BSDEs on finite and infinite horizon with timedelayed

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1 BSDEs on finie and infinie horizon wih imedelayed generaors Peng Luo a,b,1,, Ludovic Tangpi c,, June 1, 18 arxiv: v1 [mah.pr] 7 Sep 15 ABSTRACT We consider a backward sochasic differenial equaion wih a generaor ha can be subjeced o delay, in he sense ha is curren value depends on he weighed pas values of he soluions, for insance a disored recen average. Exisence and uniqueness resuls are provided in he case of possibly infinie ime horizon for equaions wih, and wihou reflecion. Furhermore, we show ha when he delay vanishes, he soluions of he delayed equaions converge o he soluion of he equaion wihou delay. We argue ha hese equaions are naurally linked o forward backward sysems, and we exemplify a siuaion where his observaion allows o derive resuls for quadraic delayed equaions wih non-bounded erminal condiions in muli-dimension. KEYWORDS: Backward sochasic differenial equaion, delay measure, Weighing funcion, infinie horizon, barrier, FBSDE. 1 Inroducion AUTHORS INFO a Shandong Universiy, 7 Shanda Nanlu, 51 Jinan, P.R. China b Universiy of Konsanz, Universiässraße. 1, D Konsanz, Germany c Universiy of Vienna, Faculy of Mahemaics, Oskar-Morgensern-Plaz 1, A-19 Wien, Ausria 1 peng.luo@uni-konsanz.de ludovic.angpi@univie.ac.a Financial suppor from China Scholarship Council (File No ), NSF of China (No ) and he Projec 111 (No. B13). Financial suppor from Vienna Science and Technology Fund (WWTF) under gran MA PAPER INFO AMS CLASSIFICATION: () 6H, 8A5. In Delong and Imkeller [8, 9], he heory of backward sochasic differenial equaions (BSDEs) was exended o BSDEs wih ime delay generaors (delay BSDEs). These are non-markovian BSDEs in which he generaor a each posiive ime may depend on he pas values of he soluions. This class of equaions urned ou o have naural applicaions in pricing and hedging of insurance conracs, see Delong [7]. The exisence resul of Delong and Imkeller [8], proved for sandard Lipschiz generaors and small ime horizon, has been refined by dos Reis e al. [1] who derived addiional properies of delay BSDEs such as pah regulariy and exisence of decoupled sysems. Furhermore, exisence of delay BSDE consrained above a given coninuous barrier has been esablished by Zhou and Ren [18] in a similar seup. More recenly, Briand and Elie [5] proposed a framework in which quadraic BSDEs wih sufficienly small ime delay in he value process can be solved. In addiion o he inheren non-markovian srucure of delay BSDEs, he difficuly in sudying hese equaions comes from ha he iner-emporal changes of he value and conrol processes always depend on heir enire pas, hence making i hard o obain boundedness of soluions or even BMO-maringale properies of he sochasic inegral of he conrol process. This suggess ha delay BSDEs can acually be solved forward and backward in ime and in his regard, share We hank Michael Kupper for helpful commens and fruiful discussions 1

2 similariies wih forward backward sochasic differenial equaions (FBSDEs), see Secion 4 for a more deailed discussion. The objec of he presen noe is o sudy delay BSDEs in he case where he pas values of he soluions are weighed wih respec o some scaling funcion. In economic applicaions, hese weighing funcions can be viewed as represening he percepion of he pas of an agen. For mulidimenional BSDEs wih possibly infinie ime horizon, we derive exisence, uniqueness and sabiliy of delay BSDE in his weighing-funcion seing. In paricular, we show ha when he delay vanishes, he soluions of he delay BSDEs converge o he soluion of he BSDE wih no delay, hence recovering a resul obained by Briand and Elie [5] for differen ypes of delay. Moreover, we prove ha in our seing exisence and uniqueness also hold in he case of reflexion on a càdlàg barrier. We observe a link beween delay BSDEs and coupled FBSDE and, based on he findings in Luo and Tangpi [13], we derive exisence of delay quadraic BSDEs in he case where only he value process is subjeced o delay. We refer o Briand and Elie [5] for a similar resul, again for a differen ype of delay and in he one-dimensional case. In he nex secion, we specify our probabilisic srucure and he form of he equaion, hen presen exisence, uniqueness and sabiliy resuls. Secions 3 and 4 are dedicaed o he sudy of refleced delay BSDEs and he link o FBSDEs, respecively. BSDEs wih ime delayed generaors We work on a filered probabiliy space(ω,f,(f ) [,T],P) wiht (, ]. We assume ha he filraion is generaed by a d-dimensional Brownian moion W, is complee and righ coninuous. Le us also assume ha F = F T. We endow Ω [,T] wih he predicable σ-algebra andr k wih is Borelσ-algebra. Unless oherwise saed, all equaliies and inequaliies beween random variables and sochasic processes will be undersood in he P -a.s. and P d-a.e. sense, respecively. For p [1, ) and m N, we denoe by S p (R m ) he space of predicable and coninuous processes X valued in R m such ha X p S := E[(sup p [,T] X ) p ] < and by H p (R m ) he space of predicable processes Z valued in R m d such ha Z p H := p E[( Z u du) p/ ] <. For a suiable inegrand Z, we denoe by Z W he sochasic inegral ( Z udw u ) [,T] of Z wih respec o W. From Proer [15], Z W defines a coninuous maringale for every Z H p (R m ). Processes (φ ) [,T] will always be exended o [,) by seing φ = for [,). We equip R wih he σ-algebra B(R) consising of Borel ses of he usual real line wih possible addiion of he poins,+, see Bogachev [4]. Le ξ be an F T -measurable erminal condiion and g an R m -valued funcion. Given wo measuresα 1 andα on[, ], and wo weighing funcionsu,v : [,T] R, we sudy exisence of he BSDE where Y = ξ + Γ(s) := g(s, Γ(s))ds u(s+r)y s+r α 1 (dr), Z s dw s, [,T], (.1) v(s+r)z s+r α (dr). (.)

3 Example BSDE wih infinie horizon: If u = v = 1 and α 1 = α = δ he Dirac measure a, hen Equaion (.1) reduces o he classical BSDE wih infinie ime horizon and sandard Lipschiz generaor.. Pricing of insurance conracs: Le us consider he pricing problem of an insurance conrac ξ wrien on a weaher derivaive. I is well known, see for insance [] ha such conracs can be priced by invesing in a highly correlaed, bu radable derivaive. In he Meron model, assuming ha he laer asse has dynamics ds = S (µ d+σ dw ), hen he insurer chooses a numberz of shares ofs o buy a imeand fixes a cosc o be paid by he clien. Hence, he seeks o find he price V such ha dv = c d+z σ (dw +θ d) wihθ = σ (σ σ ) 1 µ. I is naural o demand he cosc a imeo depend on he pas values of he insurance premium V, for insance o accoun for hisorical weaher daa. A possible cos crieria is c := M cos( π P (+s))v +sds wherep accouns for he weaher periodiciy andm is a scaling parameer. Thus, he insurance premium saisfies he delay BSDE V = ξ + M u cos( π P (u+s))v u+sds+z u σ u θ u du Z u σ u dw u..1 Exisence Our exisence resul for he BSDE (.1) is obained under he following assumpions: (A1) α 1,α are wo deerminisic, finie valued measures suppored on [,]. (A) u,v : [,T] R are Borel measurable funcions such hau L 1 (d) andv L (d). (A3) g : Ω [,T] R m R m d R m is measurable, such ha g(s,,)ds L (R m ) and saisfies he sandard Lipschiz condiion: here exiss a consan K > such ha g(,y,z) g(,y,z ) K( y y + z z ) for every y,y R m and z,z R m d. 3

4 (A4) ξ L (R m ) and is F T -measurable. Theorem.. Assume (A1)-(A4). If { K α 1 ([,]) u L 1 (d) 1 5, K α ([,]) v L (d) 1 5, (.3) hen BSDE (.1) admis a unique soluion (Y,Z) S (R m ) H (R m d ). For he proof we need he following lemma on a priori esimaes of soluions of (.1). Lemma.3 (A priori esimaion). Assume (A1)-(A3). For everyξ, ξ L (R m ),(y,z),(ȳ, z) S (R m ) H (R m d ) and (Y,Z),(Ȳ, Z) S (R m ) H (R m d ) saisfying { Y = ξ + g(s,γ(s))ds Z s dw s Ȳ = ξ + g(s, γ(s))ds Z s dw s, [,T] wih Then, one has ( γ(s) = u(s+r)y s+rα 1 (dr), γ(s) = ) v(s+r)z s+rα (dr) ) ( u(s+r)ȳ s+rα 1 (dr), v(s+r) z s+rα (dr). Y Ȳ S (R m ) + Z Z H (R m d ) K α 1 ([,]) u L 1 (d) y ȳ S (R m ) +1 ξ ξ L (R m ) +K α ([,]) v L (d) z z H (R m d ). Proof. Le(y,z) S (R m ) H (R m d ), by assumpions (A1) and (A3), usingab a +b and [1, Lemma 1.1], we have E +K g(s, γ(s))ds 3E +K E v(s+r) z s+r α (dr)ds g(s,, ) ds g(s,,) ds +K +K v(s+r) z s+r α (dr)ds u(s+r) y s+r α 1 (dr)ds u(s+r) y s+r α 1 (dr)ds 4

5 3E +K 3E g(s,, ) ds +K α ([s,]) v(s) z s ds g(s,, ) ds +3K α ([,]) Hence, i holds g(s,γ(s))ds L. Now, for [,T], we have Y Ȳ = ξ ξ + +3K α 1 ([,]) v(s) ds E α 1 ([s,]) u(s) y s ds z s ds. u(s) ds g(s,γ(s)) g(s, γ(s))ds and aking condiional expecaion wih respec of yields Y Ȳ = E ξ ξ + E g(s, γ(s)) g(s, γ(s))ds F. By Doob s maximal inequaliy and ab a +b, we obain E [ sup T Y Ȳ ] = E sup T E ξ ξ + E sup E ξ ξ + T 8E ξ ξ + [ ] sup y T Z s Z s dw s (.4) g(s, γ(s)) g(s, γ(s))ds F g(s, γ(s)) g(s, γ(s)) ds F g(s,γ(s)) g(s, γ(s)) ds. On he oher hand, for = in (.4), bringing Z s Z s dw s o he lef hand side, aking 5

6 square and expecaion o boh sides and ab a +b, we have E Z Z d = E ξ ξ + E ξ ξ + E ξ ξ + g(s, γ(s)) g(s, γ(s))ds g(s, γ(s)) g(s, γ(s))ds Y Ȳ g(s, γ(s)) g(s, γ(s)) ds. By assumpion (A3), using [1, Lemma 1.1] and he inequaliy ab a +b, we have E + g(s, γ(s)) g(s, γ(s)) ds = K E Hence, K E v(s+r) z s+r z s+r α (dr)ds α 1 ([s,]) u(s) y s ȳ s ds+ u(s+r) y s+r ȳ s+r α 1 (dr)ds α ([s,]) v(s) z s z s ds K α 1 ([,]) u L 1 (d) y ȳ S +K α ([,]) v L (d) z z H. Y Ȳ S (R m ) + Z Z H (R m d ) K α 1 ([,]) u L 1 (d) y ȳ S (R m ) 1E [ ξ ξ ] +K α ([,]) v L (d) z z H (R m d ). This concludes he proof. Proof ( ( of Theorem.). Le (y,z) S (R m ) H (R m d ) and define he process γ(s) := u(s+r)y s+rα 1 (dr), ) v(s+r)z s+rα (dr). Similar o Lemma.3, i follows from (A1)-(A4) ha E ξ + g(s, γ(s))ds <. According o he maringale represenaion heorem, here exiss a unique Z H (R m d ) such ha for all [,T], E ξ + g(s, γ(s))ds F = E ξ + g(s, γ(s))ds + Z s dw s. 6

7 Puing Y := E ξ + g(s,γ(s))ds F, T, he pair (Y,Z) belongs os (R m ) H (R m d ) and saisfies Y = ξ + g(s, γ(s))ds Z s dw s, T. Thus we have consruced a mappingφfroms (R m ) H (R m d ) o iself such haφ(y,z) = (Y,Z). Le(y,z),(ȳ, z) S (R m ) H (R m d ), and(y,z) = Φ(y,z),(Ȳ, Z) = Φ(ȳ, z). By Lemma.3, we have Y Ȳ S (R m ) + Z Z H (R m d ) 1K α 1 ([,]) u L 1 (d) y ȳ S (R m ) +1K α ([,]) v L (d) z z H (R m d ) so ha if condiion (.3) is saisfied, Φ is a conracion mapping which herefore admis a unique fixed poin on he Banach space S (R m ) H (R m d ). This complees he proof.. Sabiliy In his subsecion, we sudy sabiliy of he BSDE (.1) wih respec o he delay measures. In paricular, in Corollary.5 below we give condiions under which a sequence of soluions of BSDEs wih ime delayed generaor converges o he soluion of a sandard BSDE wih no delay. Given wo measures α and β, we wrie α β if α(a) β(a) for every measurable se A. Theorem.4. Assume (A)-(A4). For i = 1, and n N, le α n i,α i be measures saisfying (A1); wih α n i saisfying (.3) in Theorem. and such ha α n i ([,]) converges o α i ([,]). If α n 1 α 1 (or α 1 α n 1 ) and αn α (or α α n ), hen Y n Y S (R m ) and Z n Z H (R m d ), where (Y n,z n ) and (Y,Z) are soluions of he BSDE (.1) wih delay given by he measures (α n 1,αn ) and (α,α ), respecively. Proof. From Theorem., for every n, here exiss a unique soluion (Y n,z n ) o he BSDE (.1) wih delay given by he measures(α n 1,αn ). Sinceαn i,i = 1, saisfy (.3) in Theorem. and α n i ([,]) converges o α i([,]), i follows ha α i saisfy (.3) and by Theorem. here exiss a unique soluion (Y,Z) o he BSDE wih delay given by(α 1,α ). Using Y n Y = g(s,γ n (s)) g(s,γ(s))ds Z n s Z s dw s, i follows similar o he proof of Lemma.3 ha [ ] E sup Y n Y 4E g(s,γ n (s)) g(s,γ(s)) ds, T 7

8 and E Z n Z E g(s,γ n (s)) g(s,γ(s)) ds. On he oher hand, using ab a +b, we ge E K E g(s,γ n (s)) g(s,γ(s)) ds +K E u(s+r)ys+rα n n 1(dr) v(s+r)z n s+rα n (dr) u(s+r)y s+r α 1 (dr) ds v(s+r)z s+r α (dr) ds. Wihou loss of generaliy, we assume α 1 α n 1 and α α n. Hence αn i α i, i = 1,, define posiive measures saisfying (A1). Therefore, E u(s+r)ys+rα n n 1(dr) u(s+r)y s+r α 1 (dr) ds E +E u(s+r) Y n s+r Y s+r α n 1 (dr)ds Using [1, Lemma 1.1], we obain E E u(s+r) Y n s+r Y s+r u(s+r) Y s+r (α n 1 α 1)(dr)ds. α n 1 (dr)ds +E α n 1 ([s,]) u(s) Y s n Y s ds +E u(s+r) Y s+r (α n 1 α 1)(dr)ds (α n 1 α 1)([s,]) u(s) Y s ds (α n 1([,])) u L 1 (d) Y n Y S (R m ) +((αn 1 α 1 )([,])) u L 1 (d) Y S (R m ). 8

9 Similarly, for he conrol processes we have E v(s+r)zs+rα n n (dr) v(s+r)z s+r α (dr) ds (α n ([,])) v L (d) Zn Z H (R m d ) +((αn α )([,])) v L (d) Z H (R m d ). Hence Y n Y S (R m ) + Zn Z H (R m d ) K (α n 1([,])) u L 1 (d) Y n Y S (R m ) +K ((α n 1 α 1 )([,])) u L 1 (d) Y S (R m ) +K (α n ([,])) v L (d) Zn Z H (R m d ) +K ((α n α )([,])) v L (d) Z H (R m d ) 4 5 Y n Y S (R m ) Zn Z H (R m d ) +K ((α n 1 α 1 )([,])) u L 1 (d) Y S (R m ) +K ((α n α )([,])) v L (d) Z H (R m d ). Therefore, he resul follows from he convergence of α n i ([,]), i = 1,. The following is a direc consequence of he above sabiliy resul. We denoe by δ he Dirac measure a. Corollary.5. Assume (A)-(A4). For i = 1, and n N le α n i be measures saisfying (A1) and (.3) in Theorem. and such haα n i ([,]) converges o1. Ifαn 1 δ (orδ α n 1 ) and α n δ (orδ α n ), hen Y n Y S (R m ) and Z n Z H (R m d ), where(y n,z n ) is he soluion of he BSDE (.1) wih delay given by(α n 1,αn ) and(y,z) is he soluion of BSDE wihou delay. We conclude his secion wih he following counerexample which shows ha he condiion α 1 α n 1 (or αn 1 α 1) and α α n (or αn α ) is needed in he above heorem. Example.6. Assume ha m = d = 1. We denoe by δ and δ 1 he Dirac measures a and 1, respecively. I is clear ha δ ([ 1,]) = δ 1 ([ 1,]). Consider he delay BSDEs 1 1 Y = 1+ 1/5 Y s+r +Z s+r δ (dr)ds Z s dw s (.5) and 1 Ȳ = 1+ 1/5 1 1 Ȳ s+r + Z s+r δ 1 (dr)ds 1 Z s dw s. (.6) Since BSDE (.6) akes he form Ȳ = 1 1 Z u dw s, i follows ha Ȳ = 1 for all [,1]. On he oher hand, (.5) is he sandard BSDE wihou delay, is soluion can be wrien as Y = E[H1 F ], where he deflaor (Hs ) s a ime is given by dhs = H s 5 (ds + dw s). Thus, Y = exp( 1/5(1 )) and for [,1), Y < Ȳ. 9

10 3 Refleced BSDEs wih ime-delayed generaors The probabilisic seing and he noaion of he previous secion carries over o he presen one. In paricular, we fix a ime horizon T (, ] and we assume m = 1. For p [1, ), we furher inroduce he space M p (R) of adaped càdlàg processes X valued in R such ha X p M p := E[(sup [,T] X ) p ] < and by A p (R), we denoe he subspace of elemens of M p (R) which are increasing processes saring a. Le (S ) [,T] be a càdlàg adaped realvalued process. In his secion, we sudy exisence of soluions (Y,Z,K) of BSDEs refleced on he càdlàg barrier S and wih ime-delayed generaors. Tha is, processes saisfying Y = ξ + g(s,γ(s))ds+k T K Z s dw s, [,T] (3.1) Y S (3.) (Y S )dk = (3.3) wihγdefined by (.). Consider he condiion (A5) E [ sup T (S + )] < and S T ξ. Theorem 3.1. Assume (A1)-(A5). If { K α 1 ([,]) u L 1 (d) 1 36, K α ([,]) v L (d) 1 36, (3.4) hen RBSDE (3.1) admis a unique soluion (Y,Z,K) M (R) H (R d ) A (R) saisfying Y = esssupe τ T τ g(s,γ(s))ds+s τ 1 {τ<t} +ξ1 {τ=t} F, where T is he se of all sopping imes aking values in[,t] and T = {τ T : τ }. Proof. For any given (y,z) M (R) H (R d ), similar o he proof of Lemma.3, we have E ξ + g(s, γ(s))ds < wih γ defined as in Lemma.3. Hence, from [1, Theorem 3.3] for T < and [1, Theorem 3.1] for T = he refleced BSDE Y = ξ + g(s,γ(s))ds+k T K Z s dw s 1

11 wih barrier S admis a unique soluion (Y,Z,K) such ha (Y,Z) B, he space of processes (Y,Z) M (R) H (R d ) such ha Y S, and K A (R). Moreover, Y admis he represenaion Y = esssup τ T E τ g(s,γ(s))ds+s τ 1 {τ<t} +ξ1 {τ=t} F [,T]. Hence we can define a mappingφfrombobby seing Φ(y,z) := (Y,Z). Le(y,z),(ȳ, z) B and (Y,Z) = Φ(y,z), (Ȳ, Z) = Φ(ȳ, z). From he represenaion, we deduce Y Ȳ esssup τ T E τ g(s, γ(s)) g(s, γ(s)) ds F E Doob s maximal inequaliy implies ha E [ ] sup Y Ȳ 4E T Applying Iô s formula o Y Ȳ, we obain Y Ȳ + = + Z s Z s ds = + + (Y s Ȳs )d(k s K s ) g(s, γ(s)) g(s, γ(s)) ds F. g(s, γ(s)) g(s, γ(s)) ds. (Y s Ȳ s )(g(s,γ(s)) g(s, γ(s)))ds (Y s Ȳs)(g(s,γ(s)) g(s, γ(s)))ds (Y s S s )dk s (Ȳs S s )d K s. (Y s Ȳs)(Z s Z s )dw s (Y s S s )d K s (Y s Ȳs)(Z s Z s )dw s (Ȳs S s )dk s 11

12 Since (Y,K) and (Ȳ, K) saisfy (3.) and (3.3), we have Hence E Y Ȳ + Z s Z s ds E Z s Z s ds [ sup T In view of he proof of Lemma.3, we deduce (Y s Ȳs)(g(s,γ(s)) g(s, γ(s)))ds Y Ȳ ]+E Y Ȳ M (R) + Z Z H (R d ) 9E (Y s Ȳs)(Z s Z s )dw s. 18K α 1 ([,]) u L 1 (d) y ȳ M (R) g(s, γ(s)) g(s, γ(s)) ds. g(s, γ(s)) g(s, γ(s)) ds +18K α ([,]) v L (d) z z H (R d ). By condiion (3.4), Φ is a conracion mapping and herefore i admis a unique fixed poin which combined wih he associaed process K is he unique soluion of he RBSDE (3.1). 4 Link o coupled FBSDEs In his secion, we discuss he connecion beween BSDEs wih ime-delayed generaors and FBSDEs. We work in he probabilisic seing and wih he noaion of Secion. Sandard mehods o solve BSDEs wih quadraic growh in he conrol variable ofen rely eiher on boundedness of he conrol process, see for insance [16] and [6], or on BMO esimaes for he sochasic inegral of he conrol process, see for insance [17]. However, as shown in [8], soluions of BSDEs wih ime-delayed generaors do no, in general, saisfy boundedness and BMO properies so ha new mehods are required o solve quadraic BSDE wih ime-delayed generaors. Recenly, [5] obained exisence and uniqueness of soluion for a quadraic BSDE wih delay only in he value process. We show below ha using FBSDE heory, i is possible o generalize heir resuls o mulidimension and considering a differen kind of delay. Moreover, our argumen allows o solve equaions wih generaors of superquadraic growh. Le α 1 be he uniform measure on[,], α he Dirac measure a. Pu u(s) = v(s) = 1, for s [,T]. We are considering he following BSDE wih ime delay only in he value process: Y = ξ + s g(s, Y r dr,z s )ds Z s dw s, [,T]. (4.1) 1

13 We denoe by D 1, he space of all Malliavin differeniable random variables and for ξ D 1, denoe by D ξ is Malliavin derivaive. We refer o Nualar [14] for a horough reamen of he heory of Malliavin calculus, whereas he definiion and properies of he BMO-space and norm can be found in [11]. We make he following assumpions: (B1) g : [,T] R m R m d R m is a coninuous funcion such ha g i (y,z) = g i (y,z i ) and here exiss a consan K > as well as a nondecreasing funcion ρ : R + R + such ha g(s,y,z) g(s,y,z ) K y y +ρ( z z ) z z, g(s,y,z) g(s,y,z) g(s,y,z )+g(s,y,z ) K( y y + z z ) for all s [,T], y,y R m and z,z R m d. (B) ξ is F T -measurable such ha ξ D 1, (R m ) and here exis consans A ij such ha D j ξi A ij, i = 1,...,m; j = 1,...,d, for all [,T]. (B3) g : Ω [,T] R m R m d is measurable, g(s,y,z) = f(s,z)+l(s,y,z) where f andlare measurable funcions wihf i (s,z) = f i (s,z i ),i = 1,...,m and here exiss a consan K such ha f(s,z) f(s,z ) K(1+ z + z ) z z, l(s,y,z) l(s,y,z ) K y y +K(1+ z ǫ + z ǫ ) z z, f(s,z) K(1+ z ), l(s,y,z) K(1+ z 1+ǫ ), for some ǫ < 1 and for all s [,T], y,y R m and z,z R m d. (B4) ξ isf T -measurable such ha here exis a consan K such ha ξ K. (B5) g : Ω [,T] R R d R is progressively measurable, coninuous process for any choice of he spaial variables and for each fixed (s,ω) [,T] Ω, g(s,ω, ) is coninuous. g is increasing in y and for some consan K such ha for all s [,T], y R and z R d. (B6) ξ isf T -measurable such ha ξ L. g(s,y,z) K(1+ z ), 13

14 (B7) g : Ω [,T] R R d R is progressively measurable, coninuous process for any choice of he spaial variables and for each fixed (s,ω) [,T] Ω, g(s,ω, ) is coninuous. g is increasing in y and for some consan K such ha for all s [,T], y R and z R d. Proposiion 4.1. Assume T (, ). g(s,y,z) K(1+ z ), 1. If (B1)-(B) are saisfied, hen here exiss a consan C such ha for sufficienly small T, BSDE (4.1) admis a unique soluion (Y,Z) S (R m ) H (R m d ) such ha Z C.. If (B3)-(B4) are saisfied, hen here exis consans C 1,C such ha for sufficienly small T, BSDE (4.1) admis a unique soluion (Y,Z) S (R m ) H (R m d ) such ha Y C 1 and Z dw BMO C. 3. If m = d = 1 and (B5)-(B6) are saisfied, hen BSDE (4.1) admis a leas a soluion (Y,Z) S (R) H (R d ). 4. If m = d = 1 and (B4) and (B7) are saisfied, hen BSDE (4.1) admis a leas a soluion (Y,Z) S (R) H (R d ) such ha Y is bounded and Z W is a BMO maringale. Proof. Define he funcion b : R m R m by seing for y R m, b i (y) = y i, i = 1,...,m. For [,T], pu X = b(y s )ds. Thus BSDE (4.1) can be wrien as he coupled FBSDE { X = b(y s)ds, Y = ξ + g(s,x s,z s )ds (4.) Z s dw s so ha 1. and. follow from [13], and 3. and 4. from [3]. The above heorem provides an explanaion why i is no enough o solve a ime-delayed BSDE backward in ime, one acually needs o consider boh he forward and backward pars of he soluion due o he delay. References [1] K. Akdim and Y. Ouknine. Infinie horizon refleced backward sdes wih jumps and rcll obsacle. Soch. Anal. Appl., 4(6), 6. [] S. Ankirchner, P. Imkeller, and G. Reis. Pricing and hedging of derivaives based on non-radeable underlyings. Mah. Finance, ():89 31, 1. 14

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