Multidimensional Markov FBSDEs with superquadratic

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1 Mulidimenional Markov FBSDE wih uperquadraic growh Michael Kupper a,1, Peng Luo b,2, Ludovic angpi c,3 December 12, 216 arxiv: v2 [mah.pr] 9 Dec 216 ABSRAC We give local and global exience and uniquene reul for yem of coupled FBSDE in he mulidimenional eing and wih generaor allowed o grow arbirarily fa in he conrol variable. Our reul are baed on Malliavin calculu argumen and paing echnique. KEYWORDS: uniquene 1 Inroducion Markovian FBSDE, uperquadraic, exience and AUHORS INFO a Univeriy of Konanz, Univeriäraße. 1, D Konanz, Germany b EH Zürich, Rämirae 11, 892 Zürich, Swizerland c Univeriy of Vienna, Faculy of Mahemaic, Okar-Morgenern-Plaz 1, A-19 Wien, Auria 1 kupper@uni-konanz.de 2 peng.luo@mah.ehz.ch 3 ludovic.angpi@univie.ac.a PAPER INFO AMS CLASSIFICAION: (2) 6H1, 6H7, 6H3 Given a mulidimenional Brownian moion W on a probabiliy pace, conider he yem of forward and backward ochaic differenial equaion { X = x+ b (X,Y )d+ σ dw Y = h(x )+ g (X,Y,Z )d (1.1) Z dw, [,] where x i a known iniial value, > and b, σ, g and h are given funcion. In hi paper, we give condiion under which he yem admi a unique oluion in he cae where he value proce Y i mulidimenional and he generaor g can grow arbirarily fa in he conrol proce Z. he focu in hi paper i on Markovian yem, in which he funcion b, σ, g and h are deerminiic. We conider generaor ha are Lipchiz coninuou in X and Y and locally Lipchiz coninuou inz. In hi eing, and for one-dimenional value procee, he decoupled yem (wih b depending only on X) ha been olved by Cheridio and Nam [5] baed on Malliavin calculu argumen. In fac, uing ha for Lipchiz coninuou generaor he race of he Malliavin derivaive of he value proce Y i a verion of he conrol proce, hey howed ha he conrol proce can be uniformly bounded, hence enabling olvabiliy for locally Lipchiz generaor by a runcaion argumen. We make ample ue of heir mehod o deal wih he coupled yem (1.1). In hi cae, we propoe a Picard ieraion cheme which yield a equence ha can 1

2 be proved o be a Cauchy equence in an appropriae Banach pace under uniform boundedne of he conrol procee derived uing Malliavin calculu argumen provided ha he ime horizon i mall enough. Moreover uing he PDE repreenaion of Markovian Lipchiz FBSDE a developed for inance in Delarue [7] and a paing precedure, we conruc a unique global oluion for generaor wih growh pecified on he diagonal and under addiional aumpion, mainly non-degeneracy of he volailiy σ, ee heorem 2.2. Syem uch a (1.1) naurally appear in numerou area of applied mahemaic including ochaic conrol and mahemaical finance, ee Yong and Zhou [28], El Karoui e al. [11] and Hor e al. [16]. A hown for inance in Ma e al. [23] and Cheridio and Nam [5], in he Markovian cae, FBSDE can be linked o parabolic PDE. More recenly Fromm e al. [14] proved ha FBSDE can be ued in he udy of he Skorokhod embedding problem. BSDE and FBSDE wih Lipchiz coninuou generaor are well underood, we refer o El Karoui e al. [11] and Delarue [7]. If Y i one-dimenional and g i allowed o have quadraic growh in he conrol proce Z, BSDE oluion have been obained by Kobylanki [2], Barrieu and El Karoui [2] and Briand and Hu [3, 4] under variou aumpion on he erminal condiion ξ = h(x ). We furher refer o Delbaen e al. [8], Drapeau e al. [1], Cheridio and Nam [5] and Heyne e al. [15] for reul on one-dimenional BSDE and FBSDE wih uperquadraic growh. Mainly due o he abence of comparion principle, general olvabiliy of mulidimenional BSDE wih quadraic growh i le well underood. Some imporan progre have been achieved recenly for BSDE wih mall erminal condiion, ee evzadze [26] and for more recen developmen, ee Hu and ang [17], Luo and angpi [22], Jamnehan e al. [19], Cheridio and Nam [6], Frei [12] and Xing and Žikovi`c [27]. o he be of our knowledge, he only work udying well-poedne of coupled FBSDE wih quadraic growh are he aricle of Anonelli and Hamadène [1] and he preprin of Luo and angpi [22] and Fromm and Imkeller [13]. In [1] he auhor conider a one-dimenional equaion wih one dimenional Brownian moion and impoe monooniciy condiion on he coefficien o ha comparion principle for SDE and BSDE can be applied. A (non-necearily unique) oluion i hen obained by monoone convergence of an ieraive cheme. In [13], a fully coupled Markovian FBSDE i conidered wih mulidimenional forward and value procee and locally Lipchiz generaor in (Y, Z) and a exience of a unique local oluion i obained uing he echnique of decoupling field and an exenion o global oluion i propoed. Alhough he (non-markovian) yem udied in Luo and angpi [22] i he ame a he one conidered in he preen paper, he echnique here are eenially differen. Furhermore, he main reul we preen here can be exended o he non-markovian eing and o random diffuion coefficien (when σ depend on X and Y ) under ronger aumpion involving he Malliavin derivaive ofg andh. We refer o he Ph.D. hei of Luo [21] for deail. In he nex ecion, we preen our probabiliic eing and he principal reul of he paper. In Secion 3 we prove local olvabiliy of mulidimenional BSDE wih uperquadraic growh and give condiion guaraneeing global olvabiliy. Secion 4 i dedicaed o he proof of he main reul. 2

3 2 Main reul Le (Ω,F,(F ) [,],P) be a filered probabiliy pace, where (F ) [,] i he augmened filraion generaed by a d-dimenional Brownian moion W, and F = F for a finie ime horizon (, ). he produc Ω [, ] i endowed wih he predicable σ-algebra. Sube ofr k andr k k,k N, are alway endowed wih he Borelσ-algebra induced by he Euclidean norm. he inerval [,] i equipped wih he Lebegue meaure. Unle oherwie aed, all equaliie and inequaliie beween random variable and procee will be underood in he P -almo ure and P d-almo ure ene, repecively. For p [1, ] and k N, denoe by S p (R k ) he pace of all predicable coninuou procee X wih value in R k uch ha X p S p (R k ) := E[(up [,] X ) p ] <, and by H p (R k ) he pace of all predicable procee Z wih value inr k uch ha Z p H p (R k ) := E[( Z u 2 du) p/2 ] <. Le l,m N be fixed. he oluion of (1.1) wih value in R m R l R l d can be obained under he following condiion: (A1) b : [,] R l R m i a coninuou funcion and here exi k 1,k 2,λ 1 uch ha b (x,y) b (x,y ) k1 x x +k2 y y and b (x,y) λ 1 (1+ x + y ) for all x,x R m and y,y R l. (A2) σ : [,] R m d i a meaurable funcion and here i λ 2 uch ha σ λ 2 for all [,]. (A3) h : R m R l i a coninuou funcion and here exi k 5 uch ha h(x) h(x ) k5 x x for all x,x R m. (A4) g : [,] R m R l R l d R l i a coninuou funcion aifying g (,,) L 2 ([,]), and here exik 3,k 4 and a nondecreaing funcionρ : R + R + uch ha g (x,y,z) g (x,y,z) k3 x x (2.1) for all x,x R m, y R l and z R l d uch ha z M := 4λ 2 k 5 dl and g (x,y,z) g (x,y,z ) k4 y y +ρ ( z z ) z z for all x R m,y,y R l and z,z R l d. (A5) here exi a conan K uch ha g (x,y,z) g (x,y,z) g (x,y,z )+g (x,y,z ) K x x ( y y + z z ) for all [,], x,x R m,y,y R l and z,z R l d. 3

4 Our fir main reul enure local exience and uniquene for he coupled FBSDE (1.1) under he previou aumpion. he proof i given in Secion 4. heorem 2.1. Aume ha (A1)-(A5) hold. hen here exi a conan C k,λ,l,d > depending only on k 1, k 2, k 3, k 4, k 5, λ 2, l and d, uch ha he FBSDE (1.1) ha a unique oluion (X,Y,Z) S 2 (R m ) S 2 (R l ) S (R l d ) wih Z M, whenever C k,λ,l,d. Local exience reul a heorem 2.1 above have been obained in [13, heorem 3] and [22, heorem 2.1] in eenially differen eing and wih differen mehod. A naural queion i o find condiion under which he above reul can be exended o obain global olvabiliy. Fromm and Imkeller [13] propoe a concaenaion procedure allowing o prove exience of oluion o a fully coupled FBSDE on a "maximal inerval". In he preen eing, under addiional aumpion, a paing mehod baed on PDE allow o ge global exience and uniquene for he FBSDE (1.1). he proof i alo given in Secion 4. (A4 ) g : [,] R m R l R l d R l i a coninuou funcion aifying g (,,) L 2 ([,]), and here exik 3,k 4 and a nondecreaing funcionρ : R + R + uch ha g i (x,y,z) g i (x,y,z ) k3 x x +k4 y y +ρ ( z z ) z i (z ) i for all i = 1,...,l; x,x R m, y,y R l and z,z R l d. heorem 2.2. Aume ha (A1), (A2), (A3), (A4 ), and (A5) hold and here exi λ 3 > ; λ 4,λ 5 uch ha 1 b (x,y) λ 1 (1+ y ) x,σ σ x λ 3 x 2 (2.2) g (x,y,z) λ 4 (1+ y +ρ( z ) z ) h(x) λ 5 for all [,], x,x R m, y,y R l and z,z R l d, hen he FBSDE (1.1) ha a unique global oluion (X,Y,Z) S 2 (R m ) S 2 (R l ) S (R l d ) uch ha Z M for ome conan M. Remark 2.3. he following counerexample how ha he addiional condiion in heorem 2.2 canno be dropped wihou violaing global olvabiliy. Conider he FBSDE { X = Y udu Y = kx u du Z u dw u. hi equaion can be rewrien a 1 σ i he ranpoe marix of σ. Y = ky u dud Z u dw u. (2.3) 4

5 I ha been hown in [9, Example 3.2] ha if k < π 2 hen he BSDE wih ime-delayed generaor (2.3) ha a unique oluion wherea if k = π 2, (2.3) may no have any oluion and if i doe have one, here are infiniely many. heorem 2.2 exend o fully coupled generaor provided ha a more pecial rucure i aumed. he proof of he nex propoiion i given in Secion 4.3. Propoiion 2.4. If here exi an inverible marixγ R n n and λ 3 > ; λ 4,λ 5 uch ha (A1), (A2), (A3), (A4 ), (A5) and (2.2) hold for h := Γh, g (x,y,z) := Γg (x,γ 1 y,γ 1 z) and b (x,y) := b (x,γ 1 y), hen he FBSDE (1.1) ha a unique global oluion (X,Y,Z) S 2 (R m ) S 2 (R 2 ) S (R 2 d ) uch ha Z M for ome conan M. he poin of he above propoiion i ha global olvabiliy can be obained for a generaor ha doe no aify he "diagonally uperquadraic" growh condiion (A4 ) provided ha he ranformed generaor g doe. Moreover, noice ha heorem 2.2 and Propoiion 2.4 guaranee exience of a decoupling field, ee [13]. In paricular, he boundedne of Z guaranee a uniform Lipchiz propery of he decoupling field. heorem 2.1 relie on an exience reul for mulidimenional BSDE preened in Nam [24] and reviied in he nex ecion. 3 Mulidimenional BSDE wih bounded Malliavin derivaive Le u inroduce he pace of Malliavin differeniable random variable and ochaic procee D 1,2 (R l ) andl 1,2 a (R l ). For a horough reamen of he heory of Malliavin calculu we refer o Nualar [25]. Le M be he cla of mooh random variable ξ = (ξ 1,...,ξ l ) of he form ξ i = ϕ i( h i1 dw,..., h in dw ) where ϕ i i in he pace C p (Rn ;R) of infiniely coninuouly differeniable funcion whoe parial derivaive have polynomial growh, h i1,...,h in L 2 ([,];R d ) and n 1. For every ξ inmle he operaor D = (D 1,...,D d ) : M L 2 (Ω [,];R d ) be given by D ξ i := n j=1 ϕ i ( x j h i1 dw,..., h in dw ) h ij,, 1 i l, and he norm ξ 1,2 := (E[ ξ 2 + D ξ 2 d]) 1/2. A hown in Nualar [25], he operaor D exend o he cloured 1,2 (R l ) of he emwih repec o he norm 1,2. A random variableξ i Malliavin differeniable ifξ D 1,2 (R l ) and we denoe byd ξ i Malliavin derivaive. Denoe by L 1,2 a (R l ) he pace of procee Y H 2 (R l ) uch ha Y D 1,2 (R l ) for all [,], he proce DY admi a quare inegrable progreively meaurable verion and [ Y 2 L 1,2 a (R l ) := Y H 2 (R l ) +E ] D r Y 2 drd <. 5

6 We nex conider a yem of uperquadraic BSDE of he form Y = ξ + aifying he following condiion: g u (Y u,z u )du Z u dw u (3.1) (B1) g : Ω [,] R l R l d R l i a meaurable funcion and here exi a conan B R + and a nondecreaing funcion ρ : R + R + uch ha g (y,z) g (y,z ) B y y +ρ ( z z ) z z for all [,], y,y R l and z,z R l d. (B2) ξ D 1,2 (R l ) and here exi conan A ij uch ha D j ξi A ij for all i = 1,...,l, j = 1,...,d and [,]. (B3) g (,) H 4 (R l ) and here exi Borel-meaurable funcion q ij : [,] R + aifying q2 ij ()d < and for every pair (y,z) Rl R l d wih ( d ) z Q := 2 A 2 ij + qij 2()d i hold j=1 g (y,z) L 1,2 a (R l ) wih D j ug i (y,z) q ij () for all i = 1,...,l, j = 1,...,d and u [,], for almo all u [,] one ha Du g (y,z) D u g (y,z ) Ku () ( y y + z z ) for all [,],y,y R l andz,z R l d for omer + -valued adaped proce (K u ()) [,] aifying K u 4 H 4 (R) du <. he following i an exenion of Cheridio and Nam [5, heorem 2.2] o he mulidimenional cae. I wa proved in Nam [24] under lighly differen aumpion. We give he proof for he ake of compleene. log(2) heorem 3.1. Aume ha (B1)-(B3) hold and. hen he BSDE (3.1) admi 2B+ρ 2 (Q)+1 a unique oluion in S 4 (R l ) S (R l d ) and Z Q. Conider he following ronger verion of he condiion (B1) and (B3): (B1 ) g i coninuouly differeniable in(y,z) and here exi conan B R + andρ R + uch ha y g (y,z) B and z g (y,z) ρ for all [,], y,y R l and z,z R l d. 6

7 (B3 ) he condiion (B3) hold for all (y,z) R l R l d. Lemma 3.2. If (B1 ), (B2) and (B3 ) hold, hen he BSDE (3.1) admi a unique oluion (Y,Z) S 4 (R l ) H 4 (R l d ) and Z j 2 ( A 2 ij+ q 2 ij()e (2B+ρ2 +1)( ) d ) e (2B+ρ2 +1)( ) for all j = 1,...,d. Proof. By Cheridio and Nam [5, Lemma 2.5], he condiion (B2) impliee[ ξ p ] < + for all p [1, ). I follow from El Karoui e al. [11, heorem 5.1 and Propoiion 5.3] ha he BSDE (3.1) ha a unique oluion (Y,Z) S 4 (R l ) H 4 (R l d ), which i Malliavin differeniable. Moreover for every i = 1,...,l and j = 1,...,d, he proce (DrY j i,dj rz i) [,] ha a verion (U ij,r ) [,] which aifie,v ij,r U ij,r =, V ij,r =, for < r, and i he unique oluion in S 2 (R l ) H 2 (R l d ) of he BSDE U j,r = Drξ j + Applying Iô formula o U j,r 2 yield U j,r 2 = Drξ j 2 + D j r ξ 2 D j r ξ 2 y g (Y,Z )U j,r + z g (Y,Z )V j,r +Drg j (Y,Z )d 2U j,r V j,r dw (3.2) V j,r dw. 2U j,r y g (Y,Z )U j,r +2U j,r z g (Y,Z )V j,r +2U j,r D j rg (Y,Z ) V j,r 2 d 2U j,r V j,r dw + 2U j,r V j,r dw + 2B U j,r 2 +2ρ U j,r j,r V ( 2B +ρ 2 +1 ) U j,r l qij 2 ()d. Uing condiion (B3) and aking condiional expecaion in he above inequaliy yield U j,r 2 E [ A 2 ij + ( 2B +ρ 2 +1 ) U j,r 2 + qij 2() Uj,r j,r V 2 d qij 2 ()d F ]. (3.3) By El Karoui e al. [11, Propoiion 5.3] he proce Z i a verion of he race(u ) [,] of he Malliavin derivaive of Y. Hence (3.2) follow from (3.3) by applying Gronwall inequaliy. 7

8 Proof (heorem 3.1). Define he Lipchiz coninuou funcion g by { g (y,z) if z Q, g (y,z) = g (y,qz/ z ) if z > Q. By Cheridio and Nam [5, Lemma 2.5] and El Karoui e al. [11, heorem 5.1] he BSDE correponding o( g,ξ) ha a unique oluion (Y,Z) S 4 (R l ) H 4 (R l d ). Forx = (y,z) R l+l d le β C (R l+l d ) be he mollifier { ( ) λexp 1 if x < 1, 1 x β(x) := 2 oherwie, where he conanλ R + i choen uch ha R β(x)dx = 1. Seβ n (x) := n l+l d β(nx), l+l d n N\{}, and define g n (ω,x) := g (ω,x )β n (x x )dx R l+l d o ha for each n > he funcion g n aifie (B1 ) and (B3 ) wih he conan ρ replaced by ρ(q). By Lemma 3.2 he BSDE correponding o (g n,ξ) ha a unique oluion (Y n,z n ) in S 4 (R l ) H 4 (R l d ) which aifie Since Z n,j 2 A 2 ij + A 2 ij + log(2) we obain 2B+ρ 2 (Q)+1 Z n,j 2 2 A 2 ij + q 2 ij()e (2B+ρ2 (Q)+1)( ) d qij()d 2 e (2B+ρ2(Q)+1). e (2B+ρ2 (Q)+1)( ) qij()d 2 for all j = 1,...,d. hi how Z n Q. Since gn converge uniformly in (,ω,y,z) o g, uing he procedure of he proof of Cheridio and Nam [5, heorem 2.2], i follow ha (Y n,z n ) converge o (Y,Z) in S 2 (R l ) H 2 (R l d ), o ha Z Q. Since g(y,z) = g(y,z) for all (y,z) R l R l d wih z Q, i follow ha (Y,Z) i he unique oluion of he BSDE correponding o (ξ,g) in S 4 (R l ) S (R l d ). log(2) Corollary 3.3. Suppoe (B1)-(B3) hold, 2B+ρ 2 (Q)+1 and(y,z) S4 (R l ) S (R l d ) i he oluion of he BSDE (3.1). heny D 1,2 (R l ) for all [,] and for everyj = 1,...,d, one ha ( ) DrY j 2 2 A 2 ij + qij()d 2 for all r [,]. (3.5) (3.4) 8

9 Proof. Since Z Q i bounded, (Y,Z) olve he BSDE wih erminal condiion ξ and generaor g defined by (3.4). If g aifie (B1 ) and (B3 ), hen he reul follow from Lemma 3.2. Oherwie conider he equence of mooh funcion g n converging o g a defined in he proof of heorem 3.1. Le (Y n,z n ) S 4 (R l ) H 4 (R l d ) be he oluion o he BS- DE correponding o (g n,ξ), which converge o (Y,Z) in S 2 (R l ) H 2 (R l d ). By Lemma 3.2 (Y n,zn ) D1,2 (R l ) D 1,2 (R l d ) for each [,] and he argumen in he proof of heorem 3.1 imply Dr j Y n 2 2 A 2 ij + qij 2 ()d j = 1,...,d, r, [,]. Hence, up n N E[ Dj ry n 2 dr] < for each [,]. Since (Y n ) converge o Y in L 2, i follow from Nualar [25, Lemma 1.2.3] ha Y D 1,2 (R l ) and (DY n ) converge o DY in he weak opology of H 2 (R l d ). hu, D r Y aifie (3.5). A a concequence o heorem 3.1, we give a condiion for global olvabiliy of fully coupled yem of BSDE. For he remainder of hi ecion we pu n := log(2) 2B +ρ 2 (2 n Q)+1, n N. Propoiion 3.4. Aume ha (B1)-(B2) hold, ha here exi N N uch ha N n= n, and (B3) hold wih Q replaced by 2 N Q. hen he BSDE (3.1) ha a unique oluion in S 4 (R l ) S (R l d ) and Z 2 N Q. Proof. If hen he reul follow from heorem 3.1. Oherwie, if > i follow by he ame argumen a in he proof of heorem 3.1 ha he BSDE (3.1) ha a unique oluion (Y,Z ) in S 4 (R l ) S (R l d ) on he inerval [,]. Moreover, Z aifie Z Q and by Corollary 3.3 one ha Y D 1,2 (R l ) and for every r, D j r Y 2 2 A ij q ij () 2 d for all j = 1,...,d. Since g aifie (B3) for all (y,z) R l R l d uch ha z cq, again by heorem 3.1 he BSDE (3.1) wih erminal condiion Y ha a unique oluion (Y 1,Z 1 ) in S 4 (R l ) S (R l d ) on [( 1 ), ], and D j ry 1 ( 1 ) A ij 2 + Z 1 2Q, [( 1 ), ]. (2 2 +2) q ij () 2 d, for all j = 1,...,d Repeaing he previou argumen, for N 2 he BSDE (3.1) ha a unique oluion (Y N,Z N ) ins 4 (R l ) S (R l d ) on[( N n= n),( N 1 n= n) ] wih erminal condiion 9

10 Y ( N 1 n= n). Moreover, D j ry N ( N n= n) 2 Z N 2N Q, 2 N A ij 2 + [ ( Hence, he pair (Y,Z) given by N n= N ( 2 k ) q ij () 2 d for all j = 1,...,d k=1 N 1 n ),( n= ] n ). Y := Y 1 [ 1,] + Z := Z 1 [ 1,] + N n=1 N n=1 olve (3.1) and i uniquene follow from heorem 3.1. Y n 1 [( n i= i),( n 1 i= i) ] Z n 1 [( n i= i),( n 1 i= i) ] Remark 3.5. he condiion N n= n for ome N N doe no guaranee global olvabiliy of mulidimenional BSDE wih uperquadraic growh. In fac, if ρ(x) C(1+ x) for all x, hen n n <. However, i doe guaranee global olvabiliy for BS- DE whoe generaor grow lighly faer han he linear funcion. For inance, if ρ(x) C(1+ log(1+x)) one ha n= log(2) 2B +ρ 2 (2 N Q)+1 log(2) 2B +2C 2 (1+log(2 N (1+Q)))+1 n= log(2) = 2B +2C 2 (1+log(1+Q)+nlog(2))+1 =. n= 4 Coupled FBSDE wih uperquadraic growh 4.1 Proof of heorem 2.1 Sep 1: We fir aume ha h,b and g are coninuouly differeniable in all variable. Le u define C 1 k,λ,l,d := k2 5 k 2 3 log2 λ 2 k 1 k 2 M log2 2k 4 +ρ 2 (M)+1 wih M := 4k 5 λ 2 dl. We will how ha for C 1 k,λ,l,d, he equence (X n,y n,z n ) given by X =, Y =, Z = and { X n+1 Y n+1 = x+ = h(x n+1 b(xn+1 u,y n u )du+ σ udw u )+ g u (X n+1 u,y n+1 u,zu n+1 )du Z n+1 u dw u, n 1 1

11 i well defined and ha Z n M for all n N and [,]. he proce X 1 i well defined, X 1 belong o D 1,2 (R m ) for every and he proce (D r X ) [,] aifie he linear equaion D r X 1 =, < r, D r X 1 = r ( x bd r Xu 1 + y bd r Yu)du+D r r σ u dw u, r, wihd r ( r σ udw u ) = σ1 [r,], ee Nualar [25, Lemma and heorem 2.2.1]. Hence, ince b i Lipchiz coninuou, we have Dr X 1 r k 1 D r X 1 udu+σ r and Dr X 1 λ2 e k 1, where he econd eimae come from Gronwall inequaliy. We will now how ha ince Ck,λ,l,d 1, h(x1 ) and g(x1,, ) aify (B1)-(B3). In fac, ince h i coninuouly differeniable and X 1 D1,2 (R m ), i follow from he chain rule, ee for inance Nualar [25, Propoiion 1.2.4], ha h(x 1) D1,2 (R l ) and Dr(h(X j 1)) = xh(x 1)Dj rx 1 λ 2k 5 e k 1 for all r [,], j = 1,...,d. Uing log2 k 1, we deduce ha h(x 1) aifie (B2) wih A ij := 2λ 2 k 5. Similarly, by (A4) and uing ha he funcion x g(x, y, z) i coninuouly differeniable, i follow ha g. (X. 1,y,z) L 1,2 a (R l ) and D j (g ị (X 1,y,z)) λ2 k 3 e k 1, j = 1,...,d for all (y,z) R l R l d uch ha z M and, due o (A5), applying he ame argumen o ĝ (x,y,y,z,z ) := g (x,y,z) g (x,y,z ) yield D j rg (X 1,y,z) D j rg (X 1,y,z ) Kλ 2 e k 1. Uing k2 5 log2 k3 2 k 1, we deduce hag. (X. 1,y,z) aifie (B3) wihq ij = 2λ 2 k 3 andk u () := 2Kλ 2. Moreover due o (A4), he funcion (,y,z) g (X,y,z) 1 aifie (B1). log2 herefore, by 2k 4 +ρ 2 (M)+1, heorem 3.1 enure ha (Y 1,Z 1 ) exi. Conider he funcion g defined by { g (x,y,z) if z M g (x,y,z) = g (x,y,zm/ z ) if z > M. Since (Y 1,Z 1 ) alo olve he BSDE wih erminal condiion h(x 1 ) and a Lipchiz generaor g(x 1,, ), i follow from Lemma 3.2 and i proof ha (Y 1,Z1 ) D1,2 (R l ) D 1,2 (R l d ) for all [,] and D Y 1 i bounded and i hold Z 1 = D Y 1. In addiion, we have Dr X 1 4λ 2 and Dr Y 1 M. Now len N, aume ha(x n,y n,z n ) D 1,2 (R m ) D 1,2 (R l ) D 1,2 (R l d ),Z n = D Y n and D r X n 4λ 2, D r Y n M for all r, [,]. he proce X n+1 i well defined, for each ; X n+1 belong o D 1,2 (R m ) and i hold D r X n+1 =, < r, D r X n+1 = σ r + r ( x bd r Xu n+1 + y bd r Yu n )du, r. 11

12 Since x b, y b and σ are bounded by k 1, k 2 and λ 2 repecively, i follow from Gronwall inequaliy ha Dr X n+1 e k 1 λ 2 +k 2 D r Yu n du. Hence, Dr X n+1 e k 1 (λ 2 +k 2 M) < (4.1) o ha ince λ 2 k 2 M, we have D r X n+1 4λ 2. A above, h(x n+1 ) and g. (X n+1,y,z) are Malliavin differeniable and aify (B1)-(B3) wih A ij := 2λ 2 k 5, q ij = 2λ 2 k 3 and K u () := 2Kλ 2. I hen follow again from heorem 3.1 ha (Y n+1,z n+1 ) exi and Z n+1 M i bounded. Since (Y n+1,z n+1 ) alo olve he BSDE wih erminal condiion h(x n+1 ) and a Lipchiz generaor g(x n+1,, ), Lemma 3.2 and i proof guaranee ha (Y n+1,z n+1 ) D 1,2 (R l ) D 1,2 (R l d ) for all [,] and D Y n+1 i bounded and i hold Z n+1 = D Y 1, wih Dr Y n+1 M. Sep 2: Now we how ha here i a poiive conan 2 2 k,λ,l,d uch ha if 22 k,λ,l,d, hen (X n,y n,z n ) i a Cauchy equence in S 2 (R m ) S 2 (R l ) H 2 (R l d ). Uing (A1) we can eimae he norm of he difference X n+1 X n a hu X n 2 2 X n+1 up X n+1 X n 2 2 k 1 X n+1 k 1 X n+1 X d n 2 X d n k 2 Y n Y n 1 k 2 Y n d Y n 1 aking expecaion on boh ide and uing Cauchy-Schwarz inequaliy, we have E [ ] up X n+1 X n 2 2k1E 2 [ 2 2 k 2 1E X n+1 up X n+1 X n ] k2e 2 X n 2 d +2k2E 2 [ ] up Y n Y n 1 2. Chooing o be mall enough o ha 2 2 k , i follow [ [ E ] up X n+1 X n k2e 2 Y n 2. d 2 Y n 1. 2 d ] up Y n Y n 1 2. (4.2) 12

13 On he oher hand, applying Iô formula o e β Y n+1 Y n 2,β, we have e β Y n+1 Y n 2 = e β h(x n+1 +2 e β (Z n+1 e β (Y n+1 ) h(x) n 2 2 Z n ) 2 d e β (Y n+1 βe β (Y n+1 Y n ) [ g (X n+1,y n+1,z n+1 Y n )(Z n+1 Z)dW n Y n ) 2 d Hence, due o he condiion (A3) and he boundedne of (Z n ), i hold e β Y n+1 Y n 2 + e β (Z n+1 e β h(x n+1 ) h(x) n βe β (Y n+1 e β k 7 Y n+1 Z n ) 2 d Y n ) 2 d+2 e β (Y n+1 e β ρ(m) Y n+1 Y n X n+1 X n d+2 Y n )(Z n+1 Z)dW n ) g (X,Y n n,z) n ] d. Y n Z n+1 Z n d e β k 4 Y n+1 Y n 2 d. Wih ome poiive conan α 1, α 2, i follow from (A3) and Young inequaliy ha e β Y n+1 2 Y n 2 + e β (Y n+1 e β (Z n+1 Z n )2 d e β k 2 5 Xn+1 X n 2 Y n )(Zn+1 Z n )dw +α 2 ( ) (ρ(m)) k2 3 +2k 4 β α 1 α 2 e β (Y n+1 e β X n+1 Y n )2 d+α 1 X n 2 d e β Z n+1 Z n 2 d. (4.3) 13

14 Leing β = (ρ(m))2 α 1 + k2 7 α 2 +2k 8 and aking expecaion on boh ide above, we have E [ e β Y n+1 Y n 2] +E +α 1 E e β (Z n+1 e β Z n+1 Z) n 2 d e β k5e 2 [ X n+1 X n 2] Z n 2 d +α 2 E Puing α 1 = 1 2 and α 2 = 1, he previou eimae yield E e β (Z n+1 Z) n 2 d 2e β k5e 2 [ X n+1 X n 2] +2E Nex, aking condiional expecaion wih repec of in (4.3) give e β Y n+1 Y n 2 +E +α 1 E e β (Z n+1 e β Z n+1 e β X n+1 e β X n+1 X n 2 d. X n 2 d. Z) n 2 d F e β k5e 2 [ X n+1 X n 2 ] F Z n 2 d F +α 2 E e β X n+1 X n 2 d F. hu, by Burkholder-Davi-Gundy inequaliy, wih a poiive conan c 1 andα 1 = 1 2,α 2 = 1, we have [ ] E up e β Y n+1 Y n 2 c 1 e β k5e 2 [ X n+1 X n 2] 1 +c 1 2 E e β Z n+1 Z n 2 d +c 1 E e β X n+1 X n 2 d 2c 1 e β k 2 5E [ X n+1 I now follow from (4.2) ha E [ up Y n+1 Y n ]+E 2 X n 2] +2c 1 E (Z n+1 8(c 1 +1)e β (k 2 5 +)2 k 2 2 E [ e β X n+1 Z) n 2 d ] up Y n Y n 1 2. X n 2 d. 14

15 aking mall enough o ha 8(c 1 +1)e β (k 2 5 +)2 k , we obain ha (X n,y n,z n ) i a Cauchy equence in S 2 (R m ) S 2 (R l ) H 2 (R l d ). hu, i uffice o define 2 2 k,λ,d,l by he condiion { 2 2 k (c 1 +1)e β (k5 2 +)2 k By coninuiy ofb,g andhwe have he exience of a oluion(x,y,z) ins 2 (R m ) S 2 (R l ) H 2 (R l d ) of FBSDE (1.1) and i follow from he boundedne of (Z n ) ha Z M. he uniquene in S 2 (R m ) S 2 (R l ) S (R l d ) follow from he boundedne of Z and by repeaing he above argumen on he difference of wo oluion. Sep 3: If one of he funcion b, g or h i no differeniable, we apply he echnique of he proof of heorem 3.1. Namely, we ue an approximaion by he mooh funcion defined a follow: Forn N, le βn,β 1 n 2 andβn 3 be nonnegaive C funcion wih uppor on{x R m : x 1 n }, {x Rm+l : x 1 n } {x Rm+l+l d : x 1 n } repecively, and aifying R β 1 m n (r)dr = 1, R β 2 m+l n (r)dr = 1 and R β 3 m+l+l d n (r)dr = 1. We define he convoluion b n (x,y) := b (x,y )βn(x 2 x,y y)dx dy, h n (x) := h(x )βn(x 1 x)dx, R m+l R m g n (u,x,y,z) := g(u,x,y,z )βn(x 3 x,y y,z z)dx dy dz. R m+l+l d I i eay o check ha b n aifie (A1) wih he conan k 1,k 2 and 2λ 1 and ha g n and h n aify (A4) - (A5) and (A3), repecively, wih he ame conan. From Sep 1 and 2, here exi a poiive conan Ck,λ,l,d independen of n uch ha if C k,λ,l,d, FBSDE (1.1) wih parameer (b n,h n,g n ) admi a unique oluion (X n,y n,z n ) S 2 (R m ) S 2 (R l ) S (R l d ) and Z n M. By he Lipchiz coninuiy condiion on b and h and he locally Lipchiz condiion of g, he equence (b n ) and(h n ) converge uniformly obandhonr m+l andr m, repecively, and(g n ) converge o g uniformly on R m+l Λ for any compac ube Λ of R l d. Combining hee uniform convergence wih he boundedne of Z n, imilar o above, we can how ha here exi a conan Ck,λ,l,d depending only on k 1,k 2,k 3,k 4,k 5,λ 2,l,d uch ha if C k,λ,l,d, (X n,y n,z n ) i a Cauchy equence in he Banach pace S 2 (R m ) S 2 (R l ) H 2 (R l d ). In fac, for any m,n N, uing Cauchy-Schwarz inequaliy we have X n Xm 2 b n u (Xn u,y n u ) bm u (Xm u,y m u ) 2 du. 15

16 hu, aking he upremum wih repec o and hen expecaion on boh ide give X n X m 2 S 2 (R m ) 3 ( b n u (Xn u,y n u ) b u(x n u,y n u ) 2 + b m u (Xm u,y m u ) b u(x m u,y m u ) 2 + b u (Xu n,y u n ) b u(xu m,y u m ) 2 )du 3 ( b n u(x n u,y n u ) b u (X n u,y n u ) 2 + b m u (X m u,y m u ) b u (X m u,y m u ) 2 )du +3k X n X m 2 S 2 (R m ) +3k2 2 2 Y n Y m 2 S 2 (R l ) (4.4) where he econd inequaliy follow from (A1). On he oher hand, applying Iô formula a in Sep 2, one ha Y m Y n 2 + Z n u Z m u 2 du h n (X n ) h(xn ) 2 + h m (X m ) h(xm ) 2 +k 2 5 Xn Xm (Y n+1 Y n )(Z n+1 Z)dW n Y n u Y m u ( g n u(x n u,y n u,z n u) g u (X n u,y n u,z n u) + g m u (Xm u,y m u,zm u ) g u(x m u,y m u,zm u ) +k 3 X n u Xm u +k 4 Y n u Ym u +ρ(m) Zu n Zm u ). (4.5) aking expecaion, due o Young inequaliy we have Z n Z m 2 H 2 (R l d ) E[ h n (X n ) h(xn ) 2 ]+E[ h m (X m ) h(xm ) 2 ]+(k k2 3 ) Xn X m 2 S 2 (R m ) Y n Y m 2 S 2 (R l ) g n u (Xn u,y n u,zn u ) g u(x n u,y n u,zn u ) 2 + g m u (X m u,y m u,z m u ) g u (X m u,y m u,z m u ) 2 du k2 4ρ 2 (M) Y n Y m 2 S 2 (R l ) Zn Z m 2 H 2 (R l d ). On he oher hand, aking condiional expecaion in (4.5) and hen he upremum wih repec 16

17 o and hen expecaion on boh ide, we have due o Young inequaliy Y n Y m 2 S 2 (R l ) E[ h n (X n ) h(x n ) 2 ]+E[ h m (X m ) h(x m ) 2 ]+k 2 5 X n X m 2 S 2 (R m ) Y n Y m 2 S 2 (R l ) g n u(x n u,y n u,z n u) g u (X n u,y n u,z n u) 2 + gu m (Xm u,y u m,zm u ) g u(xu m,y u m,zm u ) 2 du 1 2 k2 3 Xn X m 2 S 2 (R m ) k2 4 ρ2 (M) Y n Y m 2 S 2 (R l ) Zn Z m 2 H 2 (R l d ). Combining (4.4) and (4.1) we oberve ha if i mall enough o ha { 3k k2 5 k k3 2k k2 4 ρ2 (M) 1 2 hen, he uniform convergence of (b n ), g n and (h n ) o b, g and h enure ha (X n,y n,z n ) i a Cauchy equence. he verificaion ha he limi (X,Y,Z) of he equence (X n,y n,z n ) olve he FBSDE (1.1) ue coninuiy of he funcionb,handg, and ha Z M i a conequence of he boundedne of(z n ). akingc k,λ,l,d := C k,λ,l,d C k,λ,l,d conclude he proof. Due o heorem 2.1 above, our global exience reul now follow from a paing procedure. 4.2 Proof of heorem 2.2 If C k,λ,l,d, hen he reul follow from heorem 2.1. Aume > C k,λ,l,d and le h M : R R be a coninuouly differeniable funcion whoe derivaive i bounded by 1 and uch ha h M (a) = 1 for all M a M and (M +1) if a > M +2 h M (a) = a if a M (M +1) if a < (M +2). An example of uch a funcion i given by {( M h 2 +2Ma a(a 4) ) /4 if a [M,M +2] M (a) = ( M 2 +2Ma+a(a+4) ) /4 if [ (M +2), M], ee Imkeller and Rei [18]. By he aumpion (A3) he funcion g : [,] R m R l R l d R defined by g (x,y,z) := g (x,y,h M (z)) (4.6) wih h M (z) := ( h M (z ij )) ij i Lipchiz coninuou in all variable. hu, i follow from Delarue [7, heorem 2.6] ha he equaion { X = x+ b u( X u,ỹu)du+ σ udw u Ỹ = h( X )+ g u ( X u,ỹu, Z u )du (4.7) Z u dw u, [,] 17

18 admi a unique oluion ( X,Ỹ, Z) S 2 (R m ) S (R l ) S (R l d ). Moreover, here exi a Lipchiz coninuou funcion θ : [,] R m R l bounded by a conan K uch ha Ỹ = θ(, X ) for all [,]. In fac, for everyx,x R d, [,] and,...,l we have θ(, X x x ) θ(, X ) = h i ( X x ) hi x ( X )+ gu i ( X u x,ỹ u x, Z u x ) gi u = h i ( X x ) hi ( x X )+ ( X x gu( i X u,ỹ x u x, Z u) g x u( i Z u x,i Z x i u u x x,ỹu, Z u )du x X u,ỹ u x x, Z u ) Z x,i u Z x,i u dw u ( Z x,i u Z x i u )1 { Z x,i u Z x,i } du + (g i u ( X x u,ỹ x u, Z x ) g i u ( X x u x x,ỹu, Z u ))1 { Z u x,i Z x i =}du u Z x,i u Z x,i u dw u. hu, Giranov heorem yield θ i (, X x ) θ i x (, X ) E Qi h i ( X x ) hi x + ( X ) gu i ( X u x,ỹ u x, Z u x ) gi x x x u ( X u,ỹu, Z u ) 1{ Zx,i u Z x i u =} du F E Qi k 5 X x x X + where Q i i he probabiliy meaure given by ( dq i dp = E gu i( X u x,ỹ u x, Z u x) gi u ( k 3 X ) u x x X u +k 4 Ỹ u x Ỹx u ( X x u Z u x,i Z x,i u x x,ỹu, Z u ) du F 1 W { Zx,i u Z x,i u } By (A4 ) and boundedne ofỹx andỹ x i well defined. Since by Gronwall lemma we have X x i hold θ i (, X x ) θ i (, X x ( X x x X ) E Qi e k1 (k 5 +k 3 ) X x X x +k 2 ) Ỹ x u Ỹx u du)ek 1, [,], x X +(k 2k 3 e k1 +k 4 +k 2 k 5 e k1 ). θ(u, X u x x ) θ(u, X u ) du F. 18

19 Hence, θ i (, X x) θi (, u = e k 1 (k 5 +k 3 )l X x which i given by u = e k 1 (k 5 +k 3 )l X x X x ) u where u i he oluion of he ODE x X +(k 2k 3 e k1 +k 4 +k 2 k 5 e k1 ) lu du ( ) x X exp (k 2 k 3 e k1 +k 4 +k 2 k 5 e k1 )l( ). hu, θ(, X x ) θ(, X x ) K5 X x X x, wihk 5 := le k1 (k 5 +k 3 )lexp ( (k 2 k 3 e k1 +k 4 +k 2 k 5 e k1 )l ) which how ha θ i a Lipchiz funcion and he Lipchiz coefficien doe no depend on he bound M of Z. Le C k,λ,l,d be he conan C k,λ,l,d wih k 5 replaced by K 5 and pu N = / C k,λ,l,d, where a denoe he ineger par of a, and i := i C k,λ,l,d, i =,...,N and N+1 =. Since 1 C k,λ,l,d, by heorem 2.1 he FBSDE { X = x+ b u(x u,y u )du+ σ udw u Y = θ( 1,X 1 )+ 1 g u (X u,y u,z u )du 1 Z u dw u, [, 1 ] admi a unique oluion (X 1,Y 1,Z 1 ) uch ha Z 1 M wih M = 4λ 2 K 5 dl for all [, 1 ]. herefore, (X 1,Y 1,Z 1 )1 [,1 ] = ( X,Ỹ, Z)1 [,1 ]. Similarly, we obain a family (X i,y i,z i ) of oluion of he FBSDE { X = X + i 1 i 1 b u (X u,y u )du+ i 1 σ u dw u Y = θ( i,x i )+ i g u (X u,y u,z u )du i Z u dw u, [ i 1, i ] uch ha (X i,y i,z i )1 [i 1, i ] = ( X,Ỹ, Z)1 [i 1, i ], i = 1,...,N +1. Define X := N+1 X i 1 [i 1, i ]; Y := N+1 Y i 1 [i 1, i ] and Z := N+1 Z i 1 [i 1, i ]. hen, (X,Y,Z) S 2 (R m ) S (R l ) S (R l d ) i he unique oluion of he FBSDE (1.1) aifying Z M for all [,]. In fac, i i clear ha (X,Y,Z) S 2 (R m ) S (R l ) S (R l d ) a a finie um of elemen of he ame pace. Le [,] and i = 1,...,N +1 uch ha [ i 1, i ]. We have x+ b u (X u )du+ σ u du = x+ i j j b u (Xu j )du+ σ u dw u j 1 j 1 j=1 = X i = X 19

20 and h(x )+ g u (X u,y u,z u )du = h(x N+1 )+ N+1 j=i j j 1 Z u dw u g u (X j u,y j u,z j u)du j j 1 ZudW j u = Y i = Y. ha i, (X, Y, Z) aifie Equaion (1.1). 4.3 Proof of Propoiion 2.4 By heorem 2.2 he FBSDE { X = x+ b ( X,Ỹ)d+ σ dw Ỹ = h( X )+ g ( X,Ỹ, Z )d Z dw, [,] ha a unique global oluion ( X,Ỹ, Z) S 2 (R m ) S 2 (R l ) S (R l d ) uch ha Z M for ome conan M. Le X = X,Y = Γ 1 Ỹ,Z = Γ 1 Z, we obain ha (X,Y,Z) S 2 (R m ) S 2 (R l ) S (R l d ) uch ha Z M for ome conan M. Moreover, (X, Y, Z) aifie he FBSDE (1.1). hi complee he proof. Reference [1] F. Anonelli and S. Hamadène. Exience of oluion of backward-forward SDE wih coninuou monoone coefficien. Sai. Probab. Le., 76(14): , 26. [2] P. Barrieu and N. El Karoui. Monoone abiliy of quadraic emimaringale wih applicaion o unbounded general quadraic BSDE. Ann. Probab., 41(3B): , [3] P. Briand and Y. Hu. BSDE wih Quadraic Growh and Unbounded erminal Value. Probab. heory Rela. Field, 136:64 618, 26. [4] P. Briand and Y. Hu. Quadraic BSDE wih Convex Generaor and Unbounded erminal Condiion. Probab. heory Rela. Field, 141: , 28. [5] P. Cheridio and K. Nam. BSDE wih erminal condiion ha have bounded Malliavin derivaive. J. Func. Anal., 266(3): , 214. [6] P. Cheridio and K. Nam. Mulidimenional quadraic and ubquadraic bde wih pecial rucure. Sochaic, 87(5): , 215. [7] F. Delarue. On he exience and uniquene of oluion o FBSDE in a non-degenerae cae. Soch. Proc. Appl., 99:29 286, 22. [8] F. Delbaen, Y. Hu, and X. Bao. Backward SDE wih Superquadraic Growh. Probab. heory Rela. Field, 15: , 211. ISSN [9] L. Delong and P. Imkeller. Backward Sochaic Differenial Equaion wih ime Delayed Generaor - Reul and Counerexample. Ann. Appl. Probab., 2: , 21. [1] S. Drapeau, G. Heyne, and M. Kupper. Minimal Superoluion of Convex BSDE. Annal of Probabiliy, 41(6): ,

21 [11] N. El Karoui, S. Peng, and M. C. Quenez. Backward ochaic differenial equaion in finance. Mah. Finance, 1(1):1 71, January [12] C. Frei. Spliing mulidimenional BSDE and finding local equilibria. Soch. Proc. Appl., 124(8): , 214. [13] A. Fromm and P. Imkeller. Exience, uniquene and regulariy of decoupling field o mulidimenional fully coupled FBSDE. Preprin, 213. [14] A. Fromm, P. Imkeller, and D. J. Prömel. An FBSDE approach of he Skorokhod embedding problem for Gauian procee wih non-linear drif. Elec. J. Probab., 2(127):1 38, 215. [15] G. Heyne, M. Kupper, and L. angpi. Porfolio opimizaion under nonlinear uiliy. In. J. heor. Appl. Finance, 19(5), 216. [16] U. Hor, Y. Hu, P. Imkeller, A. Réveillac, and J. Zhang. Forward Backward Syem for Expeced Uiliy Maximizaion. Soch. Proc. Appl., 124(5): , 214. [17] Y. Hu and S. ang. Muli-dimenional backward ochaic differenial equaion of diagonally quadraic generaor. Soch. Proc. Appl., 126(4): , 216. ISSN [18] P. Imkeller and G. D. Rei. Pah regulariy and explici convergence rae for BSDE wih runcaed quadraic Growh. Soch. Proc. Appl., 12: , 21. [19] A. Jamnehan, M. Kupper, and P. Luo. Solvabiliy of mulidimenional quadraic BSDE. Preprin, 216. [2] M. Kobylanki. Backward ochaic differenial equaion and parial differenial equaion wih quadraic growh. Ann. Probab, 28(2):558 62, 2. [21] P. Luo. Eay on Mulidimenional BSDE and FBSDE. PhD hei, Univeriy of Konanz, 215. [22] P. Luo and L. angpi. Solvabiliy of coupled FBSDE wih diagonally quadraic generaor. Forhcoming in Sochaic and Dynamic, 215. [23] J. Ma, P. Proer, and J. Yong. Solving Forward-Backward Sochaic Differenial Equaion Explicily - A Four Sep Scheme. Probab. heory Rela. Field, 98: , [24] K. Nam. Backward Sochaic Differenial Equaion wih Superlinear Driver. PhD hei, Princeon Univeriy, 214. [25] D. Nualar. he Malliavin Calculu and Relaed opic. Probabiliy and i Applicaion (New York). Springer-Verlag, Berlin, econd ediion, 26. ISBN ; [26] R. evzadze. Solvabiliy of backward ochaic differenial equaion wih quadraic growh. Soch. Proc. Appl., 118:53 515, 28. [27] H. Xing and G. Žikovi`c. A cla of globally olvable markovian quadraic bde yem and applicaion. Preprin, 216. [28] J. Yong and X. Y. Zhou. Sochaic conrol, Hamilonian Syem and HJB equaion, volume 43. Springer-Verlag, New York,