BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND PARTIAL DIFFERENTIAL EQUATIONS WITH QUADRATIC GROWTH. By Magdalena Kobylanski Université detours

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1 The Annals of Probabiliy 2, Vol. 28, No. 2, BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND PARTIAL DIFFERENTIAL EQUATIONS WITH QUADRATIC GROWTH By Magdalena Kobylanski Universié detours We provide exisence, comparison and sabiliy resuls for onedimensional backward sochasic differenial equaions (BSDEs when he coefficien (or generaor F Y Z is coninuous and has a quadraic growh in Z and he erminal condiion is bounded. We also give, in his framework, he links beween he soluions of BSDEs se on a diffusion and viscosiy or Sobolev soluions of he corresponding semilinear parial differenial equaions. 1. Inroducion. Backward sochasic differenial equaions (BSDE are equaions of he following ype: (1 Y = ξ + Fs Y s Z s ds Z s dw s T where W T is a sandard d-dimensional Brownian moion on a probabiliy space F F T, wih F T he sandard Brownian filraion. The random funcion F: T n n d n is generally called a coefficien, T he erminal ime, which may be a sopping ime, and he n -valued F T -adaped random variable ξ a erminal condiion; F T ξ are he parameers of (1. The ineger n is known as he dimension of he BSDE. A soluion is a couple Y Z T of processes adaped o he filraion F T, which have some inegrabiliy properies, depending on he frameworkimposed by he ype of assumpions on F. In order o simplify he noaions, we someimes wrie Y Z for he process Y Z T. Nonlinear BSDEs were firs inroduced by Pardoux and Peng [15]. When F is Lipschiz coninuous in he variables Y and Z and ξ is square inegrable, a soluion is a couple Y Z T of square inegrable adaped processes. In ha framework, Pardoux and Peng gave he firs exisence and uniqueness resuls for n-dimensional BSDEs. Since hen, BSDEs have been sudied wih grea ineres. In paricular, many effors have been made o relax he assumpions on he driver; for insance, Lepelier and San Marin [14] have proved he exisence of a soluion for one-dimensional BSDEs when he coefficien is only coninuous wih linear growh. The ineres in BSDEs comes from heir connecions wih differen mahemaical fields, such as mahemaical finance, sochasic conrol, and parial Received May AMS 1991 subjec classificaions. Primary 6H2, 6H3; secondary 35J6, 35k55. Key words and phrases. Backward sochasic differenial equaions, comparison principle, semilinear parial differenial equaions, viscosiy soluions, Feynman Kac formula. 558

2 BACKWARD SDE AND PDE 559 differenial equaions (see [1] for an exensive bibliography. We are especially concerned in his paper wih he laer connecion. Numerous resuls (for insance, [16], [1] show he connecions beween BSDEs se from a diffusion (or forward backward sysem and soluions of a large class of quasilinear parabolic and ellipic parial differenial equaions (PDEs. Those resuls may be seen as a generalizaion of he celebraed Feynman Kac formula. Through all hese resuls, a formal dicionary beween BSDEs and PDEs can be esablished, which suggess ha exisence and uniqueness resuls which can be obained on one side should have heir counerpars on he oher side. This idea is he saring poin of his work, where we consider BSDEs wih quadraic growh in Z. Indeed, and [3] have given exisence and uniqueness resuls for quasilinear PDEs se in a bounded domain when he nonlineariy has a quadraic growh in he gradien of he soluion. In his paper, we obain general exisence and uniqueness resuls for onedimensional BSDEs when he coefficien has a quadraic growh in Z, and connecions wih boh viscosiy and Sobolev soluions of PDEs when he nonlineariy has a quadraic growh in he gradien. Here, because of he quadraic growh of he coefficien, we are looking for soluions such ha Y T H T, where HT is he se of onedimensional progressively measurable processes which are almos surely bounded, for almos every (in shor, Y T is a one-dimensional bounded process while Z T remains in HT 2 d where HT 2 is he se of progressively measurable processes Z T wih values in d such ha Ɛ Z s 2 ds < (in shor, Z T is an inegrable adaped process. Alhough many ideas of or [3] are used hroughou his work, he difference of frameworkand of he sudied subjec, require addiional argumens. In fac, our resuls are no he exac counerpar of heir resuls, since hey formally correspond o PDEs se on N and no on bounded domains, a raher imporan difference for PDEs. In he firs secion, we are concerned wih general BSDEs, whose coefficien is coninuous wih quadraic growh, whose erminal ime is no necessarily deerminisic nor bounded and whose erminal condiion may be a sopping ime. We firs give exisence resuls under general assumpions. We nex give a uniqueness resul: he one-dimensional frameworkallows us o provide a comparison resul which implies uniqueness as a by-produc. However, o prove i, we use sronger assumpions on he coefficien F han for he exisence resul: he quadraic growh is mean, roughly speaking, as linear growh on he parial derivaives of F wih respec o Z. We also give a sabiliy resul: he soluions Y n Z n of BSDEs wih parameers F n ξ n converge o he unique soluion Y Z of he BSDE wih parameers F ξ under very general assumpions of he convergence of F n n o F when F saisfies he assumpions required for he uniqueness resul o hold.

3 56 M. KOBYLANSKI The second secion provides connecions beween he soluions of hose BSDEs and soluions of relaed quasilinear PDEs. Of course, when he soluion of he PDE is smooh enough, he meaning of his connecion is as usual sraighforward. Such a regulariy may be obained under srong assumpions on he coefficiens of he PDE and of he nonlineariy [15]. Conversely, if one assumes ha such assumpions hold on he coefficiens of a forward backward sysem, he flow hus defined has also a grea regulariy, and i can define a classical soluion of he PDE. However, when he hypoheses are such ha he PDE has o be solved in a weak way, difficulies arise. Differen approaches can be used relying on differen noion of weaksoluions for he associaed PDEs. A firs one consiss in using he noion of viscosiy soluions. This noion of weaksoluions was inroduced by Crandall and Lions [8] for firs-order Hamilon Jacobi equaions, and exended o second order equaions by Lions. The connecion wih BSDEs has been done by Pardoux and Peng [16] for Lipschiz coninuous coefficiens. In Secion 2.1, we give, in our framework, a proof ha he BSDEs provide a viscosiy soluion for he associaed PDE. We also prove a uniqueness resul for he viscosiy soluion of his PDE. Anoher way of defining weaksoluions of PDEs, which is more classical, is he noion of Sobolev soluions. Barles and Lesigne [2] were he firs o use his approach in order o connec he soluion of he PDE wih he associaed BSDE. I gives ineresing insighs, as he soluions of he PDE allow one o obain he whole soluion Y Z T raher han only Y T as he viscosiy soluions do. I also seems o mach beer he Hilberian aspec of sochasic inegrals heory. This is he subjec of Secion BSDEswih quadraic growh Exisence. In order o jusify he assumpions we inroduce o prove he exisence resul (more precisely, wha we undersand by quadraic growh of he coefficien on he one hand, and why we require he boundedness of he erminal condiion ξ on he oher hand we give wo examples. These examples are also an occasion o use he echniques of he exponenial change of variable and of he applicaion of Iô s formula o a well-chosen funcion, which are cenral ools hroughou his chaper. Example 1. We consider he following equaion: (2 1 T Y = ξ + 2 Z s 2 ds Z s dw s The exponenial change of variable y = expy ransforms formally his equaion a.s.: (3 T y = expξ z s dw s

4 The laer equaion being linear, we have, when BACKWARD SDE AND PDE 561 (4 expξ L 2 he exisence of a unique soluion y z HT 2 H T 2 d of (3. The process y is given explicily by T y = ƐexpξF and he process z is given by he heorem of represenaion of coninuous maringales (see, e.g., [13]. Taking ξ L is a sufficien assumpion o require on ξ in order o have (4. Indeed, if ξ L expξ L L 2 and here exiss a unique soluion of (3. Moreover, T y exp ξ and one can define T Y = lny Z = z /y I is hen easy o checkha he pair Y Z H T H T 2d is a soluion of (2. The uniqueness in H 2 T HT d comes from he fac ha he exponenial change of variable is no longer formal and from he uniqueness for equaion (3. Example 2 (a priori esimaes. Le a: + and b: + + be wo funcions and C be a posiive consan. We say ha he coefficien F saisfies condiion (H wih a b C if for all v z + d, wih H F v z =a v zv + F v z a v z a a.s. F v z b+cz 2 a.s In his example we consider a BSDE wih parameers F τ ξ where he erminal ime is a sopping ime τ and he erminal condiion ξ is bounded. In his case we call a soluion of he BSDE wih parameers F τ ξ a pair of adaped processes, such ha: Y Z H τ H 2 τ d (i Y = ξ and Z = on he se τ. (ii Ɛ τ Z 2 d. (iii For all T, Y = Y T + τ Fs Y sz s ds τ Z s dw s.

5 562 M. KOBYLANSKI Proposiion 2.1. Le Y Z Hτ Hτ 2d be a soluion of he BSDE wih parameers F τ ξ, and suppose ha F saisfies condiion (H wih a b C, such ha T> a + = maxa b L 1 T and C> Then for all T, [ Y sup ( [ resp Y inf Y T + ( Y T ] exp ] ( exp a s ds + a s ds ( s b s exp a λ dλ ds ( s b s exp a λ dλ ds as as Moreover, here exiss a consan K depending only on Y, a + L 1 τ such ha Ɛ τ Z s 2 ds K and C An immediae consequence of his proposiion is he corollary. Corollary 2.2. Le Y Z Hτ Hτ 2d be a soluion of he BSDE wih parameers F τ ξ. (i If τ is bounded τ T as, ξ L and F saisfies condiion (H wih a b C such ha a + b L 1 TC>, Y ( ξ +b L 1 T exp ( a + L 1 T (ii If τ is unbounded, ξ L and F saisfies condiion (H wih a b C such ha here exiss a consan α such ha a α <, b L + and C>, Y ξ + b α Proof of Proposiion 2.1. Le T be such ha T τ and consider he soluion ϕ of he ordinary differenial equaion [ + ϕ = supy T ] + a s ϕ s + b s ds For T, [ + ( ( s ϕ = supy T ] exp a s ds + b s exp a λ dλ ds our aim is o prove ha Y ϕ. We apply Iô s formula o he process Y ϕ

6 BACKWARD SDE AND PDE 563 and o an increasing C 2 funcion ye o be deermined: Y ϕ =Y T ϕ T + Y s ϕ s [ Fs Y s Z s a s ϕ s + b s ] ds τ 1 2 Y s ϕ s Z s 2 ds Y s ϕ s Z s dw s τ τ We se, for s T, ã s = a s Y s Z s The funcion being increasing, for all s T, wehave Y s ϕ s [ Fs Y s Z s a s ϕ s + b s ] Y s ϕ s [ ã s Y s + b s + CZ s 2 a s ϕ s + b s ] Y s ϕ s [ ã s Y s ϕ s +ã s a s ϕ s + CZ s 2] and since ã s a s ϕ s, (5 Y ϕ Y T ϕ T + + τ τ τ ã s Y s ϕ s Y s ϕ s ds C 1 2 φ Y s ϕ s Z s 2 ds Y s ϕ s Z s dw s We se M =Y +ϕ and we define on M M he funcion by { e 2Cu 1 2Cu 2C 2 u 2 for u M, u = for u M. For all u M M, one can checkeasily ha u and u= if and only if u u u u 2M + 1Cu C u 1 2 u Hence, seing k = a + 2M + 1C, he funcion k is posiive and deerminisic and, for all T, Y ϕ and herefore, Y ϕ τ k s Y s ϕ s ds k s Y s ϕ s ds τ τ Y s ϕ s Z s dw s Y s ϕ s Z s dw s as as

7 564 M. KOBYLANSKI Since Y s ϕ s s T is bounded, Y s ϕ s Z s s T HT 2d, and aking expecaions in he above inequaliy yields ƐY ϕ k s ƐY s ϕ s ds Applying Gronwall s lemma, T ƐY ϕ = herefore, since u, T Y ϕ = as and since u = if and only if u we obain T Y ϕ as The proof of [ ] ( ( s Y inf T exp a s ds b s exp a λ dλ ds relies on he same compuaions. Indeed, if ϕ is now he soluion of he ordinary differenial equaion [ ] ϕ = inf Y T + a s ϕ s b s ds applying Iô s formula o defined as above and o ϕ Y yields o ϕ Y ϕ T Y T + τ k s ϕ s Y s ds τ k s ϕ s Y s Z s dw s and he same argumen allows us o conclude. In order o esimae Ɛ τ Z s 2 ds, we use again (5 wih =, ϕ = and M =Y and defined on M M by u = 1 [ ] exp2cu + M 1 + 2Cu + M 2C 2 I is sraighforward o checkha, for u M M, u u Therefore, (5 gives u u M C e4cm 1 Y Y T u C u =1 Z s 2 ds a + s M C e4cm 1 ds Y s Z s dw s

8 BACKWARD SDE AND PDE 565 which leads o Ɛ Z s 2 ds M+ M C e4cm 1a + L 1 and he proof is compleed by leing T. The exisence and he monoone sabiliy resuls. Le α β b and c be a coninuous increasing funcion. We say ha he coefficien F saisfies condiion (H1 wih α β bc if for all v z + d, F v z =a v zv + F v z wih H1 β a v z α a.s. F v z b + cvz 2 a.s. The main resul of his secion is he following heorem. Theorem 2.3 (Exisence. Le F τ ξ be a se of parameers of BSDE (1 and suppose ha he coefficien F saisfies (H1 wih α β b, and c + + coninuous increasing, ξ L, and: (i The erminal ime τ is eiher bounded, τ T a.s. or (ii The erminal ime is such ha τ< a.s. and α <. Then he BSDE (1 has a leas one soluion Y Z in H τ H 2 τ d such ha he process Y has coninuous pahs. Moreover, here exiss a minimal soluion soluion Y Z [resp.a maximal soluion Y Z ] such ha for any se of parameers G τ ζ, if F G and ξ ζ resp F G and ξ ζ and for any soluion Y G Z G of he BSDE wih parameers G τ ζ, Y Y G resp Y Y G Before giving he proof of Theorem 2.3, we sae he following proposiion which gives he main argumen of he exisence. I is presened under general assumpions as i will also be used in he nex secion. Proposiion 2.4 (Monoone sabiliy. Le F τ ξ be a se of parameers and le F n τξ n n be a sequence of parameers such ha: (i The sequence F n n converges o F locally uniformly on + d, for each n ξ n L and ξ n n converges o ξ in L. (ii There exiss k: + + such ha for all T>k L 1 T and here exiss C> such ha (6 n u z + d F n u z k + Cz 2

9 566 M. KOBYLANSKI (iii For each n, he BSDE wih parameers F n τξ n has a soluion Y n Z n H τ H 2 τ d such ha he sequence Y n n is monoonic, and here exiss M> such ha for all n, Y n M. (iv The sopping ime τ is such ha τ< a.s. Then here exiss a pair of processes Y Z H τ H 2 τ d such ha for all T +, lim n Yn = Y uniformly on T Z n n converges o Z in Hτ 2 d and Y Z is a soluion of he BSDE wih parameers F τ ξ. In paricular, if for each n, Y n has coninuous pahs, he process Y has also coninuous pahs. Remark. The limi coefficien F saisfies he assumpion of quadraic growh, bu no necessarily a comparison principle. Hence he soluion we find here migh no be unique. Proof of Proposiion 2.4. Since for all + he sequence Y n n is monoonic and bounded, i has a limi which we denoe Y. In view of Proposiion 2.1, here exiss a consan K such ha, for all n, Ɛ τ Z s 2 ds K Therefore, here exiss a process Z H 2 τ d and a subsequence Z n j j of Z n n such ha (7 Z n j Zweakly in H 2 τ d The poin is now o show ha in fac he whole sequence converges srongly o Z in Hτ 2d. We noice ha, by inequaliy (6, seing K = 5C, F n v z F p v z 2k + K ( z z 2 +z z 2 +z 2 Sep 1. The srong convergence of Z n n in Hτ 2d. The main argumens of his sep are adaped from [5]. We have, for all n p, F n Y n Z n F p Y p Z p 2k + K ( Z n Z p 2 +Z n Z 2 +Z 2 Le us apply Iô s formula o he process Y n Y p T for n p n p, and o an increasing funcion ψ C 2 2M, such ha ψ = and ψ =.

10 BACKWARD SDE AND PDE 567 The funcion ψ is ye o be chosen: ψy n Yp = ψy n T Yp T + ψ Y n s Yp s ( F n Y n s Zn s Fp Y p s Zp s ds 1 2 As ψ Y n s Yp s, ψy n Yp ψ Y n s Yp s Zn s Zp s 2 ds ψ Y n s Yp s Zn s Zp s dw s ψy n T Yp T + ψ Y n s Yp s 1 2 [ 2k s + KZ n s Zp s 2 +Z n s Z s 2 +Z s 2 ] ds ψ Y n s Yp s Zn s Zp s 2 ds ψ Y n s Yp s Zn s Zp s dw s We now ransfer he erms in Z n s Zp s 2 and Z n s Z s 2 o he lef-hand side of he inequaliy, and we ake he expecaion. As Y n Y p is bounded, and, Ɛ ψ Y n s Yp s Zn s Zp s dw s = ƐψY n Yp + Ɛ 1 2 ψ Kψ Y n s Yp s Z n s Z p s 2 Kψ Y n s Yp s Zn s Z s 2 ds ƐψY n T Yp T +Ɛ ψ Y n s Yp s 2k s + KZ s 2 ds We wan o pass o he limi as p along he subsequence n j j defined in (7. The convergence of Y p Y being poinwise, and Y p being bounded, one has, by Lebesgue s dominaed convergence heorem, ƐψY n Y [ + lim inf Ɛ 1 2 ψ Kψ ] Y n s Y sz n s Zp s 2 ds Ɛ p p n j Kψ Y n s Y sz n s Z s 2 ds ƐψY n T Y T+Ɛ ψ Y n s Y s2k s + KZ s 2 ds

11 568 M. KOBYLANSKI and as [ lim inf Ɛ p p n j ] ψ Y n s Y sz n s Zp s 2 ds we obain, ƐψY n Y + Ɛ lim inf p p n j ƐψY n T Y T+Ɛ Ɛ ψ Y n s Y sz n s Z s 2 ds We now choose ψ such ha = 1, namely, 1 2 ψ 2Kψ Y n s Y s Z n s }{{} Zp s 2 ds = ψ Y n s Y s2k s KZ s 2 ds ψu = 1 4K e4ku 4Ku 1 I is sraighforward o checkha ψ is a C funcion, increasing on [, 2M] and such ha ψ =ψ =. Noing ha by he convexiy of he l.s.c. funcional, one has we obain Ɛ ƐψY n Y +Ɛ JZ =Ɛ Z n s Z s 2 ds ƐψY n T Y T+Ɛ Z n s Z s 2 ds lim inf Ɛ p p n j Z n s Z s 2 ds Z n s Zp s 2 ds ψ Y n s Y s2k s + KZ s 2 ds By Lebesgue s dominaed convergence heorem, he righ-hand side of his inequaliy converges o as n, as well as he firs erm of he lef-hand side. Now, passing o he limi as n, we find, for all T>, lim sup Ɛ Z n s Z s 2 ds = n Consequenly he whole sequence Z n n converges o Z in Hτ 2d. Sep 2. The uniform convergence of a subsequence of Y n n o Y. sage of he proof we know ha A his for all + lim n Y n = Y he sequence Z n n converges o Z in H 2 τ d

12 BACKWARD SDE AND PDE 569 We proceed as in [14], applying he following lemma. Lemma 2.5. There exiss a subsequence Z n j j of Z n n such ha Z n j j converges almos surely o Z and such ha Z = sup j Z n j H 2 τ. Proof. For he convenience of he reader, we skech he proof of his lemma. Exracing if necessary a subsequence, we may assume wihou loss of generaliy ha he sequence Z n n converges almos surely o Z. Since Z n n is a Cauchy sequence in Hτ 2, we can exrac a subsequence Znj j such ha Z n j+1 Z n jh 2 τ 1/2 j, for all j. Then we se g =Z n + Z n j+1 Z n j j= Because of he properies of he sequence Z n j j,wehave g H 2 τ Z n H 2 Z n H 2 < + Moreover, for any p, we also have Z n p Z n + τ + j= τ + j= Z n j+1 Z n j H 2 τ 1 2 j p Z n j+1 Z n j g Therefore, Z = sup j Z n j H 2 τ and he proof is complee. j= For he sake of simpliciy of noaions, we sill denoe by Z n n he subsequence Z n j j given by Lemma 2.5 [resp. Y n n and F n n he sequences Y n j j and F n j j ] and herefore we have Z n Z a.s. d d and Z = sup Z n Hτ 2 n Recalling ha he sequence F n n converges locally uniformly o F, we ge, for almos all ω and τ, lim n Fn Y n Zn =F Y Z Since F n saisfies condiion (6, we have, F n Y n Zn k + C sup Z n 2 = k + C Z 2 n

13 57 M. KOBYLANSKI Thus, for almos all ω and, uniformly in τ, Lebesgue s dominaed convergence heorem gives lim n τ F n s Y n s Zn s ds = Fs Y s Z s ds On he oher hand, from he coninuiy properies of sochasic inegral, we ge lim sup T τ n Z n s dw s Z s dw s = in probabiliy T τ τ Exracing a subsequence again if necessary, we may assume ha he las convergence is a.s. Finally, Y n Ym Yn T Ym T + F n s Y n s Zn s Fm s Y m s Zm s ds τ τ + Z n s dw s Z m s dw s τ Therefore aking limis on m and supremum over τ, we ge, for almos all ω, sup Y n Y Y n T Y T+ Fs Y n s Zn s Fm s Y s Z s ds T τ + sup Z n s dw s Z s dw s T from which we deduce ha Y n n converges o Y uniformly for T (in paricular Y is a coninuous process if he Y n are. We can now pass o he limi in = Y p T + F p s Y p s Zp s ds Z p s dw s Y p τ obaining ha Y Z is a soluion of he BSDE wih parameers F ξ. Proof of Theorem 2.3. The proof consiss now in finding a good approximaion of F, in order o apply he previous heorem. We use a runcaion argumen in order o conrol he growh of F in u and an exponenial change in order o conrol is growh in z. We firs suppose ha insead of (H1 he coefficien F saisfies he following condiion: here exis α β BC +, such ha, for all v z + d, wih (8 τ τ F v z =a v zv + F v z β a v z α

14 BACKWARD SDE AND PDE 571 and and moreover, ha eiher: (9 F v z B + Cz 2 as (i The erminal ime is bounded τ T a.s., or (ii The erminal ime is finie τ< a.s. and α <. Le G τ ζ be a se of parameers such ha G F and ζ ξ and suppose ha i has a soluion Y G Z G Hτ Hτ 2d. Our aim is o find a soluion Y Z Hτ Hτ 2d of he BSDE wih parameers F τ ξ such ha Y G Y By Proposiion 2.1, seing { ξ + BT expα M + T in case (i, = ξ + B/α in case (ii, for any soluion Y Z of F τ ξ, one has We define (1 Y M M = max M Y G During he proof we will use several imes C funcions φ K 1 such ha { 1 if u K, (11 φ K u = if u K + 1. Sep 1. The exponenial change. The exponenial change v = e 2Cu ransforms formally a BSDE wih parameers F τ ξ in a BSDE wih parameers f τ e 2Cξ where ( f v z =2CvF lnv 2C z 1 z 2 2Cv 2 v and ( g v z =2CvG lnv 2C z 1 z 2 2Cv 2 v We consider a funcion ψ 1 such ha { 1 if u exp 2CM exp2cm ψu = if u exp 2CM + 1 exp2cm + 1

15 572 M. KOBYLANSKI and we use he convenion = for he sake of simpliciy of noaions; we se, for v z + d, f v z =ψvf v z and g v z =ψvg v z We have, seing lv =ψvα v lnv+2cbv ha ψv (β v lnv 2CBv z2 f v z lv v We remarkha l is a Lipschiz coninuous funcion bounded from above by a consan, say, L. We also remarkha defining y G = exp2cy G and z G = 2CZ G exp2cy G he pair y G z G is a soluion of he BSDE of parameers g τ e 2Cξ. Sep 2. The approximaion. We can approximae f by a decreasing sequence of uniformly Lipschiz coninuous funcions f p p such ha for all v z + d, g v z f v z f p v z lv+ 1 2 p For insance, if f p are C funcions such ha f f p f + 1 p+1 2 p (he exisence of such funcions is given by a sandard argumen of regularizaion, and if he funcions φ p are defined as in (11 a way of obaining hem is o se, for all p, f p v z = f p v zφ p v+z + ( lv+ 1 1 φ 2 p p v+z Then classical resuls of exisence and comparison for Lipschiz coninuous coefficiens give for each p he exisence and uniqueness of a soluion y p z p of he BSDE wih parameers f p τξ, and y G y p+1 y p y 1 Moreover, in case (ii we remarkha he process e 2CM τ is he soluion of he BSDE wih parameers τe 2CM and he process e 2CM τ is he soluion o he BSDE wih parameers τe 2CM. Since for p large enough, e 2CM e 2Cξ and f p e 2CM and for all p e 2CM e 2Cξ and f p e 2CM

16 BACKWARD SDE AND PDE 573 he comparison resul for Lipschiz coninuous coefficiens (cf. [11] gives, for all p large enough, e 2CM y p+1 y p e 2CM a.s. for all + We come backo he firs problem by seing F p u z = fp e 2Cu 2Ce 2Cu z + Cz 2 2Ce 2Cu and F u z = f e 2Cu 2Ce 2Cu z 2Ce 2Cu + Cz 2 The pair Y p Z p defined by = ψe 2Cu F u z+1 ψe 2Cu Cz 2 Y p = lnyp 2C and Z p = zp 2Cy p is a soluion of he BSDE wih parameers F p τξ. Le us recapiulae wha we have obained. (i The sequence F n n converges o F locally uniformly on + d, for each n, ξ n L and ξ n n converges o ξ in L. (ii There exis K C > such ha n u z + d F n u z K + Cz 2 (iii For each n, he BSDE wih parameers F n τξ n has a soluion Y n Z n Hτ Hτ 2 d such ha he sequence Y n n is decreasing, and here exiss M> such ha for all n, Y n M [wih M = M in case (ii]. (iv For all n Y G Y n. Therefore, applying Theorem 2.4, he process Y p p converges uniformly o Y and here exiss Z, inht 2 d such ha a subsequence of Z p p converges o Z and Y Z is a soluion of Y = ξ + Moreover, we prove ha (12 Fs Y s Z s ds Y M Z s dw s In view of he remarkmade above, we only need o give he proof in case (i. Indeed, i follows from Proposiion 2.1 as F v z =ã u z+ F u z

17 574 M. KOBYLANSKI wih and ã u z =ψe 2Cu a u z α + F u z =ψe 2Cu F u z+1 ψe 2Cu Cz 2 hence F u z B + 3Cz 2 By Corollary 2.2 we have, Y ξ + BT expα + T= M M Therefore, Y Z is also a soluion of he BSDE wih parameers F τ ξ. Moreover, Y G Y and Y M Sep 3. The runcaion. We now suppose ha F saisfies (H1. Le G τ ζ be a se of parameers such ha (9 holds rue and le Y G Z G Hτ Hτ 2d be a soluion of he BSDE wih parameers G τ ζ. ForM defined by (1 we se for all v z + d, F v z =a v zv + F φ M vv z G v z =G φ M vv z The coefficien F saisfies (8, and Y G Z G is also a soluion of he BSDE wih parameers G τ ζ. Applying Seps 1 and 2 we obain a soluion Y Z of he BSDE wih parameers F τ ξ such ha Y G Y. AsY M, he process Y Z is also a soluion wih coefficien F. We have proved he exisence of a maximal soluion. The proof of he exisence of a minimal soluion relies on he same proof bu wih he change of variable v = e 2Cu Uniqueness and sabiliy. As we menioned in he inroducion, he one-dimensional frame allows us o prove a comparison principle beween suband supersoluions which implies uniqueness as a by-produc. We give i for BSDEs wih a bounded erminal condiion ξ and wih a coefficien F which is locally Lipschiz coninuous and has a quadraic growh in Z in a srong sense (i.e., he parial derivaives of F have a linear growh. We firs recall ha a supersoluion (resp. a subsoluion of a BSDE wih coefficien F and erminal condiion ξ is an adaped process Y Z C T saisfying Y = ξ + Fs Y s Z s ds Z s dw s + dc s (resp dc s where C T is a righ coninuous increasing process C RCI and where, in he classical framework(i.e., when F is Lipschiz coninuous in Y and Z, and when ξ is square inegrable, he process Y Z T is assumed

18 BACKWARD SDE AND PDE 575 o be square inegrable (his noion is inroduced in [11]. Because of he quadraic growh of he coefficien, we will assume here ha Y T is a one-dimensional bounded process and Z T is a square inegrable process Y H T Z HT 2d. We say ha he coefficien F saisfies condiion (H2 on M M wih l k and C if for all +, u M M, z d, F u z l+cz 2 as H2 F u z z k+cz as and he coefficien F saisfies condiion (H3 wih c ε and ε if for all +, v, z d, F H3 u u z l ε+εz 2 Our main resul is he following heorem. Theorem 2.6 (Comparison principle. Le F 1 τξ 1 and F 2 τξ 2 be wo ses of parameers for BSDEs and suppose ha: (i ξ 1 ξ 2 a.s. and F 1 F 2. (ii For all ε, M> here exiss l l ε L 1 τ, k L2 τ, C such ha eiher F 1 or F 2 saisfies boh condiion (H2 on M M wih l k C and saisfies boh condiion (H3 on M M wih l ε and ε. Then if Y 1 Z1 C1 T resp. Y 2 Z2 C2 T H 2 T HT d RCI is a subsoluion (resp.a supersoluion of he BSDE wih parameers F 1 τξ 1 [resp. F 2 τξ 2 ], one has + Y 1 Y2 as Remark 2.7. I holds rue if eiher F 1 Y 1 Z1 F2 Y 1 Z1 a.s. for all and F 2 saisfy (H2 and (H3, or if F 1 Y 2 Z2 F2 Y 2 Z2 a.s. for all and F 1 saisfy (H2 and (H3. as We pospone he proof o give an imporan applicaion. Le F n τξ n n be a sequence of para- Theorem 2.8 (Sabiliy of BSDEs. meers of BSDEs such ha: (i There exiss α β b and an increasing funcion c such ha for all n he coefficien F n saisfies condiion (H1 wih α β b and c. (ii For all n here exiss a soluion Y n Z n o he BSDE wih parameers F n τξ n. Le F τ ξ be a se of parameers of BSDEs such ha he coefficien F saisfies he assumpions of Theorem 2.6.

19 576 M. KOBYLANSKI Then, if he sequence F n n converges o F locally uniformly on + d, and if he sequence ξ n n converges o ξ in L, here exiss a pair of adaped processes Y Z Hτ Hτ 2d such ha sequence Y n n converges o Y uniformly on T for all T, Z n n converges o Z in Hτ 2d and Y Z is he soluion of he BSDE wih parameers F ξ. and Proof. We define G n = sup F p p n ξ n = sup ξ p p n H n = inf p n Fp ξ n = inf p n ξp and we consider he maximal soluions Y n Z n of he BSDE wih parameers H n ξ n and he minimal soluions Y n Zn of he BSDE wih parameers G n ξ n, as boh: (i The sequence ξ n n is decreasing and he sequence G n n is decreasing and converges locally uniformly o F. (ii The sequence ξ n n is increasing and he sequence H n n is increasing and converges locally uniformly o F. Then we have: (i The sequence Y n n is bounded and decreasing, and for all n Y n Y n herefore by Theorem 2.4, here exiss Y Z such ha Y n n converges uniformly o Y, and Y Z is a soluion of he BSDE wih parameers F τ ξ. (ii The sequence Y n n is bounded and decreasing, and for all n, Y n Yn herefore by Theorem 2.4, here exiss Y Z such ha Y n n converges uniformly o Y, and Y Z is a soluion of he BSDE wih parameers F τ ξ. (iii By Theorem 2.6, we have boh n Y n Yn Y n and Y = Y = Y herefore he sequence Y n n converges uniformly o Y. We now urn o he following proof. Proof of Theorem 2.6. Following he mehod used by [3] for PDEs, we firs show he comparison principle under a srucure condiion on he coefficien. We nex complee he proof by giving a change of variable ha ransforms a BSDE wih a coefficien saisfying hypoheses (H2 and (H3 ino a BSDE wih a coefficien saisfying his srucure condiion.

20 BACKWARD SDE AND PDE 577 Sep 1. The comparison resul under srucure condiion. Le a and b + be a funcion. We say ha he coefficien f saisfies condiion (STR wih a and b if for all v z + d, STR 2 f u z+a f u z u z b Proposiion 2.9. Le f 1 ξ 1, and f 2 ξ 2 be wo parameers of BSDE, and le Y 1 Z1 C1, and Y 2 Z2 C2 be associaed supersoluion and subsoluion.suppose as (13 ξ 1 ξ 2 a.s. f 1 u z f 2 u z a.s. for all u z and suppose ha here exis a> and b L 1 τ such ha eiher f1 or f 2 saisfy condiion (STR wih a, b, hen, T Y 1 Y2 a.s Remark 2.1. (i I holds rue in H p T H T 2d insead of H T H T 2d when b is bounded insead of being inegrable and when a>1/2p 1 insead of a>. (ii I sill holds rue if eiher f 1 Y 1 Z1 f2 Y 1 Z1 a.s. for all and f 2 saisfies he srucure condiion (STR, or if f 1 Y 2 Z2 f2 Y 2 Z2 a.s. for all and f 1 saisfies he srucure condiion (STR. Proof of Proposiion 2.9. We se Y = Y 1 Y2, and Z = Z 1 Z2. Tanaka s formula applied o he process Y + = maxy yields dy + = 1 Y δf + d 1 Y Z + dw + dk + + dc 1 dc2 }{{} =dc where K + is a nondecreasing process and grows only a hose poins for which Y =. Now Iô s formula gives for p, p 2, Y + p pp = p δf s Y + s p 1 ds p 1 Y + Y + s p 2 Z 2 s ds + p Y + s p 1 dk + s + p }{{} = Y + s p 1 Z s dw s Y + s p 1 dc s } {{ }

21 578 M. KOBYLANSKI where wih δf s = f 1 s Y 1 s Z1 s f2 s Y 2 s Z2 s = f 1 s Y 1 s Z1 s f2 s Y 1 s Z1 s }{{} + f 2 s Y 1 s Z1 s f2 s Y 2 s Z2 s ( 1 δf s f 2 dλ u ( 1 Y 1 s Y2 s + f 2 dλ Z 1 s z Z2 s =s λy 1 s +1 λy2 s λz1 s +1 λz2 s We hen use he well-known inequaliy 2α β α 2 +β 2 wih α = 2a f2 z Y+ s p/2 β = 1 2a Y + s p 2/2 Z s 1 Y + Hence, wih M = maxy 1 Y 2, 1 δf s Y + f 2 s p 1 u + a f 2 2 z dλy + s p + 1 4a Y+ s p 2 Z s 2 1 Y + }{{} bs Coming backo Iô s formula, Y + p + p ( p a Y Y + + s p 2 Z s 2 ds (14 bsy + s p ds p Y + s p 1 Z s dw s As Y is bounded, Y + p 1 Z HT 2 d, and aking he expecaion, ƐY + p + p ( p 1 1 Ɛ 1 2 2a Y Y + + s p 2 Z s 2 ds p bsɛy + s p ds For p large enough, p 1 1/2a. I now follows ha Gronwall s inequaliy ha for all T, ƐY + p ; herefore for all T, Y 1 Y2 Sep 2. The change of variable. We now lookfor a change of variable ha ransforms parameers of BSDE saisfying condiions (H2 and (H3 ino parameers saisfying he srucure condiion (STR. Le Y Z T be a soluion in H T H T 2 d of a BSDE wih coefficien F and bounded erminal condiion ξ. as

22 BACKWARD SDE AND PDE 579 Take M such ha Y <M, and consider he change of variable ũ = φ 1 u where φ is a regular increasing funcion ye o be chosen. Seing wu =φ ũ, Ỹ = φ 1 Y, Z = Z /wy, he couple Ỹ Z T is he soluion o he BSDE wih coefficien f and erminal condiion φ 1 ξ, where f u z = 1 φ u (F φuφ uz+ 12 φ uz 2 Wriing u for φũ, z for φ ũ z and w for wy, a sraighforward compuaion gives f ũ ũ z = w F u z+ F w u u z+1 2 w z2 + w w = 1 ( ( 1 F w 2 w z 2 + w z z F + w F u w F u zz z f ũ z = F u z+zw z z w We now show ha a good choice of φ allows f o saisfy he srucure condiion (STR. Indeed, if φ is such ha w> and w >, hen ( f ũ + a f 2 z ũ z = 1 ( ( 1 F w 2 w z 2 + w z z F + w F + a u 1 [ 1 w 2 w z 2 + w ( kz+l+2cz 2 ] ( 2 + l ε +εz 2 + a k+ (C + w z w [ 1 z 2 w 2 ] (C 2 w + w w 2C + ε + a + w w [ w +z w k+2ak [ 1 z 2 w 2 w + w w 2C + (C + w F z w ( w 2 2 ] + ε + 2a(C + w w u z+zw w ] + w w l+l ε+ak 2 + w w l+l ε+1 + 2ak 2 Thus, if we find φ saisfying all he required assumpions and such ha on M M, w 2 (15 1 w 2 w + w w 2C + ( w 2 < δ < w

23 58 M. KOBYLANSKI hen choosing a and ε small enough, he coefficien before z 2 is nonposiive for all u. Therefore (STR is saisfied. Seing φv = 1 ( e λaṽ λ ln + 1 M A a sraighforward ye edious compuaion gives wu =A exp λu + M; when A>1 and λ>, we have w>, w >, on M M and moreover as 1 w 2 w + w w 2C + ( w w 2 exp λu + M = A exp λu + M 2 [λ ( 2 A exp λu + M + λ2c ( A exp λu + M ] is nonposiive on M M for a proper choice of A and λ. The proof is now complee. 3. BSDEsand PDEs. We consider he following forward backward sysem 16a dx s x X x = bs X s x ds + σs X s x dw s for s T = x n 16b dy s x Y x T = Fs X x s = gx x T Y s x Z s x ds Z s x dw s for s T where b and σ are Lipschiz coninuous funcions on T n aking values, respecively, in n and in he space of n d marices such ha here exiss a consan K such ha for T and x y R n, H4 b x b y+σ x σ y Kx y b x 2 +σ x 2 K 2 1 +x 2 F is a real-valued coninuous funcion defined on T n d, and g is a real-valued bounded coninuous funcion defined on n. The diffusion (16a is associaed wih he second-order ellipic operaor L defined by (17 Lu = 1 2 n i j=1 2 u a ij x x i x j n i j=1 b i x u x i where a is he symmeric posiive marix defined by a = σσ T denoes he ransposed of σ. where σ T

24 BACKWARD SDE AND PDE 581 The forward backward sysem (16 [or BSDE (16b se on he diffusion (16a] is conneced o he following PDE: u + Lu F x u σ xdu = in T n (18 ut x =gx in n Indeed, suppose ha u is a soluion of (18 of class C 2 ; applying Iô s formula o us X s x gives ha he process Y s x Z s x s T defined by 19a 19b Z x s Y x s = us X s x =σ T Dus X s x is a soluion of he BSDE (16b. In his secion we generalize his connecion beween PDEs and BSDEs when he soluion u of (18 does no have such a regulariy. We firs use he noion of viscosiy soluions of PDEs. This mehod allows one only o jusify equaion (19a. We hen use he noion of Sobolev soluions of PDEs. I allows giving a meaning o boh (19a and (19b, and i is beer suied for expressing he Hilberian aspec of sochasic inegrals. Remark 3.1. (i The assumpions (H4 on b and σ give he exisence and uniqueness of he diffusion process X, as well as he coninuiy of he flow x X s x s. (ii We will precise laer he assumpions aken on he coefficien F of (16. They will assure he exisence and uniqueness of he soluion and he coninuiy of he process x s Y s x when needed. (iii We wan o emphasize ha he represenaions of soluions of semilinear by (18 hold only for hose PDEs whose nonlineariy f has he peculiar form given by f x u p =F ( x u σ xp 3.1. Viscosiy soluions. The noion of viscosiy soluion was inroduced by Crandall and Lions [8] in order o solve firs-order Hamilon Jacobi equaions, and hen exended o second-order parial differenial equaions by Lions. Consider an equaion of he form (2 u + H x u Du D2 u= in T n This equaion is said o be parabolic if H saisfies he following ellipiciy condiion: H x u p M H x u p N if M N for any T, x n, u, p n and M N S n where S n is he space of n n symmeric marices. In our case H is given by H x u p M = TraM bp F ( x u σ T xp

25 582 M. KOBYLANSKI We recall ha a lower semiconinuous (resp. upper semiconinuous funcion u is a viscosiy subsoluion (resp. viscosiy supersoluion of (2 if for any φ C 2 T n and x T n such ha φ x =u x and φ x u x [resp. φ x u x] ont n, one has φ x +H x φ x Dφ x D 2 φ x [ resp. φ ] x +H x φ x Dφ x D 2 φ x > The funcion u is a viscosiy soluion if i is boh a super and a subsoluion. For furher deails, we refer o [7]. In his secion we firs provide a proof of a uniqueness resul for viscosiy soluions of a PDE wih quadraic growh wih respec o he gradien. Then we show ha he forward backward sysem (16 provides a viscosiy soluion of ( Uniqueness for viscosiy soluions. We suppose ha F saisfies he following assumpions: here exis a posiive consan C and, for any ε> here exiss a consan c ε such ha for all T x n u and q d a F x u σ T xq C1 +σ T xq 2 b F z x u σt xq C1 +σ T xq H5 F c µ x u σt xq c ε + εσ T xq 2 d F x x u σt xq C1 +σ T xq 2 Theorem 3.2 (Uniqueness for viscosiy soluions. Under assumpions (H4 and (H5, here is a comparison resul for he viscosiy soluions of (18. More precisely, if u is a bounded upper semiconinuous viscosiy subsoluion of (18 and v a lower bounded semiconinuous viscosiy supersoluion of (18, such ha hen ut x vt x in n u v on T n Remark 3.3. Assumpions (H5(a(b are very close o condiion (H2, and (H5(c o condiion (H3. (H5(d corresponds o he assumpion we need on he coefficien of he BSDE (16 in order o prove he coninuiy of he flow x Y s x s. Proof of Theorem 3.2. For he same reasons as for he proof of he comparison resul for BSDE, we firs make a change of variable, which preserves

26 BACKWARD SDE AND PDE 583 viscosiy sub- and supersoluions and which ransforms he equaion ino an equaion easier o use. In fac, we wan o have a nonlineariy H which is increasing wih respec o u, hence F o be decreasing in ha variable. Sep 1. The change of variable. consider he real-valued funcion We se M = max u v + 1 and we φv = 1 λ ln ( e λav + 1 A φ is one-o-one from ono ln A/λ +. For A and λ such ha ln A/λ M we consider he change of variable u = φ 1 e K u M, where K is a posiive parameer and φ a posiive increasing funcion, boh ye o be chosen. We se wu =e K φ u = u/ u and p = wup. Equaion (18 gives way o he following equaion: (21 u + H x u Du D2 u= where H x u p M = Tr ( σσ T xm bxp F ( x u σ T xp wih F defined by F ( x u σ T xp = φ u σ T xp 2 K φu φ u φ u Then (22 + ek φ u F x e K φu+m e K φ uσ T xp F ( x u σ T xp = w u σ T xp 2 K u M wu + 1 wu F x u wuσt xp One has, afer some compuaion, using (H5 and supposing ha w u >, F( x u σ T xp u σt xp 2 ( w u+2cw u+εwu + σt xp c wu wu ε w u ( K 1 +M u w u + c wu ε + C w u wu

27 584 M. KOBYLANSKI Noing ha σ T xp w u wu c ε σt xp 2 w u+ w u wu wu c2 ε we obain F( x u σ T xp σt xp 2 ( w u+2c + 1w u+εwu u wu K + c ε + w u( KM u+c + c 2 wu ε In order o have F (23 u K1 +σ T xp 2 (which corresponds o a proper equaion in [7], we now choose λ and ε such ha w u +2C + 1w u +εwu δ<on M M T. Then choosing K grea enough, he second erm can also be made smaller han δ. Choosing hen A grea enough and he change of variable is valid, one has for all T, x n, u I, p n, F saisfies he following condiions: (24 a b c F( x u σ T xp K1 +σ T xp 2 u F( x u σ T xp C1 +σ T xp 2 x F( x u σ T xp C1 +σ T xp z for some posiive consans K an C. In order o simplify he saemen of he nex resul, we se, for all x y in, p q in n, x y p q =1 + σt xp σt yq 2 2 Lemma 3.4. Suppose ha F saisfies assumpion (24, hen for all T, x y n, u v and p q n, we have, if u v>, F ( x u σ T xp F ( y v σ T yq ( x y p q Ku v+cx y+c σ T xp σ T yq Proof. For all x y n, u v and p q n we have F ( x u σ T xp F ( y v σ T yq = F ( x u σ T xp ( F x u + v 2 σt xp

28 hence wih ( + F x u + v BACKWARD SDE AND PDE σt xp ( + F y u + v 2 σt yq ( F y u + v 2 σt yq F ( y v σ T yq F ( x u σ T xp F ( y v σ T yq = 1 F u 1 dλ ( u v 2 1 F + x 2x y + F z 2 ( σ T xp σ T yq dλ 1 ( F u v + u 3 dλ 2 1 = ( x λu +1 λu + v/2σ T xp 2 = ( λx +1 λy u + v/2λσ T xp +1 λσ T yq 3 = ( y λu + v/2 +1 λv σ T yq The proof is now sraighforward. We now prove he uniqueness resul wih F as nonlineariy. Sep 2. Uniqueness under he srucure condiions (24. Le u be a subsoluion and v a supersoluion. As u and v are bounded u v has a supremum M. The proof consiss in showing ha M. 2a. The regular case. Suppose ha u and v are regular C 2 T n funcions and ha here exiss x T n such ha M is reached for x. If =, hen M. Suppose now T and M>. We have u x =v x+m Du x =Dv x Then, as u is a subsoluion, u x = v x D 2 u x D 2 v x (25 u + H x u xdu xd2 u x and, as v is a supersoluion, (26 v H( x v xdv xd 2 v x

29 586 M. KOBYLANSKI Subracing (25 from (26, and using (24(a, we have, as M>, F ( x u xσ T xdu x F ( x v xσ T xdu x 1 = M F( x λu x+1 λv xσ T xdu x dλ < u which is a conradicion. 2b. The general case. The difficuly is double. Firs, he funcions u and v are only supposed o be semiconinuous and second, he maximum of u v is no necessarily reached on T n. The mehod consiss in penalizing. We define M = sup u x v x x n T he supremum of he bounded funcion u v, and also Mh = sup u x v y and M = lim Mh x y h h One has, of course, M M. One purpose is o prove ha M. Le us consider x y2 ψ ε η x y =u x v y ηx 2 +y 2 ε 2 Le M ε η be a maximum of ψ ε η and ε η x ε η y ε η he poin a which his maximum is reached. As we seeko prove ha u v, we assume o he conrary ha us z > vs z for some s and z; i follows ha M ε η us z vs z =δ> for all ε η We need he equivalen of Lemma 3.1 of [7]. For he sake of simpliciy of noaions, we wrie c insead of ε η (resp. ˆx ŷ, and we inroduce he following noaion: if a ε η is a sequence we wrie and If we wrie lim sup ε η a ε η =lim sup η [ ] lim sup a ε η ε [ lim inf a ε η =lim inf lim inf ε η η ε a ε η ] lim sup a ε η =lim inf a ε η =a ε η ε η a = lim a ε η ε η

30 Lemma 3.5. i lim M ε η = M ε η iia c BACKWARD SDE AND PDE 587 lim u c ˆx v c ŷ =M ε η lim M ε η = M b lim u ˆ ˆx v ˆ ŷ =M η ε η ε ˆx ŷ lim = d lim η ε ε η η ε ˆx2 +ŷ 2 = We pospone he proof of his lemma and coninue he main sream of our proof. Theorem 8.3 of [7] allows us o sae he following. Lemma 3.6. We se p = 2ˆx ŷ/ε 2 +2η ˆx and q = 2ˆx ŷ/ε 2 2ηŷ.There exiss X Y N such ha (27 H c ˆx u c ˆxpX H c ŷ v c ŷqy and (28 ( X Y 2 ( I ε 2 I I + 2η I ( I I ( ( Muliplying (28 by σ c ˆx from he righ side and by σ c ˆx from he lef σ cŷ σ cŷ side and aking he Trace, we find Tr ( σσ T c ˆxX Trσσ T c ŷy 2σ c ˆx σ( c ŷ ησ c ˆx 2 ε 2 2 +σ c ŷ2 2 ( 2ˆx ŷ σ 2 2 Lip + 2η ˆx 2 +ŷ 2 ε 2 In order o complee our proof we only need o show ha (27 is in conradicion wih M >. Indeed, (27 gives Tr ( σσ T ˆ ˆxX + Tr ( σσ T ˆ ŷy b ( ˆ ˆxp + b ˆ ŷ q F ( ˆ ˆx u ˆ ˆxp F ( ˆ ŷ v ˆ ŷq We use he majoraion provided by Lemma 3.4 for he righ-hand side, hence ( ( σ 2 Lip +b 2ˆx ŷ 2 Lip + 2η ( ˆx 2 +ŷ 2 ε 2 Therefore, by Lemma 3.5, ˆ ˆx ŷ p q ( Ku ˆ ˆx v ˆ ŷ o1 ˆ ˆx ŷ p q ( KM + o1

31 588 M. KOBYLANSKI As ˆ ˆx ŷ p q 1, we can divide by ˆ ˆx ŷ p q, and pass o he limi. We ge KM which is he expeced conradicion. Proof of Lemma 3.5. For all x y (29 u x v y x y2 ε 2 M ε η = u ˆ ˆx v ˆ ŷ u ˆ ˆx v ˆ ŷ ηx 2 +y 2 ˆx ŷ2 ε 2 η ˆx 2 +ŷ 2 Using he firs inequaliy of (29 wih x = y = one has ˆx ŷ 2 + η ˆx 2 +ŷ 2 2u +v ε 2 hence, wih C = 2u +v, we have he following firs esimae: (3 ˆx ŷ Cε ˆx ŷ C η Sep 1. We have, using (3, u ˆ ˆx v ˆ ŷ u ˆ ˆx v ˆ ˆx + v ˆ ˆx v ˆ ŷ }{{}}{{} M when η is fixed and ε Le h x h be a sequence such ha lim u hx h v h x h =M h Inequaliy (29 gives, when η is fixed, u h x h v h x h 2ηx h 2 M ε η u ˆ ˆx vs ŷ M + oε Leing successively ε, η and h in he inequaliy above gives boh and M lim inf M ε η lim sup M ε η M ε η ε η M lim inf u ˆ ˆx v ˆ ŷ lim sup u ˆ ˆx v ˆ ŷ M ε η ε η

32 BACKWARD SDE AND PDE 589 Sep 2. Using (3 we have u ˆ ˆx v ˆ ŷ MCε Le h x h y h be a sequence such ha x h y h h and lim u h x h v h y h =M h I gives in (29, u h x h v h y h h ε 2 ηx h 2 +y h 2 M ε η u ˆ ˆx vs ŷ MCε Leing firs η, h, hen ε, M lim inf M ε η lim sup M ε η M η ε η ε M lim inf u ˆ ˆx v ˆ ŷ lim sup u ˆ ˆx v ˆ ŷ M η ε η ε Now passing o he limi in he previous order in (29 gives he las hree resuls. This ends he proof of he lemma Exisence of viscosiy soluions given by BSDE. Le X s x s T be he diffusion defined by (16 under assumpion (H4, and suppose ha F saisfies (H5. In view of Remark3.1(i and of (H5(d, he funcion s u z Fs X x s uz converges locally uniformly o s u z Fs X x s uz as x x. The immediae applicaion of Theorem 2.8 allows us o sae he following heorem. Theorem 3.7 (Coninuiy. saisfies assumpions (H5, he flow x Y s x paricular, he deerminisic funcion x Y x The main resul of our secion is his heorem. If σ and b saisfy assumpions (H4 and F s is a.s.coninuous.in is coninuous. Theorem 3.8 (BSDE and viscosiy soluions. Under assumpions (H4 on L and (H5 on F, he funcion defined on T d by u x =Y x is a viscosiy soluion of (18. The proof of his resul does no depend on a comparison resul. I is a local proof and herefore closer o he spiri of viscosiy soluions. The main argumen is given by he following heorem. Theorem 3.9 (Touching. such ha Le Y T be a coninuous adaped process dy = b d + σ dw

33 59 M. KOBYLANSKI where b and σ are coninuous adaped processes such ha b, σ 2 are inegrable. If Y a.s. for all, hen for all, 1 Y =σ = a.s. 1 Y =b a.s. Proof. Le φ: + be a posiive C 2 bounded increasing funcion such ha φ =, φ =α>, φ =β<. Se Ỹ = φy Iô s formula gives dỹ = b d + σ dw where σ =φ Y σ and b =φ Y b+ 1 2 φ Y σ 2. Le us apply now Iô s formula o ψ ε Ỹ where ψ ε ε is a sequence of C 2 convex funcions ha converges o (wih ψ ε = 1on : ψ ε Ỹ ψ ε Ỹs = for all s. Hence (31 s s ψ ε Ỹτ bτ+ 1 2 ψ ε Ỹτ στ 2 dτ + 1 Ỹ τ = bτ dτ Taking he expecaion we ge ( Ɛ 1 bτ Ỹ τ = dτ s s 1 Ỹ τ = στ dw τ s ψ ε Ỹτ στ dw τ dividing by s and leing s and as s Ɛ1 bs Ỹ s = is lower semiconinuous, we ge, for all s T, ( Ɛ 1 bs Ys = This implies ha, for all α> and β<, ( Ɛ 1 Ys =αbs+ 1 2 βσs2 Leing β o, one ges 1 Ys =σs 2 = a.s. for all s This implies ha for all τ, one has boh 1 Ỹ τ = στ = 1 Ỹ τ = bτ =1 Ỹ τ = αbτ Backin (31, i gives he second inequaliy and complees he proof.

34 BACKWARD SDE AND PDE 591 Proof of Theorem 3.8. For he sake of simpliciy of noaions le us wrie X for he diffusion process X x saring from x a and Y Z for he soluion Y x Z x of he BSDE driven by his diffusion. Firs noice ha u X =Y. This is readily seen from he Markovian propery of he diffusion process X and from he uniqueness of he BSDE. Le φ be a C 2 funcion such ha φ x =u x and φ x u x for all x T n. Hence φ X Y. We now show ha u is a viscosiy subsoluion of (18. One has dy = FY Z d Z dw Y τ ( φ dφ X = + Lφ X d σ T Dφ X dw φτ X τ As φ X Y, Theorem 3.9 gives, for all, ( ( φ 1 φ X =Y + Lφ X FY Z a.s. 1 φ X =Y Z + σ T Dφ X 2 = a.s. As φ X =Y for =, he second equaion gives Z = σ T Dφ X, and he firs inequaliy gives he expeced resul Sobolev soluions and BSDEs Sobolev soluions for PDEs wih quadraic growh. We firs give exisence and uniqueness resuls of Sobolev soluions for a quasilinear ellipic PDE, and hen some regulariy resuls for is linear par ogeher wih properies of is firs eigenvalue. The homogeneous Dirichle problem. In his paragraph we sudy he PDE Lu fx u Du = in (32 u H 1 where is an open bounded subse of n and he operaor L is, hroughou his secion, considered in he divergence form Lu = 1 ( n a 2 x i j x u (33 i j=1 i x j where we assume wihou loss of generaliy ha a i j = a j i, and we firs recall exisence and uniqueness resuls of a soluion of (32. For he exisence resul we suppose ha a ij L and here exiss α> such ha n (34 a i j ξ i ξ j αξ 2 i j=1 i j n for all ξ n and a.e. x

35 592 M. KOBYLANSKI and ha he funcion f is a Caraheodory funcion defined on n (i.e., fx is coninuous for a.e. x and f up is measurable for all u and p n such ha for almos all x and for all u p n, wih fx u p = a xu f x u p (35 <α a x β a.e. in and f x u p C + bup 2 for a posiive consan C and an increasing funcion b. Theorem 3.1 [5]. Under assumpions (34 and (35, here exiss a leas a soluion in H 1 L of (32. For he uniqueness resul we suppose ha he funcion f defined on n is a Caraheodory funcion, such ha for a.e. x he funcion u p fx u p is locally Lipschiz coninuous wih f x u p p C u1 +p for a.e. x u p n H6 fx u C 1 u for a.e. x u p n f u x u p α <u for a.e. x u p n for a consan α and for increasing funcions C and C 1. Theorem 3.11 [3]. Assume ha (34 and (H6 hold.if u 1 and u 2 belong o H 1 L and are, respecively, a subsoluion and a supersoluion of (32 such ha u 1 u 2 + H 1, hen u 1 u 2 in In paricular (32 has a mos one soluion in H 1 L. An immediae consequence of his comparison principle is he corollary. Corollary Assume ha (34 and (H6 hold.if u H 1 L is a soluion of (32 hen u C 1 α

36 BACKWARD SDE AND PDE 593 Proof. Indeed, he funcion defined by is a supersoluion of (32, since u 1 x =C 1 /α = C Lu 1 x fx u 1 xdu 1 x = fx C C f = fx x u u C 1 +α C = In a similar way, we can show ha C is a subsoluion. Proposiion Assume ha (35 and (H6 hold, hen equaion (32 has a unique soluion in H 1 L. Proof. The proof relies on a runcaion argumen. Indeed, for all K> we consider a C 2 funcion, φ K, such ha φu =u for K u K, φ is consan when u >K+ 2 and φ Ku 1 for all u, and we se f K x u p = α u + fx φ K up+α φ K u The funcion f K obviously saisfies assumpions (H6 wih he same consans as f and saisfies also assumpions (35 wih a K x =α C K = α K+1+ C K+1+C 1 K+1 and b K u =2C K+1, hence applying Theorems 3.1 and 3.11, equaion Lu f K x u Du =has a unique soluion u K in H 1 L, and by Corollary 3.12, u K C 1 /α.asf K c u p =fx u p for all x u K and p n, u K is also he unique soluion of (32 as soon as K>C 1 /α. The nonhomogenous Dirichle problem. We now consider he following PDE: (36 Lu fx u Du = in u g H 1 where L is defined by (33, and H7 a The boundary of has a C 1 1 regulariy b g W 2p wih p>n c a ij x W 2 1 i j N and here exiss α> such ha N i j=1 a ij xξ i ξ j αξ 2 for a.e. x and ξ n Theorem 3.14 (Nonhomogenous Dirichle problem. Assume ha (H6 and (H7 hold.then here exiss a unique Sobolev soluion u in H 1 L of (36.

37 594 M. KOBYLANSKI Proof. Under assumpion (H7, he Dirichle problem Lĝ = in ĝ g H 1 has a unique soluion ĝ W 2p (cf. Theorem 9.15, page 241 of [12]. As p>n, Sobolev imbeddings show ha ĝ C 1 C 2. Consider now Lw fx ˆ w Dw = in (37 w H 1 where fˆ is defined by fx ˆ u p =fx u gxp D gx. As ĝ C 1, i is obvious ha u H 1 L is he soluion of (36 if and only if w = u ĝ is he soluion of (37, bu as fˆ saisfies he assumpions of Proposiion 3.13, (37 has a unique soluion in H 1 L, which complees he proof. (38 The linear operaor. We recall some classical resuls on he linear equaion Lu + α u = f in u = g on where L is defined by (34 and α is a nonnegaive consan. Proposiion (i If f L 2 and g =, here exiss a consan C depending only on and α such ha u H 1 C f L 2 (ii Assume a ij C 1α for all 1 i j N f C α C 1 1 g C 1α ; hen he unique soluion u of (38 belongs o C 2α C 1 and ( f u max g α in paricular, his resul holds rue under assumpions (H7. (iii Assume (34; hen he operaor L is self-adjoin sricly ellipic, is firs eigenvalue λ 1 > and he associaed eigenfuncion e 1 is unique up o a muliplicaive consan e 1 > in.moreover, if we suppose ha 2 holds rue, e 1 C 2α For he proof, see, for insance, [12] Connecions wih BSDEs. We firs define he sysem of forward backward equaions associaed wih (36.

38 The diffusion process he exi ime problem. ha (H7 holds rue, and we define b i = 1 n a ij 2 x j BACKWARD SDE AND PDE 595 j=1 From now on, we suppose and σ n n is such ha σσ T = a and, for all i j n σ i j b i W 1. L is he infiniesimal generaor of he diffusion process dx x = bxx d + σxx dw for + X x = x x We noe τ o be he firs exi ime of he process X from, τx ω =inf >X x ω As a is no degeneraed, τ is almos surely finie. In fac, he exi ime and he firs eigenvalue of L are conneced in he following way: Ɛ ( (39 e λτx < for all λ<λ1 for all x [9]. The flow of BSDEs. For all x, we define he coefficien se on he diffusion saring from x by F x v zω =fx x ωvσt X x ω 1 z for all ω τ x ωv z n, and we consider he BSDE wih parameers F x τ x gx x τ x, τ x Y x = gxx τ x+ F x s Y x s Zx s ds τx (4 Z x s dw s τ x τ x Theorem 3.16 (BSDE. Assume ha f saisfies (H6 and ha (H7 holds rue; hen for all x, here exiss a unique soluion Y x Z x τ x of (4. The proof is a direc consequence of he resuls of Secion 1, since all he required assumpions are obviously saisfied. We now give he Feynman Kac formula corresponding o equaion (38 expressed in erms of BSDEs. Lemma 3.17 (Feynman Kac formula. Assume ha (H7 holds rue and f C α, and le u be he soluion of (38.Seing Y x = uxx Z x =σt DuX x for every x, he process Y x Z x τ x is in Hτ 2 x Hτ xn and is he soluion of dy x = α Y x + f X x d Zx dw Y x τ x = gxx τ x