BSDEs with polynomial growth generators Philippe Briand IRMAR, Universite Rennes 1, Rennes Cedex, FRANCE Rene Carmona Statistics & Operations R

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1 BSDEs wh olyomal growh geeraors Phle Brad IRMAR, Uverse Rees 1, Rees Cedex, FRANCE Ree Carmoa Sascs & Oeraos Research, Prceo Uversy, Prceo NJ 8544, USA July 22, 1998 revsed December 15, 1998 Absrac I hs aer, we gve exsece ad uqueess resuls for backward sochasc dereal equaos whe he geeraor has olyomal growh he sae varable. We deal wh he case of xed ermal me as well as he case of radom ermal me. The eed for hs ye of exeso of he classcal exsece ad uqueess resuls comes from he desre o rovde a robablsc rereseao of he soluos of semlear aral dereal equaos he sr of a olear Feyma-Kac formula. Ideed may alcaos of eres, he oleary s olyomal, see e.g. he Alle-Cah equao or he sadard olear hea ad Schrodger equaos. 1 Iroduco I s by ow well-kow ha here exss a uque, adaed ad square egrable, soluo o a backward sochasc dereal equao (BSDE for shor) of ye Y = + f(s Y s s )ds ; s dw s T rovded ha he geeraor s Lschz boh he varables y ad z. We refer o he orgal work of E. Pardoux ad S. Peg [13, 14] for he geeral heory ad o N. El Karou, S. Peg ad M.-C. Queez [6] for a survey of he alcaos of hs heory ace. Sce he rs exsece ad uqueess resul esablshed by E. Pardoux ad S. Peg 199, a lo of works, cludg R. W. R. Darlg, E. Pardoux [5], S. Hamadee [8], M. Kobylask [9], J.- P. Leeler, J. Sa Mar [1, 11], see also he refereces here, have red o weake he Lschz assumo o he geeraor. Mos of hese works deal oly wh real-valued BSDEs [8, 9, 1, 11] because of her deedece o he use of he comarso heorem for BSDEs (see e.g. N. El Karou, S. Peg, M.-C. Queez [6, Theorem 2.2]). Furhermore, exce [11], he geeraor s always assumed o be a mos lear he sae varable. Le us meo everheless a exceo: [11], J.-P. Leeler ad J. Sa Mar accomodae a growh of he geeraor of he followg ye: C ; 1+jxj log jxj, C ; 1+jxj log log jxj... O he oher had, oe of he mos romsg eld of alcao for he heory of BSDEs s he aalyss of ellc ad arabolc aral dereal equaos (PDEs for shor) ad we refer o E. Pardoux [12] forasurvey of her relaoshs. Ideed, as was revealed by S. Peg [17] ad by E. Pardoux, S. Peg [14] (see also he corbuos of G. Barles, R. Buckdah, 1

2 2 Phle Brad ad Ree Carmoa E. Pardoux [1], Ph. Brad [3], E. Pardoux, F. Pradelles,. Rao [15], E. Pardoux, S. hag [16] amog ohers), BSDEs rovde a robablsc rereseao of soluos (vscosy soluos he mos geeral case) of semlear PDEs. Ths rovdes a geeralzao o he olear case of he well kow Feyma-Kac formula. I may examles of semlear PDEs, he oleary soof lear growh (as mled by a global Lschz codo) bu sead, s of olyomal growh, see e.g. he olear hea equao aalyzed by M. Escobedo, O. Kava ad H. Maao [7]) or he Alle-Cah equao (G. Barles, H. M. Soer, P. E. Sougads [2]). If oe aems o sudy hese semlear PDEs by meas of he olear verso of he Feyma-Kac formula, alluded o above, oe has o deal wh BSDEs whose geeraors wh olear (hough olyomal) growh. Uforuaely, exsece ad uqueess resuls for he soluos of BSDE's of hs ye were o avalable whe we rs sared hs vesgao, ad llg hs ga he leraure was a he org of hs aer. I order o overcome he dcules roduced by he olyomal growh of he geeraor, we assume ha he geeraor sases a kd of mooocy codo he sae varable. Ths codo s very useful he sudy of BSDEs wh radom ermal me. See he works of S. Peg [17], R. W. R. Darlg, E. Pardoux [5], Ph. Brad, Y. Hu [4] for aems he sr of our vesgao. Eve hough looks raher echcal a rs, s esecally aural our coex: deed, s la o check ha s sased all he examles of semlear PDEs quoed above. The res of he aer s orgazed as follows. I he ex seco, we x some oao, we sae our ma assumos ad we rove a echcal rooso whch wll be eeded he sequel. I seco 3, we deal wh he case of BSDEs wh xed ermal me: we rove a exsece ad uqueess resul ad we esablsh some a ror esmaes for he soluos of BSDEs hs coex. I seco 4, we cosder he case of BSDEs wh radom ermal mes. BSDEs wh radom ermal mes lay a crucal role he aalyss of he soluos of ellc semlear PDEs. They were rs roduced by S. Peg [17] ad he suded a more geeral framework by R. W. R. Darlg, E. Pardoux [5]. These equaos are also cosdered [12]. Ackowledgmes. We are graeful o Professeur Eee Pardoux for several fruful dscussos durg he rearao of hs mauscr. Also, he rs amed auhor would lke o hak he Sascs & Oeraos Research Program of Prceo Uversy for s warm hosaly. 2 Prelmares 2.1 Noao ad Assumos Le ; ( F IP) be a robably sace carryg a d-dmesoal Browa moo (W ), ad F be he lrao geeraed by (W ). As usual we assume ha each -eld F has bee augmeed wh he IP-ull ses o make sure ha ; F s rgh couous ad comlee. For y 2 IR k,we deoe by jyj s Eucldea orm ad f z belogs o IR kd, jz j deoes r(zz ) 1=2. For q>1, we dee he followg saces of rocesses: S q = H q = ( ( ) rogressvely measurable 2 IR k j j q S q := j j q < 1, T ) q=2 rogressvely measurable 2 IR kd j j q q := j j 2 d < 1,

3 BSDEs wh olyomal geeraors 3 ad we cosder he Baach sace B q = S q H q edowed wh he orm: q=2 j(y )j q q = jy j q + j j 2 d : T We ow roduce he geeraor of our BSDEs. We assume ha f s a fuco deed o [ T] IR k IR kd, wh values IR k such a way ha he rocess ; f( y z) 2[ T ] s rogressvely measurable for each (y z) IR k IR kd. Furhermore we make he followg assumo. (A 1). There exs cosas, 2 IR, C ad>1suchhaip ; a:s:, wehave: y 8(z z ) f( y z) ; f( y z ) jz ; z j z 8(y y ) (y ; y ) ; f( y z) ; f( y z) ;jy ; y j y 8z f( y z) f( z) + C ; 1+jyj ) z y 7;! f( y z) scouous. We refer o he codo (A 1).2 as a mooocy codo. Our goal s o sudy he BSDE Y = + f(s Y s s )ds ; s dw s T (1) whe he geeraor f sases he above assumo. I he classcal case = 1, he ermal codo ad he rocess ; f( ) are assumed o be square egrable. I he olear 2[ T ] case >1, we eed sroger egrably codos o boh ad ; f( ).We ose 2[ T ] ha: (A 2). s a F T -measurable radom varable wh values IR k such ha h jj 2 + f(s ) 2 ds < 1: Remark. We cosder here oly he case > 1 sce he case = 1 s reaed he works of R. W. R. Darlg, E. Pardoux [5] ade. Pardoux [12]. 2.2 A Frs a ror Esmae We ed hese relmares by esablshg a a ror esmae for BSDEs he case where ad f( ) are bouded. The followg rooso s a mere geeralzao of a resul of S. Peg [18, Theorem 2.2] who roved he same resul uder a sroger assumo o f amely, 8 y z f( y z) + jyj + jz j: Our corbuo s merely o remark ha hs roof requres oly a esmae of y f( y z) ad hus ha he resul should sll rue our coex. We clude a roof for he sake of comleeess. Prooso 2.1 Le ; (Y ) 2[ T ] 2 B 2 be a soluo of he BSDE (1). Le us assume moreover ha for each, y, z, y f( y z) jyj + jyj 2 + jyjjz j ad, j j 1 : The, for each ">, wehave,seg = " , T jy j 2 2 e T + 2 ; e T ; 1 : "

4 4 Phle Brad ad Ree Carmoa Proof. Le us x 2 [ T], wll be chose laer he roof. Alyg I^o's formula o e (s;) jy s j 2 bewee ad T,we oba: jy j 2 + e (s;); jy s j 2 + j s j 2 ds = jj 2 e (T ;) +2 e (s;) Y s f(s Y s s )ds ; M rovded we wre M for 2 e (s;) Y s s dw s. Usg he assumo o ( f) follows ha: jy j 2 + e (s;); jy s j 2 + j s j 2 ds 2 e T +2 e (s;) jy s j + jy s j 2 + jy s jj s j ds ; M : Usg he equaly 2ab a2 + b2,we oba, for ay ">, jy j 2 + e (s;); jy s j 2 + j s j 2 ds 2 e T + e (s;) 2 " +(" )jy s j 2 ds + e (s;) j s j 2 ds ; 2 e (s;) Y s s dw s ad choosg = " yelds he equaly jy j 2 2 e T + 2 ; e T ; 1 ; 2 e (s;) Y s s dw s : " Takg he codoal execao wh resec o F of boh sdes, we ge mmedaely ha: whch comlees he roof [ T] jy j 2 2 e T + 2 ; e T ; 1 " 3 BSDEs wh Fxed Termal Tmes The goal of hs seco s o sudy he BSDE (1) for xed (deermsc) ermal me T uder he assumo (A 1) ad (A 2). We rs rove uqueess, he we rove a a ror esmae ad ally we ur o exsece. 3.1 Uqueess ad a ror Esmaes Ths subseco s devoed o he roof of uqueess ad o he sudy of he egrably roeres of he soluos of he BSDE (1). Theorem 3.1 If (A 1).1{2 hold, he BSDE (1) has a mos oe soluo he sace B 2. Proof. Suose ha we have wo soluos he sace B 2,say(Y 1 1 )ad(y 2 2 ). Seg Y Y 1 ; Y 2 ad 1 ; 2 for oaoal coveece, for each real umber ad for each 2 [ T], akg execaos I^o's formula gves: h e jy j 2 + e s T j s j 2 ds = e s 2Y s ; f(s Ys 1 1 s ) ; f(s Ys 2 2 s ) ; jy s j 2 ds :

5 BSDEs wh olyomal geeraors 5 The vashg of he execao of he sochasc egral s easly jused vew of Burkholder's equaly. Usg he mooocy of f ad he Lschz assumo, we ge: h h e jy j 2 + e s j s j 2 ds 2 e s jy s jj s jds ; ( +2) e s jy s j 2 ds : Hece, we see ha h e jy j 2 + e s T j s j 2 ds (2 2 ; 2 ; ) e s jy s j 2 ds + 1 h 2 e s j s j 2 ds : We coclude he roof of uqueess by choosg =2 2 ; We close hs seco wh he dervao of some a ror esmaes he sace B 2. These esmaes gve shor roofs of exsece ad uqueess he Lschz coex. They were roduced a \ L framework " by N. El Karou, S. Peg, M.-C. Queez [6] o rea he case of Lschz geeraors. Prooso 3.2 For =1 2 we le (Y ) 2 B 2 be a soluo of he BSDE Y = + f (s Ys s)ds ; sdw s T where ( f ) sases he assumos (A 1) ad (A 2) wh cosas, ad C. Le " such ha <"<1 ad ( 1 ) 2 =" ; 2 1. The here exssacosa K " whch deeds oly o ad o " such ha: T e jy j 2 T + e j j 2 d K " e T jj 2 T + e 2 2 s jf s jds where = 1 ; 2, Y Y 1 ;Y 2, 1 ; 2 ad f f 1 ( Y 2 2 );f 2 ( Y 2 2 ). Moreover, f >( 1 ) 2 =" ; 2 1, we have also, seg = ; ( 1 ) 2 =" +2 1, e jy j 2 d K" e T jj 2 T + e 2 2 s jf s jds : Proof. As usual we sar wh I^o's formula o see ha: e jy j 2 + e s j s j 2 ds = e T jj 2 +2 where we sem =2 ; e s jy s j 2 ds ; M e s Y s ; f 1 (s Y 1 s 1 s ) ; f 2 (s Y 2 s 2 s ) ds e s Y s s dw s for each 2 [ T]. I order o use he mooocy of f 1 ad he Lschz assumo o f 1,we sl oe erm o hree ars, recsely we wre: Y s ; f 1 (s Ys 1 1 s ) ; f 2 (s Ys 2 2 s ) = Y s ; f 1 (s Ys 1 1 s ) ; f 1 (s Ys 2 1 s ) +Y s ; f 1 (s Ys 2 1 s ) ; f 1 (s Ys 2 2 s ) +Y s ; f 1 (s Y 2 s 2 s ) ; f 2 (s Y 2 s 2 s )

6 6 Phle Brad ad Ree Carmoa ad he equaly 2 1 jy s jj s j ; ( 1 ) 2 =" jy s j 2 + "j s j 2 mles ha: e jy j 2 +(1; ") e s j s j 2 ds e T jj 2 + e s ; ; ( 1) 2 " +2 e s jy s jjf s jds ; M : jy s j 2 ds Seg = +2 1 ; ( 1 ) 2 =", he revous equaly ca be rewre he followg way e jy j 2 +(1; ") e s j s j 2 ds + e s jy s j 2 ds e T jj 2 ; M +2 e s jy s jjf s jds: (2) Takg he codoal execao wh resec o F of he revous equaly, we deduce sce he codoal execao of M vashes, e jy j 2 e T jj 2 +2 ad sce >1, Doob's maxmal equaly mles: e jy j 2 T K e T jj 2 T + e s jy s jjf s jds F e s jy s jjf s jds K e T jj 2 + e (=2) jy j e (=2)s jf s jds : T where we use he oao K for a cosa deedg oly o ad whose value could be chagg from le o le. Thaks o he equaly ab a 2 =2+b 2 =2, we ge T e jy j 2 K e T jj 2 T 2 + e (=2)s jf s jds T e jy j 2 whch gves T e jy j 2 K e T jj 2 T 2 + e (=2)s jf s jds : (3) Now comg back o he equaly (2),wehave sce "<1, e s j s j 2 ds 1 e T jj 2 +2 e s jy s jjf s jds ; 2 e s Y s s dw s 1 ; " ad by Burkholder-Davs-Gudy's equaly we oba e s j s j 2 ds K " e T jj 2 T + e s jy s jjf s jds =2 +K " e 2s jy s j 2 j s j 2 ds

7 BSDEs wh olyomal geeraors 7 ad hus follows easly ha: e s j s j 2 ds K " e T jj 2 + +K " T T e (=2) jy j =2 e (=2) jy j e s j s j 2 ds e (=2)s jf s jds whch yelds he equaly, usg oe more me he equaly ab a 2 =2+b 2 =2, e s j s j 2 ds K " e T jj 2 + e jy j 2 T 2 + e (=2)s jf s jds T e s j s j 2 ds : h Takg o accou he uer boud foud for T e jy j 2 gve (3), we derve from he above equaly, 2 e s j s j 2 ds K " e T jj 2 + e (=2)s jf s jds whch cocludes he rs ar of hs rooso. ha (2) gves For he secod assero we smly remark e s jy s j 2 ds e T jj 2 +2 e s jy s jjf s jds ; 2 e s Y s s dw s A smlar comuao gves: e s jy s j 2 ds K " e T jj 2 + e jy j 2 T 2 + e (=2)s jf s jds T e s j s j 2 ds whch comlees he roof usg he rs ar of he rooso already show ad keeg md ha f >( 1 ) 2 =" ; 2 1 he >. 2 Corollary 3.3 Uder he assumos ad wh he oao of he revous rooso, here exss a cosa K, deedg oly o, T, 1 ad 1 such ha: jy j 2 T + j j 2 d K jj 2 T 2 + jf s jds : T Proof. From he revous rooso, we have (akg " = 1=2): T ad hus e ;T ; e jy j 2 T + e j j 2 d K e T jj 2 T + e 2 2 s jf s jds T jy j 2 T + j j 2 d K e T + jj 2 T 2 + jf s jds :

8 8 Phle Brad ad Ree Carmoa I s eough o se K = e jjt K o coclude he roof. 2 Remark. I s la o check ha he assumos (A1 ).3{4 are o eeded he above roofs of he resuls of Prooso 3.2 ad s corollary. Corollary 3.4 Le ; (Y ) T 2 B 2 be a soluo of he BSDE (1) ad le us assume ha 2 L 2 ad assume also ha here exss a rocess (f ) T 2 H 2 (IR k ) such ha 8(s y z) 2 [ T] IR k IR kd y f(s y z) jyjjf s j;jyj 2 + jyjjz j: The, f <"<1 ad 2 =" ; 2, here exss a cosa K " whch deeds oly o ad o " such ha: e jy j 2 T + e j j 2 d K " e T jj 2 T + e 2 2 s jf s jds T Proof. As usual we sar wh I^o's formula o see ha : e jy j 2 + e s j s j 2 ds = e T jj 2 +2 e s Y s f(s Y s s )ds ; e s jy s j 2 ds ; M rovded we se M =2 e s Y s s dw s for each 2 [ T]. Usg he assumo o y f(s y z) ad he he equaly 2jY s jj s j ; 2 =" jy s j 2 + "j s j 2,we deduce ha e jy j 2 +(1; ") e s j s j 2 ds e T jj 2 + e s ; ; " +2 e s jy s jjf s jds ; M : jy s jds Sce 2 ; 2 =", he revous equaly mles e jy j 2 +(1; ") e s j s j 2 ds e T jj 2 +2 e s jy s jjf s jds ; M : Ths equaly s exacly he same as he equaly (2). As a cosequece we ca comlee he roof of hs as he roof of Prooso Exsece I hs subseco, we sudy he exsece of soluos for he BSDE (1) uder he assumos (A 1) ad (A 2). We shall rove ha he BSDE (1) has a soluo he sace B 2. We may assume, whou los of geeraly, ha he cosa s equal o. Ideed, (Y ) 2[ T ] solves he BSDE (1) B 2 f ad oly f, seg for each 2 [ T], Y = e ; Y ad = e ; he rocess ; Y solves B 2 he followg BSDE: Y = + f(s Y s s )ds ; s dw s T

9 BSDEs wh olyomal geeraors 9 where = e ;T ad f( y z) =e ; f( e y e z)+y. Sce ; f sases he assumo (A 1) ad(a2)wh =, = ad C = C ex ; T ( ; 1) + + ; + jj, we shall assume ha = he remag of hs seco. Our roof s based o he followg sraegy: rs, we solve he roblem whe he fuco f doesodeedohevarable z ad he we use a x o argume usg he a ror esmae gve subseco 3.1, Prooso 3.2 ad Corollary 3.3. The followg rooso gves he rs se. Prooso 3.5 Le he assumos (A 1) ad (A 2) hold. Gve a rocess (V ) T he sace H 2,here exss a uque soluo ; (Y ) 2[ T ] he sace B 2 o he BSDE Y = + f(s Y s V s )ds ; s dw s T: (4) Proof. We shall wre he sequel h(s y) lace of f(s y V s ). Of course h sases he assumo (A 1) wh he same cosas as f ad ; h( ) belogs o H 2 sce f s Lschz wh resec o z ad he rocess V belogs o H 2. Wha wewould lke o do s o cosruc a sequece of Lschz (globally y uformly wh resec o (! s)) fucos h whch aroxmae h ad whch are moooe. However, we oly maage o cosruc a sequece for whch each h s moooe a gve ball (he radus deeds o ). As we wll see laer he roof, hs \ local " mooocy ssuce o oba he resul. Ths s maly due o Prooso 2.1 whose key dea ca be raced back oawork of S. Peg [18, Theorem 2.2]. We shall use a aroxmae dey. Le : IR k ;! IR + be a oegave C 1 fuco wh he u ball for or ad such ha R (u)du = 1 ad dee for each eger 1, (u) =(u). We deoe also, for each eger, by a C 1 fuco from IR k o IR + such ha 1, (u) =1forjuj ad (u) = as soo as juj +1. We se moreover 8 < = : jj f jj oherwse, 8 < ad, h ~ (s y) = : h(s y) h(s y) jh(s )j oherwse. f jh(s )j Such a h ~ sases he assumo (A 1) ad moreover we have j j ad j h ~ (s )j. Fally we se q() = 1=2 ( +2C) 1+T where [r] sads as usual for he eger ar of r ad we dee h (s ) = ; q()+1 h ~ (s ) s 2 [ T]: We rs remark ha h (s y) = wheever jyj q() + 3 ad ha h (s ) s globally Lschz wh resec o y uformly (! s). Ideed, h (s ) s a smooh fuco wh comac or ad hus wehave y2ir k rh (s y) = rh jyjq()+3 (s y) ad, from he growh assumo o f (A 1).3, s o hard o check ha j h ~ (s y)j ^jh(s )j + C ; 1+jyj whch mles ha rh (s y) + C(1 + 2 ;1 jyj ) + C2 ;1 r(u) du: As a mmedae cosequece, he fuco h s globally Lschz wh resec o y uformly (! s). I addo j j ad jh (s )j ^jh(s )j +2C ad hus Theorem 5.1 [6] rovdes a soluo (Y ) o he BSDE Y = + h (s Ys )ds ; s dw s T (5)

10 1 Phle Brad ad Ree Carmoa whch belogs acually o B q for each q>1. I order o aly Prooso 2.1 we observe ha, for each y, y h (s y) = (u) q()+1 (y ; u)y h ~ (s y ; u)du = (u) q()+1 (y ; u)y h ~ (s y ; u) ; h ~ (s ;u) du + (u) q()+1 (y ; u)y h ~ (s ;u)du: Hece, we deduce ha, sce he fuco ~ h (s ) s moooe (recall ha = ) hs seco) ad vew of he growh assumo o f we have: 8(s y) 2 [ T] y h (s y) ; ^jh(s )j +2C jyj: (6) Ths esmae wll ur ou o be very useful he sequel. Ideed, we ca aly Prooso 2.1 o he BSDE (5) o show ha, for each, choosg " =1=T, jy j( +2C)e1=2 1+T 2 : (7) T O he oher had, he equaly (6)allows oe o use Corollary 3.4, o oba, for a cosa K deedg oly o : jy j 2 T + j j 2 d K jj 2 T 2 + jh(s )j +2C ds : (8) 2IN T I s worh og ha, haks o jh(s )j jf(s )j + jv s j, he rgh had sde of he revous equaly s e. We wa o rove ha he sequece ; (Y ) coverges owards he IN soluo of he BSDE (4) ad order o do ha we rsshow ha he sequece ; (Y ) s IN acauchy sequece he sace B 2. Ths fac reles maly o he followg roery: h sases he mooocy codo he ball of radus q(). Ideed, x 2 IN ad le us ck y y such ha jyj q() adjy jq(). We have: (y ; y ) (h (s y) ; h (s y ) = (y ; y ) (u) q()+1 (y ; u) h ~ (s y ; u)du ;(y ; y ) (u) q()+1 (y ; u) h ~ (s y ; u)du: Bu, sce jyj jy jq() ad sce he or of s cluded he u ball, we ge from he fac ha q()+1 (x) =1assooasjxj q()+1, (y ; y ) (h (s y) ; h (s y )= (u)(y ; y ) ; h ~ (s y ; u) ; ~ h (s y ; u) du: Hece, by he mooocy of ~ h,wege 8y y 2 B( q()) (y ; y ) (h (s y) ; h (s y ) : (9) We ow ur o he covergece of ; (Y ). Le us x wo egers m ad such ha m. IN I^o's formula gves, for each 2 [ T], jy j 2 + j s j 2 ds = jj 2 +2 Y s ; h m (s Ys m ) ; h (s Ys ) ds ; 2 Y s s dw s

11 BSDEs wh olyomal geeraors 11 where we have se = m ;, Y Y m ; Y ad m ;. We sl oe erm of he revous equaly o wo ars, recsely we wre: Y s ; h m (s Ys m ) ; h (s Ys ) = Y s ; h m (s Ys m ) ; h m (s Ys ) + Y s ; h m (s Ys ) ; h (s Ys ) : Bu vew of he esmae (7), we have jys m jq(m) adjys jq() q(m). Thus, usg he roery (9), he rs ar of he rgh had sde of he revous equaly s o-osve ad follows ha jy j 2 + I arcular, we have j s j 2 ds jj 2 +2 jy s j h m (s Ys ) ; h (s Ys ) ds ; 2 Y s s dw s : (1) h h j s j 2 ds 2 jj 2 + ad comg back o (1), Burkholder's equaly mles jy j 2 K jj 2 + T jy s j hm (s Y s ) ; h (s Y s ) ds jy s j hm (s Y s ) ; h (s Y s ) ds + ad he usg he equaly ab a 2 =2+b 2 =2we oba he followg equaly: h T jy j 2 h K jj h T jy s j h m (s Ys ) ; h (s Ys ) ds jy j 2 + K2 2 h from whch we ge, for aoher cosa sll deoed by K, h T jy j 2 + h j s j 2 ds K jj 2 + j s j 2 ds 1=2 jy s j 2 j s j 2 ds jy s j hm (s Y s ) ; h (s Y s ) ds : Obvously, sce 2 L 2, eds o L 2 as m!1wh m. So, we have oly o rove ha h jy s j hm (s Ys ) ; h (s Ys ) ds ;! as!1: For ay oegave umber k, we wre S m T = 1 jy s j+jys mjk jy s j hm (s Ys ) ; h (s Ys ) ds R m T = 1 jy s j+jys mj>k jy s j hm (s Ys ) ; h (s Ys ) ds ad wh hese oaos we have h jy s j hm (s Ys ) ; h (s Ys ) ds = S m + R m ad hece, he followg equaly: h jy s j h m (s Ys ) ; h (s Ys ) h ds k h m (s y) ; h (s y) ds + R m : (11) jyjk

12 12 Phle Brad ad Ree Carmoa Frs we deal wh R m ad usg Holder's equaly we ge he followg uer boud: h R m T 1 jy s j+jys m Seg A m ;1 2 jk ds h jy s j 2 +1 h = jy s j 2 +1 hm (s Ys ) ; h (s Ys ) 2 +1 ds h m (s Ys ) ; h (s Ys ) 2 +1 ds +1 2 : for oaoal coveece, we have R m IP ; ;1 jys j + jy s m jk 2 +1 ds A m 2 ad Chebyshev's equaly yelds: R m k 1; 2 T ;1 2 h ;jy 2IN h s j + jys m T ;1 j ds A m 2 ;1 jy j 2 2 k 1; A m +1 2 : (12) h We have already see ha 2IN T jy j2 s e (cf. (8)) ad we shall rove ha remas bouded as m vary. To do hs, le us recall ha A m A m h = jy s j 2 +1 hm (s Ys ) ; h (s Ys ) 2 +1 ds ad usg Youg's equaly (ab 1 r ar + 1 r b r wheever 1 r + 1 r =1)whr = +1 ad r = +1 we deduce ha A m 1 +1 h jy s j 2 ds + h +1 hm (s Ys ) ; h (s Ys ) 2 ds : The rs ar h of he las uer boud remas bouded as m vary sce from (8) we kow ha 2IN T jy j2 s e. Moreover, we derve easly from he assumo (A 1) ha h (s y) ^ h(s ) +2 C ; 1+jyj, ad he, hm (s Y s ) ; h (s Y s ) 2 h(s ) C ; 1+jY s j whch yelds he equaly, akg o accou he assumo (A 1).1, h hm (s Ys ) ; h (s Ys ) h 2 ds K jf(s )j 2 + jv s j 2 +1+jYs j 2 ds : Takg o accou (8) ad he egrably assumo o boh V ad f( ), we have roved ha m A m < 1. Comg back o he equaly (12), we ge, for a cosa, R m k 1;, ad sce >1, R m ca be made arbrary small by choosg k large eough. Thus, vew of he esmae (11), remas oly o check ha, for each xed k>, h h m (s y) ; h (s y) ds jyjk,

13 BSDEs wh olyomal geeraors 13 goes o as eds o y uformly wh resec o m o ge he covergece of ; (Y ) IN he sace B 2. Bu, sce h(s ) scouous (IP ; a:s: 8s), h (s ) coverges owards h(s ) uformly o comac ses. Takg o accou ha h jyjk (s y) h(s ) +2 C ; 1+k Lebesgue's covergece heorem gves he resul. Thus, he sequece ; (Y ) coverges owards a rogressvely measurable rocess (Y ) IN he sace B 2. Moreover, sce ; (Y ) s bouded B IN 2 (see (8)), Faou's lemma mles ha (Y ) belogs also o he sace B 2. I remas o check ha (Y ) solves he BSDE (4) whch s ohg bu Y = + h(s Y s )ds ; s dw s T: Of course, we wa o ass o he lm he BSDE (5). Le us rs remark ha ;! L 2 ad ha for each 2 [ T], s dw s ;! H 2 (IR kd ). Acually, we oly eed o rove ha for 2 [ T], h (s Ys )ds ;! h(s Y s )ds as!1: s dw s sce coverges o he sace For hs, we shall see ha h ( Y )edsoh( Y ) he sace L 1 ( [ T]). Ideed, h h (s Ys );h(s Y s ) h ds h (s Ys );h(s Ys ) h ds + h(s Y s );h(s Y s ) ds : The rs erm of he rgh had sde of he revous equaly eds o as goes o 1 by he same argume we use earler he roof o see ha jy s jjh m (s Ys );h (s Ys )jds goes o. For he secod erm, we shall rsly rove ha here exss a covergg subsequece. Ideed, sce Y coverges o Y s he sace S 2, here exss a subsequece (Y j )suchhaip{a.s., 8 2 [ T] Y j ;! Y : ; Sce h( ) s couous (IP{a.s., 8), IP{a.s. 8 h( Y j ) ;! h( Y ). Moreover, sce Y 2 S 2 ad (Y ) IN s bouded S 2 ((8)), s o hard o check from he growh assumo o f ha h h(s Y j s ) ; h(s Y s ) 2 ds < 1 j2in ad he he resul follows by uform egrably of he sequece. Acually, he covergece hold for he whole sequece sce each subsequece has a covergg subsequece. Fally, we ca ass o he lm he BSDE (5) ad he roof s comlee. 2 Wh he hel of hs rooso, we ca ow cosruc a soluo (Y ) o he BSDE (1). We clam he followg resul: Theorem 3.6 Uder he assumos (A 1) ad (A 2), he BSDE (1) has a uque soluo (Y ) he sace B 2. Proof. The uqueess ar of hs saeme salready roved Theorem 3.1. The rs se he roof of he exsece s o show he resul whe T s sucely small. Accordg o

14 14 Phle Brad ad Ree Carmoa Theorem 3.1 ad Prooso 3.5, le us dee he followg fuco from B 2 o self. For (U V ) 2 B 2, (U V )=(Y ) where (Y ) s he uque soluo B 2 of he BSDE: Y = + f(s Y s V s )ds ; s dw s T: Nex we rove ha s a src coraco rovded ha T s small eough. Ideed, f ; U 1 V 1 ad ; U 2 V 2 are boh elemes of he sace B 2,wehave, alyg Prooso 3.2 for ; Y = ; U V, =1 2, T jy j 2 T 2 + j j 2 d K jf(s Ys 2 Vs 1 ) ; f(s Ys 2 Vs 2 jds where Y Y 1 ;Y 2, 1 ; 2 ad K s a cosa deedg oly o. Usg he Lschz assumo o f, (A 1).1, ad Holder's equaly we ge he equaly T jy j 2 T + j j 2 d K 2 T jvs 1 ; V s 2 j2 ds : Hece, f T s such ha K 2 T < 1, s a src coraco ad hus has a uque xed o he sace B 2 whch s he uque soluo of he BSDE (1). The geeral case s reaed by subdvdg he me erval [ T] o a e umber of ervals whose leghs are small eough ad usg he above exsece ad uqueess resul each of he subervals. 2 4 The Case of Radom Termal Tmes I hs seco, we brey exla how o exed he resuls of he revous seco o he case of a radom ermal me. 4.1 Noao ad Assumos Le us recall ha (W ) s a d-dmesoal Browa moo, deed o a robably sace ( F IP) ad ha ; F s he comlee -algebra geeraed by (W ). Le be a sog me wh resec o ; F ad le us assume ha s e IP{a.s. Le us cosder also a radom varable F {measurable ad a fuco f deed o IR + IR k IR kd wh values IR k ad such ha he rocess ; f( y z) s rogressvely measurable for each (y z). We sudy he followg BSDE wh he radom ermal me : Y = + ^ f(s Y s s )ds ; ^ s dw s : (13) By a soluo of hs equao, we always mea a rogressvely measurable rocess ; (Y ) wh values IR k IR kd such ha =f>. Moreover, sce s e IP{a.s., (13) mles ha Y = f. We eed o roduce furher oao. Le us cosder q > 1 ad 2 IR. We say ha a rogressvely measurable rocess wh values IR belogs o H q (IR )f 1 q=2 e j j 2 d < 1:

15 BSDEs wh olyomal geeraors 15 Moreover, we say ha belogs o he sace S q (IR )f e (q=2)(^ ) j j q < 1: We are gog o rove a exsece ad uqueess resul for he BSDE (13) uder assumos whch arevery smlar o hose made seco 2 for he sudy of he case of BSDEs wh xed ermal mes. Precsely, we wll ose he framework of radom ermal mes he followg wo assumos: (A 3). There exs cosas, 2 IR, C, >1ad2f 1g such haip ; a:s:, we have: y 8(z z ) f( y z) ; f( y z ) jz ; z j z 8(y y ) (y ; y ) ; f( y z) ; f( y z) ;jy ; y j y 8z f( y z) f( z) + C( + jyj ) z y 7;! f( y z) scouous, (A 4). s F -measurable ad here exss a real umber such ha > 2 ; 2 ad e + e + e jj 2 + e s f(s ) 2 ds + e (=2)s 2 f(s ) ds < 1: Remark. I he case <, whch may occur f s a ubouded sog me, our egrably codos are fullled f we assume ha e jj 2 + e (=2)s f(s ) 2 ds < 1: For oaoal coveece, we wll smly wre, he remag of he aer, S q sead of S q (IR k ) ad H q(ir kd ) resecvely. ad H q 4.2 Exsece ad Uqueess I hs seco, we deal wh he exsece ad uqueess of he soluos of he BSDE (13). We clam he followg rooso. Prooso 4.1 Uder he assumos (A3) ad (A4), here exss a mos a soluo of he BSDE (13) he sace S 2 H 2. Proof. Le (Y 1 1 ) ad (Y 2 2 )bewo soluos of (13) he sace S 2 H 2. Le us oce rs ha Y 1 = Y 2 = f ad 1 = 2 = o he se f >g. Alyg I^o's formula, we ge e (^) jy ^ j 2 + e s j s j 2 ds = 2 ^ ^ ; ^ e s Y s ; f(s Y 1 s 1 s ) ; f(s Y 2 e s jy s j 2 ds ; 2 ^ s 2 s ) ds e s Y s s dw s

16 16 Phle Brad ad Ree Carmoa where we have se Y Y 1 ; Y 2 ad 1 ; 2. I s worh og ha, sce f s Lschz z ad moooe y, wehave, for each ">, 8( y y z z ) 2(y ; y ) ; f( y z) ; f( y z ) (;2 + 2 =")jy ; y j 2 + "jz ; z j 2 : (14) Moreover, by Burkholder's equaly he couous local margale ^ s a uformly egrable margale. Ideed, D ^ E 1=2 e s Y s s dw s = ad he, D ^ E 1=2 e s Y s s dw s K2 1 1 o e s Y s s dw s 1=2 e 2s jy s j 2 j s j 2 ds 1=2 K 1=2 e jy j 2 e s j s j 2 ds e jy j 2 + e s j s j 2 ds whch s e sce (Y ) belogs o he sace S 2 H 2. Thaks o he equaly > 2 ;2, we cachoose " such ha<"<1ad> 2 =" ; 2. Usg he equaly (14), we deduce ha, he execao of he sochasc egral vashg vew of he above comuao, for each, h e (^) jy ^ j 2 +(1; ") ^ e s j s j 2 ds whch gves he resul. 2 Before rovg he exsece ar of he resul, le us roduce a sequece of rocesses whose cosruco s due o R. W. R. Darlg ad E. Pardoux [5,. 1148{1149]. Le us se = 2 =2 ; ad le ( Y b b ) be he uque soluo of he classcal (he ermal me s deermsc) BSDE o [ ] by = e ^ F + e s f(s e ;s Y b s e ;s b s ) ; Y b s ds ; b s dw s: ^ Sce he 2 jj 2 he jj 2 ad sce e 2s f(s ) 2 ds e s f(s ) 2 ds he assumo (A 4) ad Theorem 3.6 esure ha ( b Y b ) belogs o he sace B 2 (o he erval [ ]). I vew of [12, Prooso 3.1], we have by (^) = b Y ad, b =of >g: Sce e belogs o L 2 (F ) here exss a rocess () H 2 such ha =f> ad e = e + s dw s :

17 BSDEs wh olyomal geeraors 17 We roduce sll ew oao. For each >we se: ad for each oegave : by = e F = ad, b = Y = e ;(^) b Y ad, = e ;(^ ) b : Ths rocess sases Y^ = Y ad =of >g ad moreover (Y )solves he BSDE Y = + ^ f (s Y s s )ds ; ^ s dw s (15) where f ( y z) =1 f( y z)+1 > y (cf [5]). We sar wh a echcal lemma. Lemma 4.2 Le he assumos (A 3) ad (A 4) hold. The, we have, wh he oao K( f) =K e jj 2 + e (=2)s 2 f(s ) ds e (^) jy j2 + e s jys j2 ds IN ad, also, for = ; 2, 1 e (^ ) j j 2 + e s j s j 2 ds e s js j2 ds K( f) (16) h e s j s j 2 ds K e jj 2 : (17) Proof. Frsly, le us remark ha = = f > ad, sce Y = f, we have e (^) jy j2 = e jy j2. Moreover, sce > 2 we ca d " such ha <"<1 ad > 2 =" ; 2. Alyg Prooso 3.2 (acually a very mere exeso o deal wh bouded sog mes as ermal mes), we ge e jy j 2 ^ + ^ e s jys ^ j 2 ds + KE e s js j 2 ds e (^) jy (^)j 2 + ^ e (=2)s f(s ) 2 ds : We have Y^ = Y = e ;(^ ) e F^ ad he we deduce mmedaely ha, sce =2 ; > ad usg Jese's equaly, h he (^ ) jy (^)j 2 = e (=2;)(^ ) e F ^ 2 (18) he jj 2 : Hece, for each eger, e jy ^ j2 + ^ e s jys ^ j2 ds + e s js j2 ds K( f): I remas o rove hawecadhesameuerboudfor e jy j 2 + e s jys j 2 ds + e s js j 2 ds : ^< ^ ^

18 18 Phle Brad ad Ree Carmoa Bu he execao s over he se f <g ad comg back o he deo of ( Y b b )for>, s eough o check ha e (;2)(^) j j 2 + e (;2)s j s j 2 ds + e (;2)s j s j 2 ds K he jj 2 o ge he equaly (16) of he lemma ad hus o comlee he roof sce, vew of he deo of, he revous equaly s ohg bu he equaly (17). Bu, for each, ( ) solves he he followg BSDE: = e F^ ; ad by Prooso 3.2, sce = ; 2 >, ^ s dw s e j j 2 ^ + e s ^ h j s j 2 ds + e s j s j 2 ds K e (^) j ^ j 2 : We have already see (cf (18)) ha e (^) j ^ j 2 e jj 2 ad hus he roof of hs raher echcal lemma s comlee. 2 Wh he hel of hs useful lemma we ca cosruc a soluo o he BSDE (13). Ths s he am of he followg heorem. Theorem 4.3 Uder he assumos (A 3) ad (A 3), he BSDE (13) has a uque soluo (Y ) he sace S 2 H 2 whch sases moreover e (^) jy j 2 + e s jy s j 2 ds + 1 e s j s j 2 ds K( f): Proof. The uqueess ar of hs clam s already roved Prooso 4.1. We cocerae ourselves o he exsece ar. We sl he roof o he wo followg ses: rs we showha he sequece ; (Y ) s a Cauchy sequece he sace S IN 2 H 2 ad he we shall rove ha he lmg rocess s deed a soluo. Le us rs recall ha for each eger, he rocess (Y ) sases Y^ = Y ad = o f > g ad moreover solves he BSDE (15) whose geeraor f s deed he followg way: f ( y z) =1 f( y z)+1 > y. If we xm, I^o's formula gves, sce we have also Ym^ m = Ym m = Ym^ = Ym = e ;(m^ ) m,form, m^ m^ e (^ ) jy ^ j 2 + e s j s j 2 ds = 2 e s Y s ; f m (s Ys m s m ) ; f (s Ys s ) ds ^ ^ ; m^ ^ e s jy s j 2 ds ; 2 m^ ^ e s Y s s dw s where we have sey Y m ; Y, m ;. I follows from he deo of f, e (^) jy ^ j 2 + m^ ^ e s j s j 2 ds = 2 ; +2 m^ ^ m^ ^ m^ ^ e s Y s ; f(s Y m s e s jy s j 2 ds ; 2 s m ) ; f(s Ys s ) ds m^ ^ e s Y s s dw s 1 s> e s Y s ; f(s Ys s ) ; Ys ds:

19 BSDEs wh olyomal geeraors 19 Sce > 2 ; 2, we ca d " such ha < " < 1 ad = ; 2 =" +2 >. Usg he equaly (14)wh hs", we deduce from he revous equaly, m^ e (^) jy ^ j 2 +(1; ") ^ m^ e s j s j 2 ds ; +2 ^ m^ (_)^ m^ e s jy s j 2 ds ; 2 e s Y s s dw s ^ e s jy s j: f(s Ys s ) ; Ys ds: Now, usg he equaly 2ab $a 2 + b 2 =$ for he secod erm of he rgh had sde of he revous equaly, wh $<,we ge, for each m, og =m(1; " ; $) >, e (^) jy ^ j 2 + m^ ^ e s jy s j 2 + j s j 2 ds 1 $ ; 2 m^ ^ m^ ^ e s f(s Y s s ) ; Ys 2 ds e s Y s s dw s : (19) I arcular, we have, he execao of he sochasc egral vashes (cf Lemma 4.2), m^ e s m^ jy s j 2 + j s j 2 ds K e s f(s Y s s ) ; Ys 2 ds : ^ Comg back o he equaly (19), Burkholder's equaly yelds m^ m^ e jy j 2 K e s f(s Y ^ s s );Y 2 m^ ds+ s 1=2 e 2s jy s j 2 j s j 2 ds : Bu, by a argume already used, m^ 1=2 K e 2s jy s j 2 j s j 2 ds 1=2 K e m^ jy j 2 m^ 1 m^ 2 e jy j 2 + K2 2 m^ 1=2 e s j s j 2 ds e s j s j 2 ds : As a cosequece we oba he equaly: m^ m^ e jy j 2 + e s m^ jy s j 2 + j s j 2 ds K e s f(s Y ^ s s ) ; Y 2 ds ad sce Y m = Y f m, Y = o f g for each, m = = as soo as m ad =of >g we deduce from he revous equaly where we have se; = e (^ ) jy j 2 + e s jy s j 2 ds + ^ 1 (A 3).3 mles ha, u o a cosa, ; s uer bouded by e s j s j 2 ds ; (2) e s f(s Y s s ) ; Ys 2 ds. Bu he growh assumo o f e s f(s ) jys j2 + js j2 + jys j2 ds : ^ s

20 2 Phle Brad ad Ree Carmoa Sce, by assumo (A 4), e s jf(s )j 2 ds ad e are e, he rs wo erms of he ; revous uer boud eds o as goes o 1. Moreover, comg back o he deo of by b for >,wehave e s jys j 2 + js j 2 ds = e (;2)s j s j 2 + j s j 2 ds ^ ^ ad by Lemma 4.2 (cf (17)) he quay above eds also o wh gog o 1. I remas o check ha he same s rue for e s jys j 2 ds = e (;2)s j s j 2 ds ^ ^ where, le us recall, s meas e F s. By Jese's equaly, s eough o show he followg: e (;2)s e jj 2 Fs ds ;! as!1: ^ If >2, sce e 2 jj 2 e jj 2 < 1 ad e jj 2 < 1, Lemma 4.1 [5] gves e (;2)s e jj 2 Fs ds < 1 from whch we ge he resul. Now, we deal wh he case 2 whch mles < 2 <2 <. Usg oce more me Jese's equaly, wehave e (;2)s e jj Fs ^ 2 ds ^ ^ e 2 jj 2 Fs ds e (2;) e jj 2 F s ad sce >2 we have e (2;) e jj 2 F s e (2;)(s^ ) e jj 2 F s. Hece, follows, e (;2)s e jj Fs ^ 2 ds ^ e (2;)s e jj 2 Fs he jj 2 1 e (2;)s ds: Sce 2 ; <ad e jj 2 < 1, we comlee he roof of he las case. Thus we have show ha ; coverges o as eds o 1 ad comg back o he equaly (2), we ge 1 e (^) jy j 2 + e s jy s j 2 ds + e s j s j 2 ds ;! as eds o 1, uformly m. I arcular he sequece ; (Y ) s a Cauchy sequece IN S 2 H 2 ad hus coverges hs sace o a rocess (Y ). Moreover, akg o accou he equaly (16) of Lemma 4.2, Faou's lemma mles e (^) jy j e s jy s j 2 ds + e s j s j 2 ds K( f): (21) ds ds

21 BSDEs wh olyomal geeraors 21 I remasocheck ha he rocess (Y ) solves he BSDE (13). To do hs, we follow he dscusso of R. W. R. Darlg, E. Pardoux [5,. 115{1151]. Leusckarealumber such ha <^=2 ^ (hs mles ha <)adleusxaoegaverealumber. Sce (Y ) solves he BSDE (15), we have, from I^o's formula, for, e (^ ) Y = e + + ^ ^ e s Y s e s f(s Y s s ) ; Y s ds ; ; f(s Y s s ) ds ad we wa o ass o he lm hs equao kowg ha " 1 e (^ ) jy ; Y j2 + e s jy s ; Ys j2 ds + We have, e (^) Y ;! e (^ ) Y L 2. Moreover, Holder's equaly gves e s jys ; Y s jds from whch we deduce, sce 2 <, ha also ha ^ e s s dw s coverges o Usg Holder's equaly, we have ^ e s s dw s e s j s ; s j2 ds ;! : 1=2 1=2 e s jys ; Y s j 2 ds e (2;)s ds ^ ^ e s; 2 s ; s dws ^ e s Ys ds eds o e s Y s ds L 1. We remark ^ e s s dw s L 2 sce, haks o 2 <, e s js ; s j 2 ds : e s Y s ; f(s Ys s ) 1 h ds e s Y ^ ; 2 s ; f(s Ys s ) 2 1=2 ds ^ ad we have already roved ha he rgh had sde eds o (see he deo of ; ). remas o sudy he erm ^ f(s Y s s )ds. Bu, sce f s Lschz z, wehave e s f(s Y s s ) ; f(s Ys s ) h ds 1=2 e s j ^ ; 2 s ; s j 2 ds ^ ad hus goes o wh. So ow, suces o show ha e s f(s Y s s ) ; f(s Y s s ) ds ;! o corol he lm he equao. We rove hs by showg ha each subsequece has a subsequece for whch he above covergece hold. Ideed, f we ck a subsequece (sll deoed by Y ), sce we have e (^ ) jy ; Y j 2 ;! here exs a subsequece sll deoed ; he same way such ha IP{a.s. 8 Y ; ;! Y. By he couy of he fuco f, IP{a.s. 8 f( Y ) ;! f( Y ). If we rove ha e s f(s Y s s ) ; f(s Y s s ) 2 ds < 1 IN I

22 22 Phle Brad ad Ree Carmoa he he sequece jf( Y ) ; f( Y ) wll be a uformly egrable sequece for he e measure e s 1 s ds dip (remember ha <)ad hus covergg L 1 (e s 1 s ds dip) whch s he desred resul. Bu from he growh assumo o f, wehave e s f(s Y s s ) ; f(s Y s s ) 2 ds Sce >, he equales (16){(21), mles ha s e. Moreover, K +K e s jf(s )j js j2 + j s j 2 ds IN h e s jys j2 ds e s jf(s )j 2 + js j2 + j s j 2 ds e s + jys j2 + jy s j 2 ds : 1 e jy j2 e (;)s ds: Sce >, we coclude he roof of he covergece of he las erm by usg he rs ar of he equales (16){(21). Passg o he lm whe goes o y, we ge, for each, e (^) Y = e + e s f(s Y s s ) ; Y s ds ; e s s dw s : ^ ^ I he follows by I^o's formula ha (Y ) solves he BSDE (13). 2 Refereces [1] G. Barles, R. Buckdah, ad E. Pardoux, Backward sochasc dereal equaos ad egral-aral dereal equaos, Sochascs Sochascs Re. 6 (1997), o. 1-2, 57{83. [2] G. Barles, H. M. Soer, ad P. E. Sougads, Fro roagao ad hase eld heory, SIAM J. Corol Om. 31 (1993), o. 2, 439{469. [3] Ph. Brad, BSDE's ad vscosy soluos of semlear PDE's, Sochascs Sochascs Re. 64 (1998), o. 1-2, 1{32. [4] Ph. Brad ad Y. Hu, Sably of BSDEs wh radom ermal me ad homogezao of semlear ellc PDEs, J. Fuc. Aal. 155 (1998), o. 2, 455{494. [5] R. W. R. Darlg ad E. Pardoux, Backwards SDE wh radom ermal me ad alcaos o semlear ellc PDE, A. Probab. 25 (1997), o. 3, 1135{1159. [6] N. El Karou, S. Peg, ad M.-C. Queez, Backward sochasc dereal equaos ace, Mah. Face 7 (1997), o. 1, 1{71. [7] M. Escobedo, O. Kava, ad H. Maao, Large me behavor of soluos of a dssave semlear hea equao, Comm. Paral Dereal Equaos 2 (1995), o. 7-8, 1427{1452. [8] S. Hamadee, Equaos dereelles sochasques rerogrades : le cas localeme lschze, A. Is. H. Pocare Probab. Sas. 32 (1996), o. 5, 645{659.

23 BSDEs wh olyomal geeraors 23 [9] M. Kobylask, Resulas d'exsece e d'uce our des equaos dereelles sochasques rerogrades avec des geeraeurs a crossace quadraque, C. R. Acad. Sc. Pars Ser. I Mah. 324 (1997), o. 1, 81{86. [1] J.-P. Leeler ad J. Sa Mar, Backward sochasc dereal equaos wh couous coeces, Sas. Probab. Le. 32 (1997), o. 4, 425{43. [11] J.-P. Leeler ad J. Sa Mar, Exsece for BSDE wh erlear-quadrac coece, Sochascs Sochascs Re. 63 (1998), o. 3-4, 227{24. [12] E. Pardoux, Backward sochasc dereal equaos ad vscosy soluos of sysems of semlear arabolc ad ellc PDEs of secod order, Sochasc aalyss ad relaed ocs VI (The Gelo Worksho, 1996) (L. Decreusefod, J. Gjerde, B. ksedal, ad A. S. Usuel, eds.), Progr. Probab., vol. 42, Brkhauser Boso, Boso, MA, 1998,. 79{127. [13] E. Pardoux ad S. Peg, Adaed soluo of a backward sochasc dereal equao, Sysems Corol Le. 14 (199), o. 1, 55{61. [14] E. Pardoux ad S. Peg, Backward sochasc dereal equaos ad quaslear arabolc aral dereal equaos, Sochasc aral dereal equaos ad her alcaos (Charloe, NC, 1991) (B. L. Rozovsk ad R. B. Sowers, eds.), Lecure Noes Corol ad Iform. Sc., vol. 176, Srger, Berl, 1992,. 2{217. [15] E. Pardoux, F. Pradelles, ad. Rao, Probablsc erreao of a sysem of sem-lear arabolc aral dereal equaos, A. Is. H. Pocare Probab. Sas. 33 (1997), o. 4, 467{49. [16] E. Pardoux ad S. hag, Geeralzed BSDEs ad olear Neuma boudary value roblems, Probab. Theory Relaed Felds 11 (1998), o. 4, 535{558. [17] S. Peg, Probablsc erreao for sysems of quaslear arabolc aral dereal equaos, Sochascs Sochascs Re. 37 (1991), o. 1-2, 61{74. [18] S. Peg, Backward sochasc dereal equaos ad alcaos o omal corol, Al. Mah. Om. 27 (1993), o. 2, 125{144. Phle Brad Ree Carmoa Isu de Recherche Mahemaque Sascs & Oeraos Research Program Uverse Rees I Prceo Uversy Rees Cedex, Frace Prceo, NJ 8544, USA brad@mahs.uv-rees1.fr rcarmoa@rceo.edu h:// h://