 3.2.2

43 Secio 3.2. BDSD wih Lipchiz coefficie 42 Proof Uiquee.Le (Y 1, Z 1 ) ad (Y 2, Z 2 ) be wo oluio. Defie The Y = Y = Y 1 Y 2, Z = Z 1 Z 2, T [f(, Y 1, Z 1 ) f(, Y 2, Z 2 )]d + Applyig Iô formula o Y 2 yield : Y 2 + Z 2 d = 2 + Hece from (H3.1) ad he iequaliy ab 1 2(1 α) a2 + 1 α Y 2 + Z 2 d c(α) [g(, Y 1, Z 1 ) g(, Y 2, Z 2 )]db Z dw. f(, Y 1, Z 1 ) f(, Y 2, Z 2 ), Y d g(, Y 1, Z 1 ) g(, Y 2, Z 2 ) 2 d. Y 2 d + 1 α 2 2 b2, Z 2 d + α where < α < 1 i he coa appearig i (H3.1). Coequely Y α 2 Z 2 d c(α) Y 2 d. Z 2 d. From Growall lemma, ( Y 2 ) =, T, ad hece Z 2 =. xiece.we defie recurively a equece, Z ) =,1,... a follow. Le Y, Z. Give, Z ), +1, Z +1 ) i he uique oluio, coruced a i heorem (3.2.3), of he followig equaio : Y +1 = ξ + f(, Y, Z )d + g(, Y, Z )db Z +1 dw. (3.5) Le Y +1 Y +1 Y, Z +1 Z +1 Z, T. The ame compuaio a i he proof of uiquee yield : ( Y +1 2 ) + Z +1 2 d = 2 + Le β R. By iegraio by par, we deduce ( Y +1 2 e β ) + β = 2 + f(, Y, Z ) f(, Y 1, Z 1 ), Y +1 g(, Y, Z ) g(, Y 1 Y +1 2 e β d + Z +1 2 e β d f(, Y, Z ) f(, Y 1, Z 1 ), Y +1 e β d g(, Y, Z ) g(, Y 1, Z 1 ) 2 e β d., Z 1 ) 2 d.

44 Secio 3.3. Compario Theorem of BDSD 43 There exi c, γ > uch ha ( Y +1 2 e β ) + (β γ) (c Y α Z 2 )e β d. 2 Now chooe β = γ + 2c 2c, ad defie c = 1+α ( Y +1 2 e β ) α 2 I follow immediaely ha Y +1 2 e β d + 1+α, (c Y Z +1 2 )e β d (c Y 2 + Z 2 )e β d. (c Y Z +1 2 )e β d ( 1 + α ) 2 Z +1 2 e β d (c Y 2 + Z 2 )e β d. ad, ice 1+α 2 < 1,, Z ) =,1,... i a Cauchy equece i M 2 (, T ; R k ) M 2 (, T ; R k l ). I i he eay o coclude ) =,1,... i alo Cauchy i S 2 ([, T ]; R k ), ad ha olve equaio (3.1). (Y, Z ) = lim, Z ) 3.3 Compario Theorem of Backward doubly ochaic differeial equaio I hi ecio, we oly coider oe-dimeioal BDSD. We coider he followig BDSD : ( T ) Y 1 = ξ 1 + Y 2 = ξ 2 + f 1 (, Y 1, Z 1 )d + f 2 (, Y 2, Z 2 )d + g(, Y 1, Z 1 )db g(, Y 2, Z 2 )db Z 1 dw (3.6) Z 2 dw (3.7) where BDSD (3.6) ad (3.7) aify he codiio of heorem (3.2.2). The here exi wo pair of meaurable procee (Y 1, Z 1 ) ad (Y 2, Z 2 ) aifyig BDSD (3.6) ad (3.7), repecively. Aume (H3.3) { ξ 1 ξ 2, f 1 (, Y, Z) f 2 (, Y, Z), The we have he followig compario heorem. a.., a..,

45 Secio 3.3. Compario Theorem of BDSD 44 Theorem Aume BDSD (3.6) ad (3.7) aify he codiio of heorem (3.2.2), le (Y 1, Z 1 ) ad (Y 2, Z 2 ) be oluio of BDSD (3.6) ad (3.7), repecively. If (H3.3) hold, he Y 1 Y 2, a., [, T ]. Proof. The pair (Y 1 Y 2, Z 1 Z 2 ) aifie he followig BDSD. Y 1 Y 2 = (ξ 1 ξ 2 ) + + (f 1 (, Y 1, Z 1 ) f 2 (, Y 2, Z 2 ))d (g(, Y 1, Z 1 ) g(, Y 2, Z 2 ))db (Z 1 Z 2 )dw, T. Applyig Iô formula o (Y 1 Y 2 ) 2, we ge (Y 1 Y 2 ) 2 = (ξ 1 ξ 2 ) From (H3.3), we have ξ 1 ξ 2, o (Y 1 Y 2 ) (f 1 (, Y 1, Z 1 ) f 2 (, Y 2, Z 2 ))d (Y 1 Y 2 ) (g(, Y 1, Z 1 ) g(, Y 2, Z 2 ))db 1 Y 1 Y 2 g(, Y 1, Z 1 ) g(, Y 2 (Y 1 Y 2 ) (Z 1 Z 2 )dw (ξ 1 ξ 2 ) 2 =., Z 2 ) 2 d 1 Y 1 Y 2 Z1 Z 2 2 d. (3.8) Sice (Y 1, Z 1 ) ad (Y 2, Z 2 ) are i S 2 ([, T ]; R) M 2 (, T ; R d ) i eaily follow ha Le = 2 = 2 2 (Y 1 Y 2 ) (Z 1 Z 2 )dw =, (Y 1 Y 2 ) (g(, Y 1, Z 1 ) g(, Y 2, Z 2 ))db =. = 1 + 2, (Y 1 Y 2 ) (f 1 (, Y 1, Z 1 ) f 2 (, Y 2, Z 2 ))d (Y 1 Y 2 ) (f 1 (, Y 1, Z 1 ) f 1 (, Y 2, Z 2 ))d (Y 1 Y 2 ) (f 1 (, Y 2, Z 2 ) f 2 (, Y 2, Z 2 ))d

46 Secio 3.4. BDSD wih coiuou coefficie 45 where 1 = 2 2 = 2 (Y 1 Y 2 ) (f 1 (, Y 1, Z 1 ) f 1 (, Y 2, Z 2 ))d (Y 1 Y 2 ) (f 1 (, Y 2, Z 2 ) f 2 (, Y 2, Z 2 ))d. From (H3.1) ad Youg iequaliy, i follow ha 1 2C (2C + (Y 1 Y 2 ) ( Y 1 Y 2 + Z 1 Z 2 )d C2 1 α ) (Y 1 Y 2 ) 2 d + (1 α) Uig he aumpio (H3.1), agai, we deduce = C 1 Y 1 Y 2 g(, Y 1, Z 1 ) g(, Y 2, Z 2 ) 2 d 1 Y 1 Y 2 [C Y 1 Y 2 (Y 1 Y 2 ) 2 d + α Takig expecaio o boh ide of (3.8), we ge (Y 1 Y 2 ) 2 C By Growall iequaliy, i follow ha Tha i, Y 1 Y 2, a.., [, T ]. 2 + α Z 1 Z 2 2 ]d 1 Y 1 Y 2 Z1 Z 2 2 d. (Y 1 Y 2 ) 2 d. (Y 1 Y 2 ) 2 = [, T ]. 1 Y 1 Y 2 Z1 Z 2 2 d, 3.4 Backward doubly ochaic differeial equaio wih coiuou coefficie I hi ecio we udy BDSD wih coiuou coefficie. We coider coefficie (f, g) wih he followig properie: f : Ω [, T ] R R d R, g : Ω [, T ] R R d R l,

47 Secio 3.4. BDSD wih coiuou coefficie 46 uch ha here exi F -adaped procee {f, g : T } wih value i [1, + ) ad wih he propery ha for ay (, y, z) [, T ] R R d, he followig hypohee are aified for ome ricly poiive fiie coa K, L ad < α < 1 uch ha for ay (y 1, z 1 ), (y 2, z 2 ) R R d, : f (, y, z), g (, y, z) are F -meaurable procee, (H3.4) (i) f (, y, z) K (1 + y + z ), (ii) g (, y 1, z 1 ) g (, y 2, z 2 ) 2 c y 1 y α z 1 z 2 2. Theorem Uder he above hypohee (H3.4) ad if ξ L 2, here exi a oluio for he BDSD (3.1). Moreover, here i a miimal oluio (Y, Z) of BDSD (3.1) i he ee ha, for ay oher oluio (Y, Z) of BDSD (3.1), we have Y Y. We ill aume ha l = d = 1. Before givig he proof of Theorem 3.4.1, we defie, a he claical approximaio ca be proved by adapig he proof give i J. J. Aliber ad K. Bahlali [1], he equece f (, y, z) aociaed o f, f (, y, z) = if [f(, y, z) + ( y y,z Q y + z z )], he for N,f i joily meaurable ad uiformly liear growh i y, z wih coa N. We alo defie he fucio. F (, y, z) = N(1 + y + Z ) Give ξ L 2, by heorem (3.2.2), here exi wo pair of procee, Z ) ad (U, V ), which are he oluio o he followig BDSD, repecively, Y = ξ + f (, Y, Z )d + g(, Y, Z )db Z dw (3.9) U = ξ + F (, U, V )d + From heorem (3.3.1) ad lemma 1 (ee [27]), we ge g(, U, V )db V dw (3.1) m N, Y m Y U, d dp a.. (3.11) Lemma There exi a coa A > depedig oly o N, C, α, T ad ξ, uch ha N, Y A, Z A; U A, V A

48 Secio 3.4. BDSD wih coiuou coefficie 47 Proof. Fir of all, we prove ha U ad V are all bouded. Clearly, from 3.11 here exi a coa B depedig oly o N, C, α, T ad ξ, uch ha ( Applyig Iô formula o U 2, we have Y 2 d) 1/2 B, ( U 2 d) 1/2 B, V B. U 2 = ξ U V dw + U F (, U, V )d + 2 g(, U, V ) 2 d U g(, U, V )db From (H3.1), for all α < α < 1, here exi a coa C(α ) > uch ha From 3.12 ad 3.13, i follow ha U 2 + V 2 d (3.12) g(, u, v) 2 C(α )( u 2 + g(,, ) 2 ) + α v 2 (3.13) V 2 d ξ 2 + 2N 2 U V dw + C(α ) U (1 + U + V )d + 2 ξ 2 + N 2 (T ) + C(α ) α 2 V 2 d + (1 + 2N + C(α ) + 2N 2 ) 1 α + 2 U g(, U, V )db 2 Takig expecaio, we ge by Youg iequaliy, U g(, U, V )db ( U 2 + g(,, ) 2 )d + α V 2 d g(,, ) 2 d U 2 d U V dw. U α 2 V 2 d ( ξ 2 + N 2 T + C(α ) g(,, ) 2 d) + (1 + 2N + C(α ) + 2N 2 ) U 1 α 2 d + 2 up U g(, U, V )db + 2 up T T U V dw. (3.14)