Multidimensional Markovian FBSDEs with superquadratic

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1 Mulidimenional Markovian FBSDE wih uperquadraic growh Michael Kupper a,1, Peng Luo b,, Ludovic angpi c,3 December 14, 17 ABSRAC We give local and global exience and uniquene reul for mulidimenional coupled FBSDE for generaor wih arbirary growh in he conrol variable. he local exience reul i baed on Malliavin calculu argumen for Markovian equaion. Under addiional monooniciy condiion on he generaor we conruc global oluion by a paing echnique along PDE oluion. AUHORS INFO a Univeriy of Konanz, Univeriäraße. 1, D Konanz, Germany b EH Zürich, Rämirae 11, 89 Zürich, Swizerland c Univeriy of Vienna, Faculy of Mahemaic, Okar- Morgenern-Plaz 1, A-19 Wien, Auria 1 kupper@uni-konanz.de peng.luo@mah.ehz.ch 3 ludovic.angpi@univie.ac.a PAPER INFO AMS CLASSIFICAION: 6H1, 6H7, 6H3 1 Inroducion Given a mulidimenional Brownian moion W on a probabiliy pace, we conider he yem of forward and backward ochaic differenial equaion { X Y = x + b X, Y d + σ dw = hx + g X, Y, Z d Z dw,, ] where x i he iniial value, > i a finie ime horizon, 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. Our focu 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 in Z. For one-dimenional value procee he decoupled yem wih b depending only on X ha been olved by Cheridio and Nam 6] 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 how ha he conrol proce can be uniformly bounded, hence enabling olvabiliy for locally Lipchiz generaor by a runcaion argumen. o olve 1.1 in he mulidimenional cae we propoe a Picard ieraion cheme which yield a Cauchy equence in an appropriae Banach pace under uniform boundedne of he conrol procee. Uing Malliavin calculu argumen he boundedne i guaraneed if he ime horizon i mall enough. Here we make ample ue of he mehod in Cheridio and Nam 6]. Moreover uing he PDE repreenaion of Markovian Lipchiz FBSDE a developed for inance in Delarue 8] and a paing procedure, we conruc a unique global oluion for generaor wih an addiional monooniciy-ype condiion and non-degeneracy of he volailiy σ, ee heorem

2 Syem uch a 1.1 naurally appear in numerou area of applied mahemaic including ochaic conrol and mahemaical finance, ee e.g. Yong and Zhou 3], El Karoui e al. 1], Hor e al. 17], Kramkov and Pulido ] and Bielagk e al. 3]. A hown for inance in Ma e al. 5] and Cheridio and Nam 6], in he Markovian cae, FBSDE can be linked o parabolic PDE. More recenly i i hown in Fromm e al. 15] 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. 1] and Delarue 8]. If Y i one-dimenional and g ha quadraic growh in he conrol proce Z, BSDE oluion have been obained by Kobylanki 1], Barrieu and El Karoui ] and Briand and Hu 4, 5] under differen aumpion on he erminal condiion ξ = hx. We furher refer o Delbaen e al. 9], Drapeau e al. 11], Cheridio and Nam 6] and Heyne e al. 16] 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. Under mallne of he erminal condiion olvabiliy i hown in evzadze 8], ee alo Hu and ang 18], Luo and angpi 4], Jamnehan e al. ], Cheridio and Nam 7], Frei 13] and Xing and Žiković 9] for more recen developmen. o he be of our knowledge, Anonelli and Hamadène 1], Luo and angpi 4] and Fromm and Imkeller 14] are he only work udying well-poedne of coupled FBSDE wih quadraic growh. In Anonelli and Hamadène 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. hi approach canno be ranfered o he preen mulidimenional cae ince comparion reul are no available. Fromm and Imkeller 14] conider fully coupled Markovian FB- SDE wih mulidimenional forward and value procee and locally Lipchiz coninuou generaor in Y, Z. Uing he echnique of decoupling field hey obain exience of a unique local oluion and provide an exenion o a maximal ime inerval. Compared o Fromm and Imkeller 14], we ue an eenially differen echnique baed on Malliavin calculu which guaranee he exience of a uniformly Lipchiz decoupling field and Malliavin differeniabiliy of oluion. Moreover, we alo conruc a global oluion. Alhough he non-markovian yem udied in Luo and angpi 4] i he ame a he one conidered here, he echnique are eenially differen. In paricular, he growh condiion in he preen paper are weaker and we do no impoe any diagonally quadraic condiion. Our main reul 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 of g and h, for deail we refer o he Ph.D. hei of Luo 3]. he paper i organized a follow. In he nex ecion, we preen he eing and main reul. 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. 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 of R k and R k k, k N,

3 are alway endowed wih he Borel σ-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, we denoe by S p R k he pace of all predicable coninuou procee X wih value in R k uch ha X S p R k := up, ] X p <, and by H p R k he pace of all predicable procee Z wih value in R k uch ha Z H p R k := Z u du 1/ p <. Here, p denoe he L p -norm. Le l, m N be fixed. A oluion of 1.1 wih value in R m R l R l d can be obained under he following condiion: A1 b :, ] R m R l R m i a meaurable funcion, b, i coninuou for each, ], and here exi k 1, k, λ 1 uch ha b x, y b x, y k 1 x x + k y y and b x, y λ x + y for all x, x R m and y, y R l. A σ :, ] R m d i a meaurable funcion and here i λ uch ha σ λ for all, ]. A3 h : R m R l i a coninuou funcion and here exi k 5 uch ha hx hx k 5 x x for all x, x R m. A4 g :, ] R m R l R l d R l i a meaurable funcion and g,, d < +. Moreover, he funcion g,, i coninuou for each, ] and here exi k 3, k 4 and a nondecreaing funcion ρ : R + R + uch ha g x, y, z g x, y, z k 3 x x.1 for all x, x R m, y R l and z R l d uch ha z M := 8λ k 5 dl and g x, y, z g x, y, z k 4 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. Our main reul enure local exience and uniquene for he coupled FBSDE 1.1 under he previou aumpion. he proof i given in Secion 4. 3

4 heorem.1. Aume ha A1-A5 hold. hen here exi a conan C > depending on k 1, k, k 3, k 4, k 5, λ, l and d, uch ha he FBSDE 1.1 ha a unique oluion X, Y, Z S R m S R l S R l d wih Z M, whenever C. Local exience reul have been obained in 14, heorem 3] and 4, heorem.1] in eenially differen eing and wih differen mehod. Le u menion ha our echnique allow o obain exience of oluion of coupled FBSDE wih Burger ype nonlineariie a lea for mall enough ime horizon. Example.. Aume ha i mall enough, b, σ and h aify A1-A3, wih h λ 5 for ome λ 5. hen for each k 1 he FBSDE { X Y = x + b X, Y d + σ dw = hx + Y Z k d Z dw,, ] admi a oluion X, Y, Z S R m S R l S R l d. he deail are given in Subecion 4.. Remark.3. he condiion A5 i he minimal condiion needed o enure Lipchiz coninuiy in y, z of he Malliavin derivaive of g X, y, z for a given SDE oluion X, ee e.g. El Karoui e al. 1] and Cheridio and Nam 6] for deail. When he generaor g i of he form g x, y, z := f 1 x + f y + f 3 z for ome funcion f 1, f and f 3, hen A5 i aified. Moreover, le u menion ha an advanage of our mehod i ha i implie Malliavin differeniabiliy of he forward proce X and he value proce Y in he oluion X, Y, Z of he FBSDE 1.1 obained in heorem.1. We refer o Subecion 4.3 for deail. Remark.4. he following counerexample how ha in general, even in he one-dimenional cae, coupled yem do no have a unique global oluion. Conider he FBSDE { X = hi equaion can be rewrien a Y = Y = Y u du kx u du Z u dw u. ky u du d. Z u dw u..3 I i hown in 1, Example 3.] ha if k < π hen he BSDE wih ime-delayed generaor.3 ha a unique oluion wherea if k = π, he equaion.3 may no have any oluion and if i ha one, here are infiniely many. Nex, we would like o find condiion under which heorem.1 can be exended o obain global olvabiliy. In he preen eing, under addiional aumpion, a paing mehod baed on PDE allow o ge global exience and uniquene for he FBSDE

5 A6 here exi K 4 aifying K 4 e k 1 k k 5 + k 3 + k 3 + ρ M dl wih M = 8λ K 5 dl and K5 = k 5 + k 3 e k 1 uch ha y i y i gx, i y, z gx, i y, z K 4 y y for all x R m, y, y R l and z R l d. A7 here exi K 1, K 4 aifying K 1 k 5 K 4 ρ M dl k k 5 + k 3 uch ha m x i x i b i x, y b i x, y K 1 x x, y i y i gx, i y, z gx, i y, z K 4 y y for all x, x R m, y, y R l and z R l d. heorem.5. Aume ha A1-A5 hold and here exi λ 3, λ 4, λ 5 > uch ha 1 b x, y λ y x, σ σ x λ 3 x g x, y, z λ y + ρ z z hx λ 5.4 for all, ], x, x R m, y, y R l and z, z R l d. hen, if A6 repecively A7 i aified, he FBSDE 1.1 ha a unique global oluion X, Y, Z S R m S R l S R l d uch ha Z M repecively Z M. Remark.6. Aumpion A6 and A7 can be underood a monooniciy condiion on he generaor and he drif coefficien. hee condiion are aified for inance when he componen of g rep. b are linear in y rep. x. In fac, if here i a funcion f wih value in R l uch ha g x, y, z := K 4 y + f x, z, hen g aifie A6, and uiable condiion on f guaranee ha g aifie A4 and A5 a well. Noice ha heorem.5 yield exience of a decoupling field, ee 14]. In paricular, he boundedne of Z yield uniform Lipchiz coninuiy of he decoupling field. heorem.1 relie on an exience reul for mulidimenional BSDE preened in Nam 6] and reviied in he nex ecion. 1 σ i he ranpoe marix of σ. 5

6 3 Mulidimenional BSDE wih bounded Malliavin derivaive Le u inroduce he pace of Malliavin differeniable random variable and ochaic procee D 1, R l and La 1, R l. For a horough reamen of he heory of Malliavin calculu we refer o Nualar 7]. 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 R n ; R of infiniely coninuouly differeniable funcion whoe parial derivaive have polynomial growh, h i1,..., h in L, ]; R d and n 1. For every ξ in M le he operaor D = D 1,..., D d : M L 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, := E ξ + D ξ d] 1/. A hown in Nualar 7], he operaor D exend o he cloure D 1, R l of he e M wih repec o he norm 1,. A random variable ξ i Malliavin differeniable if ξ D 1, R l and we denoe by D ξ i Malliavin derivaive. Denoe by La 1, R l he pace of procee Y H R l uch ha Y D 1, R l for all, ], he proce DY admi a quare inegrable progreively meaurable verion and Y L 1, a R l := Y H R l + E We nex conider a yem of uperquadraic BSDE of he form aifying he following condiion: Y = ξ + g u Y u, Z u du ] D r Y dr d <. 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. B ξ D 1, R l and here exi conan A ij uch ha D j ξi A ij for all i = 1,..., l, j = 1,..., d and, ]. 6

7 B3 g, H 4 R l and here exi Borel-meaurable funcion q ij :, ] R + aifying q ij d < uch ha for every pair y, z Rl R l d wih d z Q := A ij + qij d one ha j=1 g y, z L 1, a R l wih Dug j y, i 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 and z, z R l d for ome R + -valued adaped proce K u, ] aifying K u 4 H 4 R du <. he following i an exenion of Cheridio and Nam 6, heorem.] o he mulidimenional cae. I wa proved in Nam 6] under lighly differen aumpion. For inance, we do no aume a monooniciy-ype condiion on y g y, z for every z. Our reul rely on he echnique of 6]. For he ake of compleene we give he proof. heorem 3.1. Aume ha B1-B3 hold and oluion in S 4 R l S R l d and Z Q. Conider he following ronger verion of he condiion B1 and B3: log. hen he BSDE 3.1 admi a unique B+ρ Q+1 B1 g i coninuouly differeniable in y, z and here exi conan B R + and ρ 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. B3 he condiion B3 hold for all y, z R l R l d. Lemma 3.. If B1, B 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 A ij + q ije B+ρ +1 d e B+ρ +1 for all j = 1,..., d. 3. Proof. By B, each componen ξ i of ξ ha bounded Malliavin derivaive, which implie by 6, Lemma.5] ha E ξ i p ] < for all p 1. I follow from El Karoui e al. 1, 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, DrZ j i, ] ha a verion U ij,r, V ij,r, ] which aifie U ij,r =, V ij,r =, for < r, 7

8 and i he unique oluion in S R l H R l d of he BSDE U j,r = Drξ j + y g Y, Z U j,r Applying Iô formula o U j,r yield U j,r = Drξ j + D j rξ D j rξ U j,r V j,r dw U j,r y g Y, Z U j,r U j,r V j,r dw + U j,r V j,r dw + + z g Y, Z V j,r + U j,r z g Y, Z V j,r B U j,r + Drg j Y, Z d + U j,r + ρ U j,r V j,r B + ρ + 1 U j,r + V j,r dw. Drg j Y, Z V j,r d + l qijd. Uing condiion B3 and aking condiional expecaion in he above inequaliy yield U j,r E A ij + B + ρ + 1 U j,r + q ij U j,r V j,r d qijd F ]. 3.3 By El Karoui e al. 1, Propoiion 5.3] he proce Z i a verion of he race U, ] of he Malliavin derivaive of Y. Hence 3. follow from 3.3 by applying Gronwall inequaliy. 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. 3.4 By Cheridio and Nam 6, Lemma.5], he condiion B implie E ξ p ] < + for all p 1,. hu, ξ L p for all p 1. herefore, i follow from El Karoui e al. 1, heorem 5.1] ha he BSDE correponding o g, ξ ha a unique oluion Y, Z S 4 R l H 4 R l d. For x = y, z R l+l d le β C R l+l d be he mollifier { λ exp 1 if x < 1, 1 x βx := oherwie, 8

9 where he conan λ R + i choen uch ha R βxdx = 1. Se β n x := n l+l d βnx, 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 ρq. By Lemma 3. 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 A ij + A ij + log we obain B+ρ Q+1 Z n,j A ij + qije B+ρ Q+1 d e B+ρ Q+1 qijd e B+ρQ+1. qijd for all j = 1,..., d. hi how Z n Q. Since g n converge uniformly in, ω, y, z o g, uing he procedure of he proof of Cheridio and Nam 6, heorem.], i follow ha Y n, Z n converge o Y, Z in S R l H R l d, o ha Z Q. Since gy, z = gy, 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 Corollary 3.3. Suppoe B1-B3 hold, B+ρ Q+1 and Y, Z S4 R l S R l d i he oluion of he BSDE 3.1. hen Y D 1, R l for all, ] and for every j = 1,..., d, one ha DrY j A ij + q ijd for all r, ]. 3.5 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.. 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 BSDE correponding o g n, ξ, which converge o Y, Z in S R l H R l d. By Lemma 3. Y n, Z n D 1, R l D 1, R l d for each, ] and he argumen in he proof of heorem 3.1 imply DrY j n A ij + qijd j = 1,..., d, r,, ]. 9

10 Hence, up n N E Dj ry n dr] < for each, ]. Since Y n converge o Y in L, i follow from Nualar 7, Lemma 1..3] ha Y D 1, R l and D r Y n converge o D r Y in he weak opology of H R l d. hu, D r Y aifie 3.5. A a conequence 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 B + ρ n Q + 1, n N. Propoiion 3.4. Aume ha B1-B hold, ha here exi N N uch ha N n= n, and B3 hold wih Q replaced by N Q. hen he BSDE 3.1 ha a unique oluion in S 4 R l S R l d and Z 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, R l and for every r, D j ry A ij + q ij 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 + Z 1 Q, 1, ]. + q ij d, for all j = 1,..., d Repeaing he previou argumen, for N he BSDE 3.1 ha a unique oluion Y N, Z N in S 4 R l S R l d on N n= n, N 1 n= n ] wih erminal condiion Y N 1 Moreover, D j ry N N n= n Z N N Q, N A ij + N n= N k q ij d for all j = 1,..., d k=1 N 1 n, n= ] n. n= n. 1

11 Hence, he pair Y, Z given by 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 ρx C1 + x for all x, hen n n <. However, i doe guaranee global olvabiliy for BSDE whoe generaor grow lighly faer han he linear funcion. For inance, if ρx C1 + log1 + x one ha n= log B + ρ N Q + 1 log B + C 1 + log N 1 + Q + 1 n= log = B + C 1 + log1 + Q + n log + 1 =. n= Noe alo ha global olvabiliy of ric ubquadraic yem ha been eablihed by Cheridio and Nam 7]. 4 Coupled FBSDE wih uperquadraic growh 4.1 Proof of heorem.1 Sep 1: We fir aume ha h, b and g are coninuouly differeniable in all variable. Le u define C 1 := k 5 k3 log λ k 1 k M log k 4 + ρ M + 1 wih M := 8k 5 λ dl. We will how ha for C1, he equence X n, Y n, Z n given by X =, Y =, Z = and { X n+1 Y n+1 = x + = hx n+1 bxn+1 u, Yu n du + σ u dw u u, Zu n+1 + g u X n+1 u, Y n+1 du Z n+1 u dw u, n 1 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, 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, 11

12 wih D r r σ u dw u = σ1 r,], ee Nualar 7, Lemma..1 and heorem..1]. Hence, ince b i Lipchiz coninuou, we have Dr X 1 r k 1 D r X 1 u du + σ r and Dr X 1 λ e k 1, where he econd eimae come from Gronwall inequaliy. We will now how ha ince C 1, hx 1 and gx1,, aify B1-B3. In fac, ince h i coninuouly differeniable and X 1 D 1, R m, i follow from he chain rule, ee for inance Nualar 7, Propoiion 1..4], ha hx 1 D 1, R l and DrhX j 1 = xhx 1 Dj rx 1 λ k 5 e k 1 for all r, ], j = 1,..., d. Uing log k 1, we deduce ha hx 1 aifie B wih A ij := λ k 5. Similarly, by A4 and uing ha he funcion x gx, y, z i coninuouly differeniable, i follow ha g. X. 1, y, z La 1, R l and D j g ị X 1, y, z λ 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 e k 1. Uing k 5 log k3 k 1, we deduce ha g. X. 1, y, z aifie B3 wih q ij = λ k 3 and K u := Kλ. Moreover due o A4, he funcion, y, z g X 1, y, z aifie B1. log herefore, by k 4 +ρ 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 hx 1 and a Lipchiz generaor gx1,,, i follow from Lemma 3. and i proof ha Y 1, Z 1 D 1, R l D 1, R l d for all, ] and D Y 1 i bounded and i hold Z 1 = D Y 1. In addiion, we have D r X 1 4λ and D r Y 1 M. Now le n N, aume ha X n, Y n, Z n D 1, R m D 1, R l D 1, R l d, Z n = D Y n and D r X n 4λ, 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, R m and i hold D r X n+1 =, < r, D r X n+1 = σ r + r x bd r X n+1 u + y bd r Y n u du, r. Since x b, y b and σ are bounded by k 1, k and λ repecively, i follow from Gronwall inequaliy ha D r X n+1 e k 1 λ + k D r Yu n du. Hence, Dr X n+1 e k 1 λ + k M < 4.1 1

13 o ha ince λ k M, we have D r X n+1 4λ. A above, hx n+1 and g. X n+1, y, z are Malliavin differeniable and aify B1-B3 wih A ij := 4λ k 5, q ij = λ k 3 and K u := Kλ. 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 hx n+1 and Lipchiz coninuou generaor g X n+1,,, Lemma 3. and i proof guaranee ha Y n+1, Z n+1 D 1, R l D 1, R l d for all, ] and D Y n+1 i bounded and i hold Z n+1 = D Y n+1, wih D r Y n+1 M. Sep : Now we how ha here i a poiive conan C depending on k 1, k, k 3, k 4, k 5, λ, l and d, uch ha if C, hen X n, Y n, Z n i a Cauchy equence in S R m S R l H R l d. Uing A1 we can eimae he norm of he difference X n+1 X n a hu X n X n+1 up X n+1 X n k 1 X n+1 k 1 X n+1 X n d X n d + + k Y n k Y n Y n 1 aking expecaion on boh ide and uing Cauchy-Schwarz inequaliy, we have E ] up X n+1 X n k1e k 1E X n+1 ] up X n+1 X n + ke Chooing o be mall enough o ha k1 1, i follow E ] up X n+1 X n 4 ke On he oher hand, applying Iô formula o e β Y n+1 e β Y n+1 Y n = e β hx n+1 hx n + e β Z n+1 e β Y n+1 X n d + ke Z n d ] up Y n Y n 1. d Y n 1 Y n. d Y n 1. d ] up Y n Y n Y n, β, we have 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 n dw Y n d g X n, Y n, Z n ] d. 13

14 Hence, due o he condiion A3 and he boundedne of Z n, i hold e β Y n+1 Y n + e β Z n+1 e β hx n+1 hx n + βe β Y n+1 e β k 7 Y n+1 Z n d Y n d + e β Y n+1 Y n X n+1 X n e β ρm Y n+1 Y n Z n+1 Z n dw d + Y n Z n+1 Z n e β k 4 Y n+1 Wih ome poiive conan α 1, α, i follow from A3 and Young inequaliy ha e β Y n+1 Y n + e β Y n+1 e β Z n+1 Z n d e β k 5 X n+1 X n Y n Z n+1 Z n dw + α ρm + + k 3 + k 4 β α 1 α e β Y n+1 e β X n+1 Y n d + α 1 X n d Leing β = ρm α 1 + k 3 α + k 4 and aking expecaion on boh ide above, we have E e β Y n+1 Y n ] + E e β Z n+1 + α 1 E d Y n d. Z n d e β k5e X n+1 X n ] e β Z n+1 Puing α 1 = 1 and α = 1, he previou eimae yield E e β Z n+1 Z n d + α E Z n d e β k5e X n+1 X n ] + E e β Z n+1 Z n d. 4.3 e β X n+1 e β X n+1 X n d. X n d. 14

15 Nex, aking condiional expecaion wih repec o F in 4.3 give e β Y n+1 Y n + E + α 1 E e β Z n+1 Z n d F e β k5e X n+1 X n ] F e β Z n+1 Z n d F + α E e β X n+1 X n d F. hu, by Burkholder-Davi-Gundy inequaliy, wih a poiive conan c 1 and α 1 = 1, α = 1, we have ] E up e β Y n+1 Y n c 1 e β k5e X n+1 X n ] 1 + c 1 E e β Z n+1 Z n d + c 1 E e β X n+1 X n d I now follow from 4. ha E aking mall enough o ha c 1 e β k 5E X n+1 ] up Y n+1 Y n + E X n ] + c 1 E 8c 1 + 1e β k 5 + k E Z n+1 8c 1 + 1e β k 5 + k 1, e β X n+1 Z n d ] up Y n Y n 1. X n d. we obain ha X n, Y n, Z n i a Cauchy equence in S R m S R l H R l d. hu, i uffice o define C by he condiion { k 1 1 8c 1 + 1e β k 5 + k 1. By coninuiy of b, g and h we have he exience of a oluion X, Y, Z in S R m S R l H R l d of FBSDE 1.1 and i follow from he boundedne of Z n ha Z M. he uniquene in S R m S R l S R l d follow from he boundedne of Z and by repeaing he above argumen on he difference of wo oluion. 15

16 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: For n N, le β 1 n, β n and β 3 n be nonnegaive C funcion wih uppor on {x R m : x 1 n }, {x Rm+l : x 1 n } and {x Rm+l+l d : x 1 n } repecively, and aifying R m β 1 nrdr = 1, R m+l β nrdr = 1 and R m+l+l d β 3 nrdr = 1. We define he convoluion b n x, y := b x, y βnx x, y ydx dy, h n x := hx βnx 1 xdx, R m+l R m g n u, x, y, z := gu, x, y, z βnx 3 x, y y, z zdx dy dz. R m+l+l d I i eay o check ha b n aifie A1 wih he conan k 1, k and λ 1 and ha g n and h n aify A4 - A5 and A3, repecively, wih he ame conan. From Sep 1 and, here exi a poiive conan C independen of n uch ha if C, FBSDE 1.1 wih parameer b n, h n, g n admi a unique oluion X n, Y n, Z n S R m S 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 o b and h on R m+l and R 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, here exi a conan C depending on k 1, k, k 3, k 4, k 5, λ, l and d uch ha if C, X n, Y n, Z n i a Cauchy equence in he Banach pace S R m S R l H R l d. In fac, for any m, n N, uing Cauchy-Schwarz inequaliy we have X n X m b n ux n u, Y n u b m u X m u, Y m u du. hu, aking he upremum wih repec o and hen expecaion on boh ide give X n X m S R m 3 b n uxu n, Yu n b u Xu n, Yu n + b m u Xu m, Yu m b u Xu m, Yu m + b u X n u, Y n u b u X m u, Y m u du 3 b n uxu n, Yu n b u Xu n, Yu n + b m u Xu m, Yu m b u Xu m, Yu m du + 3k 1 X n X m S R m + 3k Y n Y m S R l

17 where he econd inequaliy follow from A1. On he oher hand, applying Iô formula a in Sep, one ha Y m Y n + Z n u Z m u du h n X n hx n + h m X m hx m + k 5 X n X m + Y n+1 Y n Z n+1 Z n dw Y n u Y m u g n ux n u, Y n u, Z n u g u X n u, Y n u, Z n u + g m u X m u, Y m u, Z m u g u X m u, Y m u, Z m u + k 3 X n u X m u + k 4 Y n u Y m u + ρm Z n u Z m u du. 4.5 aking expecaion, due o Young inequaliy we have Z n Z m H R l d E h n X n hx n ] + E h m X m hx m ] + k k 3 X n X m S R m + 1 Y n Y m S R l + 1 g n ux n u, Y n u, Z n u g u X n u, Y n u, Z n u + g m u X m u, Y m u, Z m u g u X m u, Y m u, Z m u du + 1 k 4ρ M Y n Y m S R l + 1 Zn Z m H R l d. On he oher hand, aking condiional expecaion in 4.5 and hen he upremum wih repec o and hen expecaion on boh ide, we have due o Young inequaliy Y n Y m S R l E h n X n hx n ] + E h m X m hx m ] + k 5 X n X m S R m + 1 Y n Y m S R l + 1 g n ux n u, Y n u, Z n u g u X n u, Y n u, Z n u + g m u X m u, Y m u, Z m u g u X m u, Y m u, Z m u du + 1 k 3 X n X m S R m + 1 k 4ρ M Y n Y m S R l + 1 Zn Z m H R l d. Combining 4.4 and 4.1 we oberve ha if i mall enough o ha { 3k k 5 k k 3 k + 1 k 4 ρ M 1 17

18 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 funcion b, h and g, and ha Z M i a conequence of he boundedne of Z n. aking C := C C conclude he proof. 4. Proof of Example. λ 5e Mk Le k 1, gy, z = y z k and R := wih < 1 log 1 1. he funcion g 1 M k e Mk M k M k rericed o he ball {y : y R} {z : z M} i Lipchiz coninuou, i.e. gy, z gy, z M k y y + RM k 1 z z for all y, y R and z, z M. hu, i can be exenden o a Lipchiz coninuou funcion g wih he ame Lipchiz conan on R l R l d. In paricular, g aifie he condiion of heorem.1. hu, he FBSDE. wih generaor g replaced by g admi a unique oluion, Ỹ, Z S R m S R l S R l d wih Z M. Moreover, one ha Ỹ λ 5 + E M k Ỹ + RM k 1 Z d F o by Gronwall inequaliy and boundedne of Z, i follow ha Ỹ λ 5 + RM k e M k = R. ha i, gỹ, Z = gỹ, Z, howing ha, Ỹ, Z olve he FBSDE. wih generaor g. 4.3 Proof of Remark.3 By conrucion, for ufficienly mall, here i a equence X n, Y n, Z n S R m S R l S R l d converging in S R m S R l S R l d o he oluion X, Y, Z of he FBSDE 1.1. Moreover, for each, ] i hold X n, Y n, Z n D 1, R m D 1, R l D 1, R l d wih D r X n 4λ and D r Y n M for all r, ]. For, ] one ha X n X L + Y n Y L. hu, i follow from 7, Propoiion 1..3] ha X, Y D 1, R m D 1, R l. 4.4 Proof of heorem.5 Fir aume ha A6 i aified. If C, hen he reul follow from heorem.1. Aume > C and le h M : R R be a coninuouly differeniable funcion whoe derivaive i bounded by 1 and uch ha h Ma = 1 for all M a M and M + 1 if a > M + h Ma = a if a M M + 1 if a < M +. 18

19 An example of uch a funcion i given by { M h Ma + = Ma aa 4 /4 if a M, M + ] M + Ma + aa + 4 /4 if M +, M], ee Imkeller and do Rei 19]. By he aumpion A3 he funcion g :, ] R m R l R l d R l defined by g x, y, z := g x, y, h Mz 4.6 wih h Mz := h Mz ij ij being Lipchiz coninuou in all variable. hu, i follow from Delarue 8, heorem.6] ha he equaion { = x + b u u, Ỹu du + σ u dw u Ỹ = h + g u u, Ỹu, Z u du Z u dw u,, ] 4.7 admi a unique oluion, Ỹ, Z S R m S R l S R l d. Moreover, here exi a Lipchiz coninuou funcion θ :, ] R m R l bounded by K 5 uch ha Ỹ = θ, for all, ]. In fac, for every x, x R d,, ] and i = 1,..., l, i follow by Iô formula ha θ, x θ, = h x h + h x h + θ i u, x u θ i u, θu, x u θu, K 4 θu, x u θu, θ i u, x u θ i u, u Zx,i u Z,i u dw u u gu i u, x Ỹ u x, Z u x gu i u, Ỹ u, Z u du θ i u, x u θ i u, u k 3 u x u + Z x u u Zx,i u Z,i u dw u u + ρ M dl Z x u Z u du Z u du h x h + k 3 x u θ i u, x u θ i u, u Zx,i u Z,i u dw u u K 4 k 3 ρ M dl θu, u x θu, u du 19

20 Since by Gronwall lemma we have x x i hold θ, x θ, E e k1 k5 + k 3 x hu, + k + e k1 k k5 + k 3 + k 3 + ρ M ] dl K 4 e k 1 k 5 + k 3 x. θ, x θ, Ỹ x u Ỹ u due k 1,, ], K 5 x θu, x u θu,. u du F ] Le C := Ck 1, k, k 3, k 4, K 5, λ, l, d and pu N = / C, where a denoe he ineger par of a, and i := i C, i =,..., N and N+1 =. Since 1 C, by heorem.1 he FBSDE { X = x + b ux u, Y u du + σ u dw u Y = θ 1, X 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 = 8λ K 5 dl for all, 1 ]. herefore, X 1, Y 1, Z 1 1,1 ] =, Ỹ, Z1,1 ]. Similarly, we obain a family X i, Y i, Z i of oluion of he FBSDE { 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 ] =, Ỹ, Z1 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 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 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=1 = X i = X j j 1 b u X j u du + j j 1 σ u dw u

21 and hx + g u X u, Y u, Z u du = hx 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 Zu j dw u = Y i = Y. ha i, X, Y, Z aifie Equaion 1.1. In he cae where A7 i aified, he proof i imilar and we only need o provide he argumen for he Lipchiz coninuiy of θ. If C, hen he reul follow from heorem.1. Aume > C and le h M : R R be a uiable runcaing funcion and g :, ] R m R l R l d R l defined by g x, y, z := g x, y, h M z 4.8 wih h M z := h M z ij ij. I follow from Delarue 8, heorem.6] ha he equaion { = x + b u u, Ỹu du + σ u dw u Ỹ = h + g u u, Ỹu, Z u du Z u dw u,, ] 4.9 admi a unique oluion, Ỹ, Z S R m S R l S R l d. Moreover, here exi a Lipchiz coninuou funcion θ :, ] R m R l bounded by k 5 uch ha Ỹ = θ, for all, ]. In fac, for every x, x R d,, ] and i = 1,..., l applying Iô formula we have θ, x θ, = h x h + h x h + θ i u, x u θ i u, θu, x u θu, K 4 θu, x u θu, θ i u, x u θ i u, u Zx,i u Z,i u dw u u gu i u, x Ỹ u x, Z u x gu i u, Ỹ u, Z u du θ i u, x u θ i u, u k 3 u x u + Z x u u Zx,i u Z,i u dw u u + ρm ld Z x u Z u du Z u du 1

22 h x h + Since for, ], x i hold θ, x θ, hu, k 3 θu, x u θu, E k5 x x + θ i u, x u θ i u, u x u + K 1 k 5 x u u Zx,i u Z,i u dw u u K 4 ρ M ld θu, u x θu, K 1 x u + ρ M ld K 4 θu, u x θu, k 5 x. θ, x θ, u + k x u u + k k 5 + k 3 x u u du F ] k 5 x. u Ỹ x u Ỹ u du, u du u Ỹ x u Ỹ u du Having proved hi Lipchiz coninuiy propery, he re of he proof i exacly he ame a in he fir par. Remark 4.1. Wih he echnique preened above, i i hard o conider yem where he drif b depend on z, ince we do no have good enough eimae on he Malliavin derivaive of he conrol proce Z. Similarly, when he diffuion coefficien σ i a funcion of x, y or z, we looe he eimae on he Malliavin derivaive of he oluion. hee cae even he in non-markovian cae can neverhele be udied under ronger aumpion, we refer o 3] for deail. Reference 1] F. Anonelli and S. Hamadène. Exience of oluion of backward-forward SDE wih coninuou monoone coefficien. Sai. Probab. Le., 7614: , 6. ] P. Barrieu and N. El Karoui. Monoone abiliy of quadraic emimaringale wih applicaion o unbounded general quadraic BSDE. Ann. Probab., 413B: , ] J. Bielagk, A. Lionne, and G. do Rei. Equilibrium pricing under relaive performance concern. SAIM Fin. Mah., 81:435 48, 17.

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