Quadratic Backward Stochastic Volterra Integral Equations

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1 Quadraic Backward Sochaic Volerra Inegral Equaion Hanxiao Wang Jingrui Sun Jiongmin Yong Ocober 25, 218 arxiv: v1 mah.pr 24 Oc 218 Abrac. Thi paper i concerned wih backward ochaic Volerra inegral equaion (BSVIE, for hor) wih generaor having quadraic growh. The exience and uniquene for boh he o-called adaped oluion and adaped M-oluion are obained. The comparion heorem for adaped oluion i eablihed a well. A applicaion of uch BSVIE, coninuou-ime equilibrium dynamic rik meaure and equilibrium recurive uiliy procee are preened. Key word. Backward ochaic Volerra inegral equaion, quadraic generaor, comparion heorem, equilibrium dynamic rik meaure, equilibrium recurive uiliy proce. AMS ubjec claificaion. 6H1, 6H2, 91B3, 91B7, 91G8. 1 Inroducion Le (Ω, F, P) be a complee probabiliy pace on which a one-dimenional andard Brownian moion W = {W(); < } i defined, wih F = {F } being he naural filraion of W augmened by all he P-null e in F. Le g :,T 2 R R R Ω R, ψ :,T Ω R be wo given random field. We are concerned wih he following backward ochaic Volerra inegral equaion (BSVIE, for hor): Y() = ψ()+ g(,,y(),z(,),z(,))d Z(,)dW(),,T. (1.1) By an adaped oluion o BSVIE (1.1), we mean an (R R)-valued random field (Y, Z) = {(Y(), Z(,));, T} uch ha (i) Y( ) i F-progreively meaurable (no necearily coninuou), (ii) for each fixed T, Z(, ) i F-progreively meaurable, and (iii) equaion (1.1) i aified in he uual Iô ene for Lebegue meaure almo every,t. School of Mahemaical Science, Fudan Univeriy, Shanghai 2433, China (hxwang14@fudan.edu.cn). Thi auhor i uppored in par by he China Scholarhip Council, while viiing Univeriy of Cenral Florida. Correponding auhor. Deparmen of Mahemaic, Univeriy of Cenral Florida, Orlando, FL 32816, USA (jr@mail.uc.edu.cn). Deparmen of Mahemaic, Univeriy of Cenral Florida, Orlando, FL 32816, USA (jiongmin.yong@ucf.edu). Thi auhor i uppored in par by NSF Gran DMS

2 Condiion (ii) implie ha for any,t), he random variable Z(,) i F -meaurable for any,t. In (1.1), g and ψ are called he generaor and he free erm, repecively. Le u poin ou ha in hi paper, we only udy he BSVIE wih Y( ) being one-dimenional. The cae ha Y( ) being higher dimenional will be ignificanly differen in general, and will be inveigaed in he near fuure. However, he Brownian moion W( ) aumed o be one-dimenional i ju for convenience of our preenaion. When Z(,) i aben, (1.1) i reduced o he form Y() = ψ()+ g(,,y(),z(,))d Z(,)dW(),,T, (1.2) which i a naural exenion of he o-called backward ochaic differenial equaion (BSDE, for hor) of he following (inegral) form: Y() = ξ + g(,y(),z())d Z()dW(),,T. (1.3) BSVIE of form (1.2), referred o a Type-I BSVIE, wa firly udied by Lin 29, followed by everal oher reearcher: Aman and N Zi 3, Wang and Zhang 43, Djordjević and Janković 15, 16, Hu and Økendal 2. BSVIE of he form (1.1) (conaining Z(, )) were firly inroduced by Yong 45, 47, moivaed by he udy of opimal conrol for forward ochaic Volerra inegral equaion (FSVIE, for hor). We call (1.1) a Type-II BSVIE o diinguih i from Type-I BSVIE. Type-II BSVIE (1.1) ha a remarkable feaure ha i adaped oluion, imilarly defined a ha for Type-I BSVIE, migh no be unique due o lack of rericion on he erm Z(,) (wih T). Suggeed by he naure of he equaion from he adjoin equaion in he Ponryagin ype maximum principle, Yong 47 inroduced he noion of adaped M-oluion: A pair (Y( ),Z(, )) i called an adaped M-oluion o (1.1), if in addiion o (i) (iii) aed above, he following condiion i alo aified: Y() = EY()+ Z(,)dW(), a.e.,t, a.. (1.4) Under uual Lipchiz condiion, well-poedne wa eablihed in 47 for he adaped M-oluion o Type-II BSVIE of form (1.1). Thi imporan developmen ha riggered exenive reearch on BSVIE and heir applicaion. For inance, Anh, Greckch and Yong 4 inveigaed BSVIE in Hilber pace; Shi, Wang and Yong 36 udied well-poedne of BSVIE conaining mean-field (of he unknown); Ren 34, Wang and Zhang 44 dicued BSVIE wih jump; Overbeck and Röder 32 even developed a heory of pah-dependen BSVIE; Numerical apec wa conidered by Bender and Pokalyuk 6; relevan opimal conrol problem were udied by Shi, Wang and Yong 37, Agram and Økendal 2, Wang and Zhang 42, and Wang 39; Wang and Yong 4 eablihed variou comparion heorem for boh adaped oluion and adaped M-oluion o BSVIE in muli-dimenional Euclidean pace. Recenly, inpired by he Four-Sep Scheme in he heory of forward-backward ochaic differenial equaion(fbsde, for hor) (31) and he ime-inconien ochaic opimal conrol problem (48), Wang and Yong 41 eablihed a repreenaion of adaped oluion o Type-I BSVIE and adaped M-oluion o Type-II BSVIE in erm of he oluion o a yem of (non-claical) parial differenial equaion and he oluion o a (forward) ochaic differenial equaion. Noe ha in all he above-menioned work on BSVIE, he generaor g(,,y,z,z ) of he BSVIE (1.1) aifie a uniform Lipchiz condiion in (y,z,z ) o ha he generaorha a linear growh in (z,z ). 2

3 From he exiing lieraure, ay, for inance 45, 47, one ee ha here are wo obviou moivaion for inroducing and udying BSVIE: a naural exenion of BSDE, and o mee he need of he aemen of Ponryagin ype maximum principle for opimal conrol of (forward) ochaic Volerra inegral equaion. I urn ou ha here are ome oher inereing moivaion for BSVIE from finance and economic. Le u now briefly elaborae ha. Le ξ be a (random) payoff a ome fuure ime T of cerain European ype coningen claim. Le Y( ) olve he following equaion: ( Y() = E ξ + f(c(),y()) A(Y())Z() 2) d,,t, (1.5) where, and hereafer, E = E F, f : R R R i a given map, called he aggregaor, c( ) i a conumpion rae, Z() 2 = d d Y (), wih Y () being he quadraic variaion of Y( ), and A(Y()) i called he variance muliplier. Such defined Y( ) i called a recurive uiliy proce (which ha been alo called ochaic differenial uiliy proce) of he payoff ξ. Thi noion wa firly inroduced by Duffie and Epein 17 in A lile more generally, we may conider he following equaion: Y() = E e δ(t ) ξ + e δ( ) f(,y(),z())d,,t, (1.6) where δ > i called a dicoun rae, and f(,y,z) = f(c(),y) A(y)z 2, (,y,z),t R R. I i eay o ee ha (Y( ),Z( )) olve (1.6) if and only if i i an adaped oluion o he following BSDE: ( ) T Y() = ξ + δy()+ f(,y(),z()) d Z()dW(),,T, (1.7) which i a pecial cae of (1.3). Becaue of he above obervaion, recurive uiliy proce wa laer exended o he adaped oluion of general BSDE (ee 28, 27). A for BSDE (1.3), when (y,z) g(,y,z)aifiea uniform Lipchiz condiion, wih g(,,)being L p -inegrable (wih ome p > 1), for any F T -meaurable L p -inegrable random variable ξ, BSDE (1.3) admi a unique adaped oluion(y( ), Z( )) (33, 31, 49) which could be called a recurive uiliy proce for ξ. On he oher hand, we ee ha in order (1.3) include (1.7), we need he generaorg(,y,z) of (1.3) allowing a quadraic growh in z. For convenience, when z g(,y,z) ha an up o quadraic growh, he BSDE (1.3) i called a quadraic BSDE (QBSDE, for hor). In 2, Kobylanki 24 eablihed he well-poedne of QBSDE wih ξ being bounded. Since hen, ome effor have been made by reearcher o relax he aumpion on he generaor a well a he erminal value ξ. Among relevan work, we would like o menion Briand and Hu 7, 8, Hu and Tang 21, Briand and Richou 9, and Zhang 5, Chaper 7. Furher, BSDE wih uperquadraic growh wa inveigaed by Delbaen, Hu and Bao 1, where ome general negaive reul concerning he well-poedne can be found. Therefore, one can ay ha he heory of recurive uiliy for erminal payoff ξ ha reached a prey maure age. Now, ifineadofξ, we haveanf T -meaurableproceξ(), nonecearilyf-adaped, whichcould be called an anicipaed cah flow proce. For example, i could be an anicipaed received dividend proce of a ock (which depend on he performance of he company, and i i uncerain), anicipaed received morgage paymen (for a bank, ay, wih an uncerainy of defaul or prepaymen), anicipaed claim 3

4 paymen of an inurance policy, he random mainenance co of an owned faciliy, ec. To calculae he recurive uiliy for uch a proce, mimicking (1.6), we migh formally olve he following equaion a he curren ime,t: Y(;r) = E r e δ(t r) ξ()+ e δ( r) f(,y(;),z(;))d, r,t, (1.8) r wih he curren ime being a parameer. Furher, from he udy of ime-inconien opimal conrol (ee 48 and reference cied herein), i i known ha mo people over-weigh he immediae fuure uiliy. To decribe uch a ime-preference, one could ue non-exponenial dicouning, namely, replace e δ in he above by a general decreaing funcion λ(). Thu, inead of (1.8), we hould olve he following: Y(;r) = E r λ(t r)ξ() + λ( r) f(,y(;),z(;))d, r,t. (1.9) r Thi i equivalen o he following BSDE: Y(;r)=λ(T r)ξ()+ r λ( r) f(,y(;),z(;))d r Z(;)dW(), r,t. (1.1) Inuiively, he curren uiliy hould be given by Y(;). However, by aking r = in he above, we obain Y(;)=λ(T )ξ()+ λ( ) f(,y(;),z(;))d Z(;)dW(),,T, (1.11) which i no an equaion for he proce Y(;) ince Y(;) appear in he righ-hand ide of he above. Apparenly, Y(; r) defined by (1.1) ha ome ime-inconien naure. Therefore, Y(; ) defined aboveeem no obe agood candidaeofhe recuriveuiliy proceforhe proceξ( ). Bu, ifinead, we are able o olve he following Type-I BSVIE: Y() = λ(t )ξ()+ λ( ) f(,,y(),z(,))d Z(,)dW(),,T, (1.12) hen Y( ) i ime-conien and i hould be a perfec candidae for he recurive uiliy (or ochaic differenial uiliy) proce for he anicipaed cah flow proce ξ( ). We call uch a Y( ) an equilibrium recurive uiliy proce for he anicipaed cah flow proce ξ( ). Now, we reurn o Type-I BSVIE (1.2). Similar o he BSDE cae, in order (1.2) o include he above (1.12), we hould allow quadraic growh of Z(,) g(,,y(),z(,)). We call uch kind of BSVIE Type-I quadraic BSVIE (QBSVIE, for hor). We can define Type-II QBSVIE imilarly. The purpoe of hi paper i o eablih well-poedne of QBSVIE, under cerain condiion. By chooing uiable pace of procee and uing he mehod inroduced by Yong 47, combining hoe found in Briand Hu 7, 8, we ge he exience and uniquene of adaped oluion and adaped M- oluion for boh Type-I and Type-II QBSVIE. Conequenly, we will eablih he heory of equilibrium recurive uiliy procee. In addiion, ome comparion heorem for adaped oluion of Type-I QBSVIE will be eablihed, and he o-called coninuou-ime equilibrium dynamic rik meaure will be inveigaed a an applicaion of he reul obained for QBSVIE. See Yong 46 and Wang Yong 4, Agram 1 for ome earlier work. See alo Di Perio 14 for ochaic differenial uiliy, and Kromer Overbeck 26 for dynamical capial allocaion by mean of BSVIE. The re of hi paper i organized a follow. In Secion 2, we inroduce ome preliminary noaion and definiion, and preen ome lemma which are of frequen ue in he equel. Secion 3 i devoed o he udy of exience and uniquene of adaped oluion for Type-I QBSVIE, and Secion 4 i 4

5 devoed o he udy of exience and uniquene of adaped M-oluion for Type-II QBSVIE. Some comparion heorem for adaped oluion o Type-I QBSVIE (1.2) will be eablihed in Secion 5, and an applicaion of QBSVIE o coninuou-ime rik meaure will be preened in Secion 6. Some concluion remark will be colleced in Secion 7. 2 Preliminarie For a < b T, we denoe by B(a,b) he Borel σ-field on a,b and define he following e: a,b { (,) a b }, c a,b { (,) a < b }, a,b 2 { (,) a, b } = a,b c a,b, a,b c a,b. Noe ha a,b i a lile differen from he complemen c a,b of a,b in a,b 2, ince boh a,b and a,b conain he diagonalline egmen. In he equel we hall deal wih varioupace of funcion and procee, which we collec here fir for he convenience of he reader: { L 1 (a,b) = h : a,b R h( ) i B(a,b)-meaurable, b a }, h() d < { } L F b (Ω) = ξ : Ω R ξ i F b -meaurable and bounded, { } L F b (a,b) = ϕ : a,b Ω R ϕ( ) i B(a,b) F b -meaurable and bounded, { L 2 F (a,b) = ϕ : a,b Ω R ϕ( ) i F-progreively meaurable, E } b a ϕ() 2 d <, { L F (a,b) = ϕ( ) L 2 F (a,b) ϕ( ) i bounded }, { L 2 F(Ω;Ca,b) = ϕ : a,b Ω R ϕ( ) i coninuou, F-adaped, E up ϕ() 2 } <, a b L F {ϕ( ) (Ω;Ca,b) = L 2 F (Ω;Ca,b) } upa b ϕ() L F b (Ω), { L F b (Ω;C U a,b) = ϕ( ) L F b (a,b) here exi a modulu of coninuiy ρ :, ), ) } uch ha ϕ() ϕ() ρ( ), (,) a,b, a.., L 2 F {ϕ ( a,b) = : a,b Ω R ϕ(, ) i F-progreively meaurable on,b, a.e. a,b, E b } b a ϕ(,) 2 dd <, { L 2 F (a,b2 ) = ϕ :a,b 2 Ω R ϕ(, ) i F-progreively meaurable on a,b, a.e. a,b, E b b a a ϕ(,) 2 dd < }, H 2 a,b = L 2 F (a,b) L2 F (a,b2 ). Now, we recall he definiion of adaped oluion and adaped M-oluion for BSVIE (1.1) inroduced in 47. Definiion 2.1. A pair of procee (Y( ),Z(, )) H 2,T i called an adaped oluion of BSVIE (1.1) if (1.1) i aified in he uual Iô ene for Lebegue meaure almo every,t. Definiion 2.2. For any r,t), a pair of procee (Y( ),Z(, )) H 2 r,t i called an adaped M-oluion of BSVIE (1.1) on r,t if (1.1) i aified in he uual Iô ene for Lebegue meaure almo every r,t and, in addiion, he following hold: Hereafer, we recall E r = F r. Y() = E r Y()+ r Z(,)dW(), a.e. r,t. (2.1) 5

6 Le M 2 r,t be he e of all (y( ),z(, )) H 2 r,t aifying (2.1). Clearly, M 2 r,t i a cloed ubpace of H 2 r,t. Furher, for any (y( ),z(, )) M 2 r,t, we have I follow ha E y() 2 = E Er y() 2 +E (y( ),z(, )) 2 H 2 r,t E r r z(,) 2 d E y() 2 d+ = E y() 2 d+ E 2 r r y() 2 d+2 r r r r r r z(,) 2 d, a.e. r,t. z(,) 2 dd z(,) 2 dd+ z(,) 2 dd 2 (y( ),z(, )) 2 M 2 r,t 2 (y( ),z(, )) 2 H 2 r,t, which implie ha M 2 r,t i an equivalen norm of H 2 r,t on M 2 r,t. r z(,) 2 dd Nex, we recall he Giranov heorem, which will play an imporan role in our ubequen analyi. We refer he reader o Karaza Shreve 22 for a proof. Le X = {X,F ; T} be a meaurable, adaped proce aifying We e P X() 2 d < = 1. E{X} e X()dW() 1 2 X() 2d,,T, (2.2) which i called he Doléan-Dade exponenial of X. Define a probabiliy meaure P on F T by Then, he Giranov heorem can be aed a follow. dp = E{X} T dp. (2.3) Lemma 2.3. Aume ha E{X} defined by (2.2) i a maringale, hen he proce W = {W(),F T} defined by W() W() i a andard Brownian moion on (Ω,F T,P). X()d, T (2.4) In order o give a ufficien condiion under which E{X} defined by (2.2) i a maringale, we need he following reul. We refer he reader o Kazamaki 23 for he deail. Definiion 2.4. Le M = {M(),F ; T}be auniformly inegrablemaringalewih M() =. Proce M( ) i called a BMO maringale on,t, if M( ) 2 BMO(,T) up E τ M(T) M(τ) 2 <, τ T,T where T,T i he e of all opping ime τ valued in,t. Someime, he norm BMO(,T) i wrien a BMOP(,T), indicaing he dependence on he probabiliy P. Lemma 2.5. If M( ) i a BMO maringale on,t, hen Ē{M} e M() 1 2 M () ; T i a uniformly inegrable maringale, where M ( ) i he quadraic variaion proce of M( ). 6

7 Remark 2.6. By Lemma 2.3, if X()dW(); T i a BMO maringale on,t, hen E{X} defined by (2.2) i a uniformly inegrable maringale. Thu, by Giranov heorem (Lemma 2.3), W( ) defined by (2.4) i a one-dimenional Brownian moion on (Ω,F T,P). Nex, we inroduce he following pace: Le a < b < c T, and { BMO(a,b) = ϕ : a,b Ω R ϕ( ) L 2 F (a,b), BMO( a, b) = ϕ( ) 2 BMO(a,b) up τ T a,b b E τ { ϕ : a,b Ω R ϕ(, ) L 2 F ( a,b), ϕ(, ) 2 BMO( a,b ϕ(, ) 2 BMO(a,b b,c) ) eup a,b eup a,b τ up τ T,b } ϕ() d 2 <, τ T b,c b E τ BMO ( a,b b,c ) { = ϕ : a,b b,c Ω R ϕ(, ) L 2 F (a,b b,c), c up E τ τ } ϕ(,) d 2 <, τ } ϕ(,) d 2 <. We noe ha for ϕ( ) BMO(a,b), if we le ϕ(),,a), hen ϕ(r)dw(r); b i a BMO maringale on,b. Similarly, for ϕ(, ) BMO( a,b), if we le ϕ(,),,), hen ϕ(,r)dw(r); b i a BMO maringale on,b for almo all a,b). The iuaion for BMO ( a,b b,c ) i alo imilar. The following lemma play a baic role in our ubequen argumen. we refer he reader o 23, Theorem 3.3 for he proof and deail. Lemma 2.7. For K >, here are conan c 1,c 2 > depending only on K uch ha for any BMO maringale M( ), we have for any one-dimenional BMO maringale N( ) uch ha N( ) BMO(,T) K, c 1 M( ) BMOP (,T) M( ) BMOP (,T) c 2 M( ) BMOP (,T), where M( ) M( ) M,N ( ) and dp = Ē{N( )} T dp. We now conider he following BSDE: Y() = ξ + Le u inroduce he following hypohei. f(,y(),z())d Z()dW(),,T. (2.5) (A). Le he generaor f :,T R R Ω R be B(,T R R) F T -meaurable uch ha f(,y,z) i F-progreively meaurable for all (y,z) R R. There exi conan β, γ, L and a funcion h( ) L 1 (,T) uch ha f(,y,z) h()+β y + γ 2 z 2, (,y,z),t R R; (2.6) f(,y 1,z 1 ) f(,y 2,z 2 ) L y 1 y 2 +L(1+ z 1 + z 2 ) z 1 z 2, (,y i,z i ),T R R, i = 1,2. (2.7) Lemma 2.8. Le (A) hold. Then, for any ξ L F T (Ω), BSDE (2.5) admi a unique adaped oluion (Y( ),Z( )) L F (Ω; C,T) BMO(,T). Moreover, e γ Y() E e γeβ(t ) ξ +γ T h() eβ( ) d. (2.8) Proof. By 5, Theorem 7.3.3, BSDE (2.5) admi a unique adaped oluion (Y( ), Z( )) L F (Ω;C,T) L2 F (,T). Then, by 5, Theorem 7.2.1, we ee ha he adaped oluion (Y( ), Z( )) L F (Ω;C,T) BMO(,T). Furher, by 8, Propoiion 1, we have inequaliy (2.8). 7

8 3 Adaped Soluion o Type-I BSVIE In hi ecion, we will eablih he exience and uniquene of he adaped oluion o Type-I BSVIE. Fir, le u look a he following imple example. Example 3.1. Conider he one-dimenional BSVIE: Y() = ψ()+ Z(,) 2 d Z(, )dw(), (3.1) where ψ( ) L F T (,T), and W( ) i a one-dimenional andard Brownian moion. In order o olve equaion (3.1), we inroduce a family of BSDE parameerized by,t: η(,) = ψ()+ ζ(,r) 2 dr ζ(,r)dw(r),,t. (3.2) By Lemma 2.8, BSDE (3.2) admi a unique adaped oluion (η(, ),ζ(, )) L F (Ω;C,T) BMO(,T). Le Y() = η(,) and Z(,) = ζ(,), (,),T, hen Y() = ψ()+ Z(,) 2 d Z(,)dW(),,T, which implie ha (Y( ),Z(, )) i an adaped oluion o BSVIE (3.1). The uniquene of he oluion o BSVIE (3.1) can be obained by he following Theorem 3.2. From he above example, we ee ha BSVIE (3.1) can be fully characerized by a family of BSDE (3.2). The main reaon i ha he generaor of equaion (3.1) i independen of y. Thi ugge u fir conider a pecial cae of Type-I BSVIE (1.2). 3.1 A pecial cae Conider he following BSVIE: Y() = ψ()+ g(,,z(,))d Z(, )dw(), (3.3) where he generaor g :,T R Ω R and he free erm ψ :,T Ω R are given map. We adop he following aumpion concerning g( ), which i comparable wih (A). (A1). Le he generaor g :,T R Ω R be B(,T R) F T -meaurable uch ha g(,,z) i F-progreively meaurable on,t, for all (,z),t) R. There exi wo conan γ, L and a funcion h( ) L 1 (,T;R) uch ha g(,,z) h()+ γ 2 z 2, (,,z),t R; g(,,z 1 ) g(,,z 2 ) L(1+ z 1 + z 2 ) z 1 z 2, (,,z i ),T R, i = 1,2. Now, we ae he following exience and uniquene reul of BSVIE (3.3). Theorem 3.2. Le (A1) hold. Then for any ψ( ) L F T (,T), Type-I BSVIE (3.3) admi a unique adaped oluion (Y( ),Z(, )) L F (,T) BMO(,T). 8

9 Proof. We fir how he exience of he adaped oluion o BSVIE (3.3). Conider he following BSDE parameerized by,t: η(,) = ψ()+ g(,r,ζ(,r))dr ζ(,r)dw(r),,t. (3.4) For almo all, T, by Lemma 2.8, under (A1), BSDE (3.4) admi a unique adaped oluion (η(, ),ζ(, )) L F (Ω;C,T) BMO(,T). Le Y() = η(,), Z(,) = ζ(,), (,),T, hen (Y( ),Z(, )) L F (,T) BMO(,T) and Y() = ψ()+ g(,,z(,))d Z(,)dW(),,T, which implie ha (Y( ),Z(, )) i an adaped oluion for BSVIE (3.3). The uniquene will follow from he nex heorem. Conider he following BSVIE: For i = 1,2, Y i () = ψ i ()+ g i (,,Z i (,))d Z i (,)dw(),,t. (3.5) We have he following comparion heorem. L F Theorem 3.3. Le g 1 ( ) and g 2 ( ) aify (A1), ψ 1 ( ),ψ 2 ( ) L F T (,T). Le (Y i ( ),Z i (, )) (,T) BMO(,T) be he adaped oluion of correponding BSVIE (3.5). Suppoe ψ 1 () ψ 2 (), g 1 (,,z) g 2 (,,z), a.., a.e. (,,z),t R, (3.6) hen we have Y 1 () Y 2 (), a.., a.e.,t. (3.7) In paricular, if g 1 ( ) = g 2 ( ) and ψ 1 ( ) = ψ 2 ( ), he comparion implie he uniquene of adaped oluion o Type-I BSVIE (3.3). Proof. We noe ha Y 1 () Y 2 () = ψ 1 () ψ 2 ()+ Define he proce θ(, ) uch ha g 1 (,,Z 1 (,)) g 2 (,,Z 2 (,))d Z 1 (,) Z 2 (,)dw(). (3.8) θ(,) =, (,),T; (3.9) θ(,) C(1+ Z 1 (,) + Z 2 (,) ), (,),T; (3.1) g 1 (,,Z 1 (,)) g 1 (,,Z 2 (,)) = Z 1 (,) Z 2 (,) θ(,), (,),T. (3.11) Hereafer, C > and for a generic conan which could be differen from line o line. Then, for almo all,t, W(; ) defined by W(;) W() 9 θ(,r)dr,,t (3.12)

10 i a Brownian moion on,t under he equivalen probabiliy meaure P defined by dp E{θ(, )} T dp. The correponding expecaion i denoed by E P. Thu, by (3.8) and (3.12), we have Y 1 () Y 2 () = ψ 1 () ψ 2 ()+ Z 1 (,) Z 2 (,)dw(;). g 1 (,,Z 2 (,)) g 2 (,,Z 2 (,))d Taking he condiional expecaion wih repec o P on he boh ide of he above equaion and hen by (3.6), we have Y 1 () Y 2 () = E P ψ 1 () ψ 2 ()+ Hence, (3.7) follow. g 1 (,,Z 2 (,)) g 2 (,,Z 2 (,))d, a.. Remark 3.4. Theorem 3.2 and Theorem 3.3 are boh concerned wih he BSVIE (3.3), a very pecial cae of Type-I BSVIE (1.2), in which, he generaor g( ) i independen of he variable y. Thi make he BSVIE (3.3) much eaier o handle. Even hough, Theorem 3.2 and Theorem 3.3 erve a a crucial bridge o he proof of he reul for general Type-I BSVIE. 3.2 The general cae In hi ubecion, we will conider he following Type-I BSVIE: Y() = ψ()+ g(,,y(),z(,))d We fir inroduce he following aumpion, which i alo comparable o (A). Z(, )dw(). (3.13) (A2). Le he generaor g :,T R R Ω R be B(,T R R) F T -meaurable uch ha g(,,y,z) i F-progreively meaurable on,t for all (,y,z),t R R. There exi wo conan L and γ uch ha: g(,,y,z) L(1+ y )+ γ 2 z 2, (,,y,z),t R R; g(,,y 1,z 1 ) g(,,y 2,z 2 ) L { y 1 y 2 +(1+ z 1 + z 2 ) z 1 z 2 }, Now, we ae he main reul of hi ubecion. (,,y i,z i ),T R R, i = 1,2. Theorem 3.5. Le (A2) hold. Then for any ψ( ) L F T (,T), Type-I BSVIE (3.13) admi a unique adaped oluion (Y( ),Z(, )) L F (,T) BMO(,T). L F We will prove Theorem 3.5 by mean of conracion mapping heorem. For any (U( ), V(, )) (,T) BMO(,T), conider he following BSVIE: Y() = ψ()+ g(,,u(),z(,))d Z(, )dw(). (3.14) By Theorem 3.2, BSVIE (3.14) admi a unique adaped oluion (Y( ),Z(, )) L F (,T) BMO(,T). Thu, he map Γ(U( ),V(, )) (Y( ),Z(, )), (U( ),V(, )) L F (,T) BMO(,T) (3.15) i well-defined. In order o prove Theorem 3.5, we preen he following lemma. 1

11 Lemma 3.6. Le (A2) hold and ε (, 1 2L. Then for any ψ( ) L F T (,T), he map Γ(, ) defined by (3.15) aifie he following: Γ(B ε ) B ε, (3.16) where B ε i defined by he following: B ε { (U( ),V(, )) L F (T ε,t) BMO( T ε,t) U( ) L F (,T) 2 ψ( ) +1, V(, ) 2 BMO(,T) A }, (3.17) wih A = 2 γ 2eγ ψ( ) + 1 γ e2(γ+1) ψ( ) +γ+2. Proof. For any (U( ),V(, )) B ε, conider a family of BSDE (parameerized by,t): η(,) = ψ()+ g(,r,u(r),ζ(,r))dr ζ(,r)dw(r),,t. (3.18) Noe ha U( ) i bounded. For almo all T ε,t, by Lemma 2.8, he above BSDE admi a unique adaped oluion (η(, ),ζ(, )) L F (Ω;C,T) BMO(,T). Le Y() = η(,), Z(,) = ζ(,), (,) T ε,t. (3.19) Then by Theorem 3.2, (Y( ),Z(, )) L F (,T) BMO(,T) i he unique adaped oluiono BSVIE (3.14). The re of he proof i divided ino hree ep. Sep 1: Eimae of Y( ). For BSDE (3.18), by (A2), we have g(,r,u(r),ζ) L ( 1+ U(r) ) + γ 2 ζ 2. 1 Thu, noe ha ε (, 2L, by Lemma 2.8 wih h() = L(1+ U() ), γ = γ and β =, we have ( e γ η(,) E e γ ψ() +L ) ( ) T (1+ U(r) )dr e γ ψ( ) +Lε 1+ U( ) L (,T) F e γ(2 ψ( ) +1), T ε T, (3.2) which i equivalen o η(,) 2 ψ( ) +1, T ε T. (3.21) Conequenly, noing Y() = η(,), one ha Sep 2: Eimae of Z(, ) 2 BMO(,T). Y( ) L F (,T) 2 ψ( ) +1. Define Then, we have φ(y) γ 2( e γ y γ y 1 ) γ n 2 y n = ; y R. (3.22) n! n=2 φ (y) = φ (y) = n=2 n=1 γ n 2 y n 2 y (n 1)! = n=1 γ n 1 y n 1 y n! γ n 1 ( ) y n 1 +(n 1) y n 2 ygn(y) n! = γ 1 e γ y 1gn(y), = n=1 γ n 1 y n 1 (n 1)! = e γ y, (3.23) 11

12 which lead o φ (y) = γ φ (y) +1. Applying Iô formula o φ(η(,)), we have φ(ψ()) φ(η(,)) = + φ (η(,r))g(,r,u(r),ζ(,r))dr φ (η(,r))ζ(,r)dw(r),,t. Taking condiional expecaion on he boh ide of (3.24) and by (A2), we have φ(η(,))+ 1 2 E φ( ψ( ) )+LE φ (η(,r)) ζ(,r) 2 dr Combining hi wih (3.23), one obain φ(η(,))+ 1 2 E φ (η(,r)) ζ(,r) 2 dr (3.24) φ (η(,r)) ( 1+ U(r) ) dr + γ 2 E φ (η(,r)) ζ(,r) 2 dr. ζ(,r) 2 T dr φ( ψ( ) )+LE φ (η(,r)) (1+ U(r) )dr. (3.25) Then, noing ha φ(η(,)), we imply drop i o ge Hence, E Z(,r) 2 T dr 2φ( ψ( ) )+2LE φ (η(,r)) (1+ U(r) )dr 2 2eγ ψ( ) + 2L εeγ(2 ψ( ) +1) e 2( ψ( ) +1) 2 2eγ ψ( ) + 1 e2(γ+1) ψ( ) +γ+2. γ γ γ γ Thi prove our claim. Z(, ) 2 BMO(,T) 2 γ 2eγ ψ( ) + 1 γ e2(γ+1) ψ( ) +γ+2 = A. (3.26) The nex reul i concerned wih he local oluion of BSVIE (3.13). Propoiion 3.7. Le (A2) hold and he map Γ(, ) be defined by (3.15). Then here i ε > uch ha Γ(, ) i a conracion on B ε, where B ε i defined by (3.17). Thi implie ha BSVIE (3.13) admi a unique adaped oluion on T ε,t. Proof. Le ε (, 1 2L. For any (U( ),V(, )),(Ũ( ),Ṽ(, )) B ε, e ha i, (Y( ),Z(, )) = Γ(U( ),V(, )) and (Ỹ( ), Z( )) = Γ(Ũ( ),Ṽ(, )); (3.27) and η(,) = ψ()+ η(,) = ψ()+ g(,r,u(r),ζ(,r))d g(,r,ũ(r), ζ(,r))dr ζ(, r)dw(r), (3.28) ζ(, r)dw(r), (3.29) Y() = η(,), Ỹ() = η(,), Z(,r) = ζ(,r), Z(,r) = ζ(,r). (3.3) By Lemma 3.6, (Y( ),Z(, )) and (Ỹ( ), Z(, )) B ε. By (A2), for almo all,t, we can define he proce θ(, ) in an obviou way uch ha: θ(,) =, (,) T ε,t,, (3.31) 12

13 θ(,) L(1+ ζ(,) + ζ(,) ), (,) T ε,t, (3.32) g(,,ũ(),ζ(,)) g(,,ũ(), ζ(,)) = ζ(,) ζ(,)θ(,). (3.33) Noe ha (Y( ),ζ(, )),(Ỹ ( ), ζ(, )) B ε. Thu, by (3.31) (3.32), θ(, ) 2 BMO(,T) 3L2 T +3L 2 ζ(, ) 2 BMO(,T) +3L2 ζ(, ) 2 BMO(,T) 3L 2 T +6L 2 A. (3.34) Thu, for almo all T ε,t, θ(,r)dw(r); T i a BMO maringale and 2 θ(, r)dw(r) 3L 2 T +6L 2 A. (3.35) By Lemma 2.3 and Lemma 2.5, W(; ) defined by W(;) W() BMO(,T) θ(,r)dr,,t (3.36) i a Brownian moion on,t under he equivalen probabiliy meaure P, which i defined by dp E{θ(, )} T dp. (3.37) Denoe he expecaion in P by E P. Combining (3.28), (3.29), and (3.33) (3.36), we have η(,) η(,)+ = ζ(,r) ζ(,r)dw(;r) g(,r,u(r),ζ(,r)) g(,r,ũ(r),ζ(,r)) dr. (3.38) Taking quare and hen aking condiional expecaion wih repec o P on he boh ide of he above equaion, we have (noing T ε T) η(,) η(,) 2 T +E P { = E P E P { ζ(,r) ζ(,r) 2 dr (g(,r,u(r),ζ(,r)) g(,r,ũ(r),ζ(,r)) ) dr (L U(r) Ũ(r) ) 2 } dr 2 } L 2 (T ) 2 U( ) Ũ( ) 2 L F (,T) L2 ε 2 U( ) Ũ( ) 2 L F (,T). Le =, by (3.3) and (3.39), we have (3.39) Y( ) Ỹ( ) 2 L F (,T) L2 ε 2 U( ) Ũ( ) 2 L F (,T). (3.4) Alo, by (3.3), (3.39), (3.35), and Lemma 2.7, here i a conan C (which i depending on ψ( ) and i independen of ) uch ha Thu, up E Z(,r) Z(,r) 2 dr = up E ζ(,r) ζ(,r) 2 dr,t,t C up E P ζ(,r) ζ(,r) 2 dr CL 2 ε 2 U( ) Ũ( ) 2 L F,T Z(, ) Z(, ) 2 BMO(,T) CL2 ε 2 U( ) Ũ( ) 2 L F (,T). (3.41) (,T). (3.42) Combining (3.4) (3.42), we ee ha for ome mall ε >, he map Γ(, ) i a conracion on he e B ε. Hence, BSVIE (3.13) admi a unique adaped oluion on T ε,t. 13

14 Le u make ome commen on he above local exience of he unique adaped oluion. T 2 1,T ε, (Figure 1) T We have een ha (Y(),Z(,)) i defined for (,) T ε,t, he region marked 1 in he above figure. Now, for any,t ε, we can rewrie our Type-I BSVIE a follow: where Y() = ψ ()+ ε g(,,y(),z(,))d ε Z(,)dW(),,T ε, (3.43) ψ () = ψ()+ g(,,y(),z(,))d Z(,)dW(),,T ε. (3.44) If ψ ( ) L F (,T ε), hen (3.43) i a BSVIE on,t ε. However, unlike BSDE, having (Y(),Z(,)) defined on T ε,t, ψ ();,T ε ha ill no been defined ye. Since, on he righ-hand ide of (3.44), alhough Y() wih T ε,t ha already been deermined, Z(,) ha no been defined for (,),,T, he region marked 2 in he above figure, which i needed o define ψ (). Moreover, we need ha ψ () i F -meaurable (no ju F T -meaurable). Hence, (3.44) i acually a ochaic Fredholm inegral equaion (SFIE, for hor) o be olved o deermine ψ ();,T ε. Now, we are a he poiion o prove Theorem 3.5. Proof of Theorem 3.5. The proof will be divided ino hree ep. Sep 1: Eimae of Y( ) 2. For given ψ( ) L F T (,T), we can find a conan C > uch ha ψ( ) 2 C and (by (A2)) 2xg(,,y,) C + C x 2 + C y 2, (,,x,y),t R R. (3.45) Le u conider he following (inegral form of) ordinary differenial equaion: α() = C + Cα()d + Cα()+1d,,T. (3.46) I i eay o ee ha he unique oluion o he above ordinary differenial equaion i given by 1 ) α() = ( C + e 2 C(T ) 1 2 2,,T, which i a (coninuou) decreaing funcion. Thu, ψ( ) 2 C = α(t) α(). 14

15 By Propoiion 3.7, here exi an ε > (depending on ψ( ) ) uch ha Γ(, ) defined by (3.15) i a conracion on B ε. Therefore, a Picard ieraion equence converge o he unique adaped oluion (Y( ),Z(, )) of he BSVIE on T ε,t. Namely, if we define: { (Y ( ),Z (, )) =, (3.47) (Y k+1 ( ),Z k+1 (, )) = Γ(Y k ( ),Z k (, )), k ; ha i, hen (Y ( ),Z (, )) =, η k+1 (,) = ψ()+ g(,r,y k (r),ζ k+1 (,r))dr ζ k+1 (,r)dw(r), Y k+1 () = η k+1 (,), Z k+1 (,) = ζ k+1 (,), (,) T ε,t, lim (Y k ( ),Z k (, )) (Y( ),Z(, )) k L F (,T) BMO(,T) =. (3.48) Nex, for almo all T ε,t, imilar o (3.32), (3.33), (3.36), and (3.37), here exi a proce θ k+1 (, ) uch ha and g(,r,y k (r),ζ k+1 (,r)) g(,r,y k (r),) = ζ k+1 (,r)θ k+1 (,r), (3.49) W k+1 (;) W() θ k+1 (,r)dr,,t (3.5) i a Brownian moion on,t under he correponding equivalen probabiliy meaure P k+1 P k+1 = E{θ k+1 (, )} T dp. defined by For impliciy, we denoe P k+1 by P k+1 here, uppreing he ubcrip. The correponding expecaion i denoed by E k+1. I follow ha η k+1 (,) = ψ()+ g(,r,y k (r),ζ k+1 (,r))dr ζ k+1 (,r)dw(r), = ψ()+ g(,r,y k (r),)dr ζ k+1 (,r)dw k+1 (;r). (3.51) Applying he Iô formula o he map η k+1 (,) 2 and aking condiional expecaion E k+1 τ = E k+1 F τ for any τ T ε,, by (3.45), we have E k+1 τ η k+1 (,) 2 +E k+1 τ = E k+1 τ ψ() 2 +E k+1 τ C + C ζ k+1 (,r) 2 dr E k+1 τ η k+1 (,r) 2 dr + C We now prove he following inequaliy by inducion: 2η k+1 (,r)g(,r,y k (r),)dr { E k+1 τ Y k (r) 2 } +1 dr. (3.52) Y k () 2 α(), T ε,t, for any k. (3.53) In fac, by (3.47), i i obviou o ee Y () 2 = α(). Suppoe Y k () 2 α() for any T ε,t, hen E k+1 τ η k+1 (,) 2 C + C E k+1 τ η k+1 (,r) 2 dr+ C α(r) + 1dr. (3.54) 15

16 In ligh of (3.46), by he comparion heorem of ordinary differenial equaion, we have E k+1 τ η k+1 (,) 2 α(). (3.55) Le τ = and =, we have Y k+1 () 2 α(), T ε,t. (3.56) Thu, by inducion, (3.53) hold. Then by (3.48), we have Y() 2 α(), T ε,t. (3.57) Sep 2: A relaed ochaic Fredholm inegral equaion i olvable on,t ε. We now olve SFIE (3.44) on,t ε. Le u inroduce a family of BSDE parameerized by,t ε: η(,) = ψ()+ g(,r,y(r),ζ(,r))dr ζ(,r)dw(r), T ε,t. (3.58) By Lemma 2.8, he above BSDE admi a unique adaped oluion (η(, ),ζ(, )) on T ε,t. Noe ha (3.57), imilar o (3.55), we have Similar o (3.26), we have η(,) 2 α(), T ε,t. (3.59) eup ζ(, ) 2 <. (3.6) BMO(,T),T ε Le ψ () = η(,t ε) and Z(,) = ζ(,), we have (ψ ( ),Z(, )) L F (,) BMO(,T ε T ε,t) and (ψ ( ),Z(, )) i a oluion o SFIE (3.44). Moreover, by (3.59), we have ψ () 2 = η(,t ε) 2 α(t ε) α(),,t ε. (3.61) Nex, we will prove he oluion o SFIE (3.44) i unique. Le (ψ ( ),Z(, )), ( ψ ( ), Z(, )) L F (,T ε) BMO(,T ε T ε,t). be wo oluion o SFIE (3.44). Then ψ () ψ () = g(,,y(),z(,)) g(,,y(), Z(,)) d (3.62) Z(,) Z(,) dw(),,t ε. For almo all,t ε, imilar o (3.32), (3.33), (3.36), and (3.37), here i a proce θ(, ) uch ha: g(,,y(),z(,)) g(,,y(), Z(,)) = Z(,) Z(,) θ(,), (3.63) and W(;) W() θ(,r)dr,,t (3.64) i a Brownian moion on,t under he correponding equivalen probabiliy meaure P. The correponding expecaion i denoed by E P. Combining (3.62) (3.64), we have ψ () ψ () = Z(,) Z(,) dw(;),,t ε. (3.65) 16

17 Taking condiional expecaion E P E P F on he boh ide of he equaion (3.65), we have ψ () ψ () =,,T ε. (3.66) E P Noe ha ψ () i F -adaped for any,t ε. I follow ha By (3.65) (3.67), we have which implie ψ () = ψ (), a..,,t ε. (3.67) Z(,) Z(,) dw(;) =,,T ε, (3.68) Z(,) = Z(,), a.., (,),T ε T ε,t. (3.69) Combining (3.67) (3.69), SFIE (3.44) admi a unique oluion. Sep 3: Complee he proof by inducion. Combining Sep 1 and 2, we have uniquely deermined Y(), T ε,t, Z(,), (,) T ε,t ) (,T ε T ε,t. (3.7) Now, we conider BSVIE (3.43) on,. By (3.61), we ee ha he above procedure can be repeaed. We poin ou ha he inroducion of α( ) i o uniformly conrol he erminal ae ψ(t ε), ψ(t 2ε), ec. Then we can ue inducion o finih he proof of he exience and uniquene of adaped oluion o BSVIE (3.13). We now would like o look ome beer regulariy for he adaped oluion of BSVIE under addiional condiion. More preciely, we inroduce he following aumpion. (A3). Le g :,T 2 R R Ω R be meaurable uch ha for every (,y,z),t R R, g(,,y,z) i F-progreively meaurable. There exi a modulu of coninuiy ρ :, ), ) (a coninuou and monoone increaing funcion wih ρ() = ) uch ha g(,,y,z) g(,,y,z) ρ( ),,,,T, (y,z) R R. Noe ha in (A3), he generaor g(,,y,z) i defined for (,) in he quare domain,t 2 inead of he riangle domain,t. Theorem 3.8. Le (A2) (A3) hold. Then for any ψ( ) L F T (Ω;C U,T), BSVIE (3.13) admi a unique adaped oluion (Y( ),Z(, )) L F (Ω;C,T) BMO(,T). Proof. Wihou lo of generaliy, le u aume ha ψ( ) ψ() ρ( ),,,T, wih he ame modulu of coninuiy ρ( ) given in (A3). By Theorem 3.5, BSVIE (3.13) admi a unique adaped oluion (Y( ),Z(, )) L F (,T) BMO(,T). We ju need o prove ha Y( ) L F (Ω;C,T), i.e., Y( ) i coninuou. Conider he following family of BSDE (parameerized by,t): η(,) = ψ()+ g(,r,y(r),ζ(,r))dr ζ(,r)dw(r),,t. (3.71) 17

18 By Lemma 2.8, for any,t, BSDE (3.71) admi a unique adaped oluion (η(, ),ζ(, )) L F (Ω;C,T) BMO(,T). By Theorem 3.5, we have Y() = η(,), Z(,) = ζ(,) for any (,),T. Now, le < T. Similar o (3.32), (3.33), (3.36), and (3.37), here i a proce θ(, ; ) uch ha g(,,y(),ζ(,)) g(,,y(),ζ(,)) = ζ(,) ζ(,)θ(, ;), (3.72) and W(, ;) W() θ(, ;r)dr,,t (3.73) i a Brownian moion on,t under he correponding equivalen probabiliy meaure P,. The correponding expecaion i denoed by E P,. Combining (3.71), (3.72), and (3.73), we have η(,) η(,) = ψ() ψ( ) + ζ(,r) ζ(,r)dw(, ;r) g(,r,y(r),ζ(,r)) g(,r,y(r),ζ(,r))dr. Taking condiional expecaion E P, E P, F on he boh ide of he above equaion, we have ( ) η(,) η(,) = E P, ψ() ψ( )+ g(,r,y(r),ζ(,r)) g(,r,y(r),ζ(,r)) dr. Combining hi wih (A3), we have η(,) η(,) E P, ψ() ψ( ) + g(,r,y(r),ζ(,r)) g(,r,y(r),ζ(,r)) dr (T +1)ρ( ). Thi lead o lim up,t η(,) η(,) =, a.. On he oher hand, ince η(, ) L F (Ω;C,T) for any,t, one ha I follow ha (,) η(,) i coninuou, i.e., Conequenly, η(,) = Y() i coninuou. lim η(,) η(, ) =,,T, a.. (3.74) lim (, ) (,) η(, ) η(,) =, (,),T 2, a.. 4 Adaped M-oluion o Type-II BSVIE We now conider he following one-dimenional Type-II BSVIE: Y() = ψ()+ g(,,y(),z(,),z(,))d Z(,)dW(),,T. (4.1) Since Z(, ) i preened in he generaor g( ), we hall conider he adaped M-oluion. Le u fir inroduce he following aumpion: 18

19 (A4). Le he generaor g :,T R R R Ω R be B(,T R R R) F T -meaurable uch ha g(,,y,z,z ) i F-progreively meaurable on,t for all (,y,z,z ),T R R R. There exi wo conan L and γ uch ha: g(,,y,z,z ) L(1+ y )+ γ 2 z 2, (,,y,z,z ),T R R R; g(,,y 1,z 1,z 1 ) g(,,y 2,z 2,,z 2 ) L ( y 1 y 2 +(1+ z 1 + z 2 ) z 1 z 2 + z 1 z 2 ), (,,y i,z i,z i),t R R R, i = 1,2. Noe ha in (A4), we have aumed ha z g(,,y,z,z ) i bounded. Thi will allow u o ue he reul for Type-I BSVIE. The main reul of hi ecion i he following. Theorem 4.1. Le (A4) hold. Then for any ψ( ) L F T (,T), Type-II BSVIE (4.1) admi a unique adaped M-oluion (Y( ),Z(, )) M 2,T ( L F (,T) BMO(,T)). Proof. For any (y( ),z(, )) M 2,T, conider he following BSVIE: Y() = ψ()+ g(,,y(),z(,),z(,))d Z(,)dW(),,T. (4.2) In ligh of (A4), by Theorem 3.5, BSVIE (4.2) admi a unique adaped oluion (Y( ),Z(, )) L F (,T) BMO(,T). Deermine Z(,);(,),T by maringale repreenaion heorem, i.e., Y() = EY()+ Z(,)dW(),,T. Thi mean ha BSVIE (4.2) admi a unique adaped M-oluion (Y( ),Z(, )) M 2,T. Thu he map Γ(y( ),z(, )) (Y( ),Z(, )), (y( ),z(, )) M 2 (,T) (4.3) i well-defined. In order o prove BSVIE (4.1) admi a unique adaped M-oluion, we need o prove ha Γ(, ) ha a fixed poin in M 2,T. The proof i divided ino wo ep. Sep 1. There i an ε > uch ha Γ(, ) i a conracion on M 2 T ε,t and hence BSVIE (4.1) admi a unique adaped M-oluion on T ε,t. For any (y( ),z(, )),(ỹ( ), z(, )) M 2 T ε,t, wih ε > undeermined, e ha i, for T ε,t, and (Y( ),Z(, )) = Γ(y( ),z(, )), (Ỹ( ), Z(, )) = Γ(ỹ( ), z(, )); (4.4) Y() = ψ()+ Ỹ() = ψ()+ g(,,y(),z(,),z(,))d g(,,ỹ(), Z(,), z(,))d Y() = EY() F + Ỹ() = EỸ() F + Z(, )dw(), (4.5) Z(, )dw(), (4.6) Z(,)dW(), T ε,t, (4.7) Z(,)dW(), T ε,t. (4.8) Similar o Lemma 3.6, noing ha z g(,,y,z,z ) i bounded, here i an ε > uch ha Γ(y( ),z(, )) B ε for any (y( ),z(, )) M 2 (T ε,t), where B ε i defined by (3.17). Thu, we have (Y( ),Z(, )), (Ỹ( ), Z(, )) B ε. (4.9) 19

20 By (A4), for any T ε,t, here i a proce θ(, ) uch ha: Similar o (3.35), we have θ(,) =, T ε,t,,, (4.1) θ(,) L(1+ Z(,) + Z(,) ), (,) T ε,t, (4.11) g(,,ỹ(),z(,), z(,)) g(,,ỹ(), Z(,), z(,)) = Z(,) Z(,)θ(,), (,) T ε,t. (4.12) θ(, ) 2 BMO(,T) 3L2 T +6L 2 A. (4.13) For almo all T ε,t, by Lemma 2.3 and Lemma 2.5, W(; ) defined by W(;) W() θ(,r)dr,,t (4.14) i a Brownian moion on,t under he equivalen probabiliy meaure P, which i defined by dp E{θ(, )} T dp. (4.15) The correponding expecaion i denoed by E P. Combining (4.5) (4.6) and (4.12) (4.14), we have Y() Ỹ()+ Z(,) Z(,)dW(,) = g(,,y(),z(,),z(,)) g(,,ỹ(),z(,), z(,)) d. (4.16) Taking quare and hen aking he condiional expecaion E P = E P F, we have Y() Ỹ() 2 +E P = E P L 2 E P Z(,) Z(,) 2 d (g(,,y(),z(,),z(,)) g(,,ỹ(),z(,), z(,)) ) d ( Y() Ỹ() + z(,) z(,) ) d 2. (4.17) By (Y( ),Z(, )),(Ỹ( ), Z(, )) B ε and Lemma 2.7, here i a conan C > (which i depending on ψ( ) and i independen of ) uch ha Y() Ỹ() 2 +E CE C(T )E Thu, inegraing he above on T ε,t, we obain Z(,) Z(,) 2 d ( Y() Ỹ() + z(,) z(,) ) d 2 ( Y() Ỹ() 2 + z(,) z(,) 2) d. (4.18) E Y() Ỹ() 2 d+e Z(,) Z(,) 2 dd CεE Y() Ỹ() 2 + z(,) z(,) 2 dd, (4.19) wih a poible differen conan C >. By he variaion of conan formula, we obain E Y() Ỹ() 2 d+e 2 Z(,) Z(,) 2 dd 2

21 CεE z(,) z(,) 2 dd CεE y() ỹ() 2 d. (4.2) The conan appear above i generic (only depend on he conan L, γ, T, and ψ( ), and i independen of ε > ). Therefore, when ε i mall enough, Γ(, ) i a conracion on M 2 (T ε,t). Conequenly, BSVIE (4.1) admi a unique adaped oluion on T ε,t. Furher, by (4.9), he unique adaped M-oluion (Y( ),Z(, )) alo belong o L F (T ε,t) BMO( T ε,t). T 2,T ε, (Figure 2) T The above deermined Y() for T ε,t and deermined Z(,) for (,) T ε,t (he region marked 1 in he above figure) by uing Type-I BSVIE, and for (,) T ε,t (he region marked 3 in he above figure) by uing maringale repreenaion. Sep 2. BSVIE (4.1) admi a unique adaped M-oluion on,t. By Sep 1, BSVIE (4.1) admi a unique oluion on T ε,t. For almo every T ε,t, E Y() L 2 F (Ω), by maringale repreenaion heorem, here i a unique Z(, ) L 2 (T ε,t;l 2 F (,T ε)) uch ha: E Y() = EY() + ε Z(,)dW(), T ε,t. (4.21) Hence, we have uniquely deermined (Y(),Z(,)) for (,) T ε,t,t (he region marked 1, 3 and 4 ) and he following i well-defined: g (,,z) = g(,,y(),z,z(,)), (,),T ε T ε,t. (4.22) Noe ha,,t i he region marked 2 in he above Figure 2. Now, conider he following SFIE: ψ () = ψ()+ g (,,Z(,))d Z(,)dW(),,T ε. (4.23) Similar o he Sep 2 of he proof of Theorem 3.5, SFIE (4.23) admi a unique oluion (ψ ( ),Z(, )) on,t ε T ε,t and he following eimae hold: ψ () 2 α(),,t ε, (4.24) where α( ) olve an equaion imilar o (3.46). The above uniquely deermined Y(), T ε,t, ( ) ( ) Z(,), (,) T ε,t,t,t ε T ε,t. (4.25) 21

22 Now, we conider Y() = ψ ()+ ε g(,,y(),z(,),z(,))d ε Z(, )dw() (4.26) on,t ε. Since ψ ( ) L F (,T ε), (4.26) i a BSVIE on,t ε. Then he above procedure can be repeaed. Since he ep-lengh ε > can be fixed, we hen could ue inducion o complee he proof. 5 Comparion Theorem for Adaped Soluion o Type-I BSVIE Conider he following BSVIE: For i = 1,2, Y i () = ψ i ()+ g i (,,Y i (),Z i (,))d Z i (,)dw(),,t. (5.1) We aume ha he generaor g i ( ), i = 1,2 of BSVIE (5.1) aify (A2). Then by Theorem 3.5, BSVIE (5.1) admi a unique adaped oluion (Y i ( ),Z i (, )) L F (,T) BMO(,T) for any ψi ( ) L F T (,T). In order o udy he comparion heorem of he oluion o BSVIE (5.1), we inroduce he following BSVIE: Ȳ() = ψ()+ ḡ(,,ȳ(), Z(,))d Z(,)dW(),,T, (5.2) wih he generaor ḡ( ) alo aifie (A2). Furher, we adop he following aumpion. (C). Le he generaor ḡ :,T R R Ω R aify ha y ḡ(,,y,z) i nondecreaing for any (,,z),t R. We preen he comparion heorem for BSVIE (5.1) now. Theorem 5.1. Le g 1 ( ),g 2 ( ) and ḡ( ) aify (A2) and le ḡ( ) aify (C). Suppoe g 1 (,,y,z) ḡ(,,y,z) g 2 (,,y,z), (y,z) R R, a.., a.e. (,),T. (5.3) Then for any ψ 1 ( ),ψ 2 ( ) L F T (,T) aifying ψ 1 () ψ 2 (), a.., a.e.,t, (5.4) he correponding unique adaped oluion (Y i ( ),Z i (, )), i = 1,2 of BSVIE (5.1) aify Proof. Le ψ( ) L F T (,T) uch ha Y 1 () Y 2 (), a.., a.e.,t. (5.5) ψ 1 () ψ() ψ 2 (), a.., a.e.,t. (5.6) Wihou lo of generaliy, le ψ( ) L, (5.7) where ψ( ) = ψ 1 ( ),ψ 2 ( ), ψ( ). By Theorem 3.5, BSVIE (5.1) admi a unique adaped oluion (Y 1 ( ),Z 1 (, )) L F (,T) BMO(,T) for i = 1. Se Ỹ( ) = Y 1 ( ) and conider Ỹ 1 () = ψ()+ ḡ(,,ỹ(), Z 1 (,))d Z 1 (,)dw(),,t. (5.8) 22

23 By Theorem 3.2, here i a unique adaped oluion (Ỹ1( ), Z 1 (, )) L F (,T) BMO(,T) o he above BSVIE. By (5.3), we have g 1 (,,Ỹ(),z) ḡ(,,ỹ(),z), z R, a.., a.e. (,),T. (5.9) Combining hi and (5.6), by Theorem 3.3, for almo all,t, here exi a meaurable e Ω 1 Ω aifying P(Ω 1 ) = uch ha Nex, we conider he following BSVIE Ỹ () = Y 1 () Ỹ1(), ω Ω\Ω 1, a.e.,t. (5.1) Ỹ 2 () = ψ()+ ḡ(,,ỹ1(), Z 2 (,))d Z 2 (,)dw(),,t. (5.11) Le (Ỹ2( ), Z 2 (, )) be he unique oluion o he above equaion. Since y ḡ(,,y,z) i nondecreaing, by (5.1), we have ḡ(,,ỹ(),z) ḡ(,,ỹ1(),z), z R, a.., a.e. (,),T. (5.12) Similar o he above, for almo everywhere,t, here exi a meaurable e Ω 2 Ω aifying P(Ω 2 ) = uch ha Ỹ 1 () Ỹ2(), ω Ω\Ω 2, a.e.,t. (5.13) By inducion, we can conruc a equence (Ỹk( ), Z k (, )) and Ω k aifying P(Ωk ) = uch ha and Ỹ k+1 () = ψ()+ ḡ(,,ỹk(), Z k+1 (,))d ( Y 1 () = Ỹ() Ỹ1() Ỹ2(), ω Ω\ Noe ha PΩ\( k 1 Ωk ) =. We may aume ha Z k+1 (,)dw(),,t, (5.14) k 1 Ω k ), a.e.,t. (5.15) ψ() α(),,t, (5.16) where ψ( ) = ψ 1 ( ),ψ 2 ( ), ψ( ) and α( ) olve an ODE of form (3.46). By Propoiion 3.7, here i an ε > uch ha Ỹ k ( ) i Cauchy in L F (T ε,t) and Combining (5.15) and (5.17), we have Nex, conider he following SFIE: ψ 1, () = ψ 1 ()+ ψ () = ψ()+ lim k Ỹk( ) Ȳ( ) L F (,T) =. (5.17) Y 1 () Ȳ(), a.., a.e. T ε,t. (5.18) g 1 (,,Y 1 (),Z 1 (,))d ḡ(,,ȳ(), Z(,))d Z 1 (,)dw(),,t ε; (5.19) Z(,)dW(),,T ε. (5.2) Similar o he Sep 2 in Theorem 3.5, he above SFIE (5.19) and (5.2) admi unique oluion (ψ 1, ( ),Z 1 (, )), ( ψ ( ), Z(, )) L F (,T ε) BMO(,T ε T ε,t), repecively. Similar o (3.61), we have ψ 1, () α(), ψ () α(),,t ε. (5.21) 23

24 For almo all,t ε, imilar o (3.32) (3.33) and (3.36) (3.37), here i a proce θ(, ) uch ha: and g 1 (,,Y 1 (),Z 1 (,)) g 1 (,,Y 1 (), Z(,)) = Z 1 (,) Z(,) θ(,), (5.22) W(;) W() θ(,r)dr,,t (5.23) i a Brownian moion on,t under he correponding equivalen probabiliy meaure P. The correponding expecaion i denoed by E P. Combining (5.19) (5.2) and (5.22) (5.23), we have ψ 1, () ψ () = ψ 1 () ψ()+ g 1 (,,Y 1 (), Z(,)) ḡ(,,ȳ(), Z(,)) d Z 1 (,) Z(,) dw(;),,t ε. (5.24) Since y ḡ(,,y,z) i nondecreaing for any (,,z),t R, by (5.18), we have ḡ(,,y 1 (),z) ḡ(,,ȳ(),z), (,,z),t T ε,t R. (5.25) Taking condiional expecaion E P E P, on he boh ide of (5.24), by (5.3), (5.25) and (5.18), we have ψ 1, () ψ () ψ 1 () ψ()+ = E P E P ψ 1 () ψ()+,,t ε. g 1 (,,Y 1 (), Z(,)) ḡ(,,ȳ(), Z(,)) d g 1 (,,Y 1 (), Z(,)) ḡ(,,y 1 (), Z(,)) d (5.26) Now, we conider he following BSVIE: y 1 () = ψ 1, ()+ ȳ() = ψ ()+ ε ε g 1 (,,y 1 (),z 1 (,))d ḡ(,,ȳ(), z(,))d ε ε z 1 (,)dw(),,t ε; (5.27) z(,)dw(),,t ε. (5.28) By Theorem 3.5, he above equaion (5.27), (5.28) admi unique oluion (y 1 ( ),z 1 (, )), (ȳ( ), z(, )) (,T ε) BMO(,T ε), repecively. By he Sep 3 in he proof of Theorem 3.5, we have L F y 1 () = Y 1 (), z 1 (,) = Z 1 (,), (,),T ε; (5.29) ȳ() = Ȳ(), z(,) = Z(,), (,),T ε. (5.3) Hence, by inducion, we have Similarly, we can prove ha Y 1 () Ȳ(), a.., a.e.,t. (5.31) Ȳ() Y 2 (), a.., a.e.,t. (5.32) Thu, he inequaliy (5.5) hold. Noe ha he inequaliy (5.5) in Theorem 5.1 hold for almo everywhere,t. We may renghen he above reul under a ronger condiion. More preciely, we have he following heorem which give he comparion for all,t. 24

25 Theorem 5.2. Le he generaor g 1 ( ), g 2 ( ) and ḡ( ) aify (A2) (A3) and furher le ḡ( ) aify (C). Suppoe g 1 (,,y,z) ḡ(,,y,z) g 2 (,,y,z), Then for any ψ 1 ( ),ψ 2 ( ) L F T (Ω;C U,T) aifying (,y,z),t R R, a.., a.e.,t. (5.33) ψ 1 () ψ 2 (),,T, a.., (5.34) he correponding unique adaped oluion (Y i ( ),Z i (, )), i = 1,2 of BSVIE (5.1) aify Proof. By Theorem 5.1, we have Y 1 () Y 2 (),,T, a.. (5.35) Y 1 () Y 2 (), a..,,t. (5.36) Le { k } k 1,T be all he raional number in,t. For any fixed k, by (5.36), here i a Ω k Ω aifying P(Ω k ) = uch ha: Y 1 ( k ) Y 2 ( k ), ω Ω\Ω k. (5.37) Le Ω = k 1 Ω k, hen P( Ω) =. By (5.37), we have Y 1 () Y 2 (), { k } k 1, ω Ω\ Ω. (5.38) By Theorem 3.8, here i a Ω Ω aifying P( Ω) = uch Y i (,ω), i = 1,2 are coninuou for any ω Ω\ Ω. For any fixed ω Ω\( Ω Ω), by (5.38), we have Y 1 (,ω) Y 2 (,ω), { k } k 1. (5.39) Since Y i (,ω), i = 1,2 are coninuou on,t and { k } k 1,T i dene on,t, we have Noe ha P(Ω\( Ω Ω)) =, we have Thi complee he proof. Y 1 (,ω) Y 2 (,ω),,t. (5.4) Y 1 () Y 2 (),,T, a.. (5.41) 6 Coninuou-Time Equilibrium Dynamic Rik Meaure We have een he o-called equilibrium recurive uiliy proce in he inroducion ecion, which erve a a very imporan moivaion of udying BSVIE. In hi ecion, we will look anoher cloely relaed applicaion of BSVIE. Saic rik meaure have been udied by many reearcher. Among many of hem, we menion Arzner Delbaen Eber Heah 5, Föllmer Schied 19, and he reference cied herein. For dicreeime dynamic rik meaure, we menion Riedel 35 and Delefen Scandolo 13, and he reference cied herein. We now look a coninuou-ime dynamic rik meaure. Any ξ L F T (Ω) repreen he payoff of cerain European ype coningen claim a he mauriy ime T. According o El Karoui Peng Quenez 18, we inroduce he following definiion. 25

26 Definiion 6.1. A map ρ :,T L F T (Ω) R i called a dynamic rik meaure if he following are aified: (i) (Adapivene) For any ξ L F T (Ω), ρ(;ξ) i F-adaped; (ii) (Monooniciy) For any ξ, ξ L F T (Ω) wih ξ ξ, one ha ρ(;ξ) ρ(; ξ), for all,t; (iii) (Tranlaion Invarian) For any σ L F T (Ω) and c R, ρ(;+c) = ρ(;ξ) c. Furher, ρ i aid o be convex if he following hold: (iv) (Convexiy): ξ ρ(; ξ) i convex; and ρ i aid o be coheren if he following are aified: (v) (Poiive Homogeneiy): For any ξ L F T (Ω) and λ, ρ(;λξ) = λρ(;ξ); (vi) (Subaddiiviy): For any ξ, ξ L F T (Ω), ρ(;ξ + ξ) ρ(;ξ)+ρ(; ξ). Each iem in he above definiion can be naurally explained. For example, (ii) mean ha beween wo gain, he one dominanly larger one ha a maller rik; (vi) mean ha combining wo invemen will have maller rik. The following i quoed from 18. Propoiion 6.2. Le g :,T R R be meaurable uch ha z g(,z) i convex and grow a mo quadraically. Then for any ξ L F T (Ω), he following BSDE: Y() = ξ + g(, Z())d Z()dW(),,T, (6.1) admi a unique adaped oluion (Y( ),Z( )) (Y( ;ξ),z( ;ξ)). Le ρ :,T L F T (Ω) R be defined by he following: ρ(,ξ) = Y(;ξ), (,ξ),t L F T (Ω). Then ρ i a dynamic convex rik meaure. One of he mo inereing example i he following. Y() = ξ + 1 T 2γ Z() 2 d Z()dW(),,T. The above admi a unique adaped oluion (Y( ), Z( )), and ρ(,ξ) Y() = γlne e ξ γ F e γ, (ξ),,t, i called a dynamic enropic rik meaure for ξ. Now, if we havean anicipaed cah flow proce ψ( ) inead of ju a erminal payoffξ, hen formally, he correponding dynamic rik hould be meaured via he following parameerized BSDE: Y(,r) = ψ()+ g(,y(,),z(,))d Z(,)dW()), (r,),t, r r and he curren dynamic rik hould be Y(; ). Bu, imilar o he inroducion ecion, imply aking r = in he above lead o he following: Y(,) = ψ()+ g(,y(,),z(,))d Z(,)dW()),,T, 26

27 which i no a cloed form equaion for he pair (Y(,),Z(,)) of procee. A we indiced in he inroducion, Y(, r) above ha ome hidden ime-inconiency naure. One expec ha he dynamic rik meaure hould be ime-conien. Namely, he value of he rik oday (for a proce ψ( )) hould mach he one ha one expeced yeerday. Therefore, i i naural o ue BSVIE o decribe/meaure he dynamic rik of he proce ψ( ). We now make hi precie. We call ψ( ) L F T (,T) a poiion proce (a name borrowed from 35), and ψ() could repreen he oal (nominal) value of cerain porfolio proce which migh be a combinaion of cerain (ay, European ype) coningen claim (which are maure a ime T, hu hey are uually only F T -meaurable), ome curren cah flow (uch a dividend o be received, premia o be paid), poiion of ock, muual fund, and bond, and oon, a ime he curren ime. Thu, he poiion proceψ( ) i merely F T -meaurable (no necearily F-adaped). Now, mimicking Definiion 6.1, we inroduce he following. Definiion 6.3. A map ρ : L F T (,T) L F (,T) i called an equilibrium dynamic rik meaure if he following hold: (i) (Pa Independence) For any ψ 1 ( ),ψ 2 ( ) L F T (,T), if for ome,t), hen ψ 1 () = ψ 2 (), a.., a.e.,t, ρ(;ψ 1 ( )) = ρ(;ψ 2 ( )), a.. (ii) (Monooniciy) For any ψ 1 ( ),ψ 2 ( ) L F T (,T), if for ome,t), hen ψ 1 () ψ 2 (), a.., a.e.,t, ρ(;ψ 1 ( )) ρ(;ψ 2 ( )), a..,,t. (iii) (Tranlaion Invariance) There exi a deerminiic inegrable funcion r( ) uch ha for any ψ( ) L F T (,T), ρ(;ψ( )+c) = ρ(;ψ( )) ce r()d, a..,,t. Furher, ρ i aid o be convex if he following hold: (iv) (Convexiy) For any ψ 1 ( ),ψ 2 ( ) L F T (,T) and λ,1, ρ(;λψ 1 ( )+(1 λ)ψ 2 ( )) λρ(;ψ 1 ( ))+(1 λ)ρ(;ψ 2 ( )), a..,,t. And ρ i aid o be coheren if he following are aified: (v) (Poiive Homogeneiy) For any ψ( ) L F T (,T) and λ >, ρ(;λψ( )) = λρ(;ψ( )), a..,,t. (vi) (Subaddiiviy) For any ψ 1 ( ),ψ 2 ( ) L F T (,T), ρ(;ψ 1 ( )+ψ 2 ( )) ρ(;ψ 1 ( ))+ρ(;ψ 2 ( )), a..,,t. The word equilibrium indicae he ime-coniency of he rik meaure ρ which i ome kind of modificaion of he naive one. Similar iuaion ha happened in he udy of ime-inconien opimal conrol problem (ee 48). Le u now look a he following Type-I BSVIE: Y() = ψ()+ We have he following reul. g(,,y(),z(,))d Z(,)dW(),,T. (6.2) 27

28 Propoiion 6.4. Le he generaor be given by g(,,y,z) r()y +g (,,z);(,,y,z),t R R, and aify (A2), where r( ) i a non-negaive deerminiic funcion. Then he following are rue: (i) The map ψ( ) ρ(; ψ( )) i ranlaion invarian. (ii) Suppoe z g (,,z) i convex, o i ψ( ) ρ(;ψ( )). (iii) Suppoe z g (,,z) i poiively homogeneou and ub-addiive, o i ψ( ) ρ(;ψ( )). By Theorem 5.1, he proof of Propoiion 6.4 i very imilar o 46, Corollary 3.4, Propoiion 3.5, we omi hem here. By Propoiion 6.4, we can conruc a large cla of equilibrium dynamic rik meaure by chooing uiable generaor g( ) of BSVIE (6.2). More preciely, we have he following reul. Theorem 6.5. Le he generaor g(,,y,z) r()y+g (,,z);(,,y,z) R R aify (A2), where r( ) i a non-negaive deerminiic funcion and z g (,,z) i convex, hen ψ( ) ρ(;ψ( )) i an equilibrium dynamic convex rik meaure. If z g (,,z) i poiively homogeneou and ub-addiive, hen ψ( ) ρ(; ψ( )) i an equilibrium dynamic coheren rik meaure. From Propoiion 6.4, he proof of he above reul i obviou. According o he above reul, we can have ome example of equilibrium dynamic rik meaure by he choice of g (,,z): If g (,,z) = ḡ(,) z, ḡ(,), hen, i i ub-addiive and poiively homogeneou in z. The correponding equilibrium dynamic rik meaure i coheren. If g (,,z) = ḡ(,) 1+ z 2, ḡ(,), hen, i i convex in z. The correponding equilibrium dynamic rik meaure i convex. If g (,,z) = ḡ(,) z 2, ḡ(,), hen one ha an enropy ype equilibrium dynamic rik meaure. 7 Concluding Remark Recurive uiliy proce (or ochaic differenial uiliy proce) and dynamic rik meaure for erminal payoff can be decribed by he adaped oluion o proper BSDE. For F T -meaurable poiion proce ψ( ), inead of he erminal payoff ξ, one could alo ry o find i recurive uiliy proce and/or dynamic rik. One poibiliy i again ue BSDE. However, one immediaely find ha he reuling proce (recurive uiliy or dynamic rik meaure) are kind of ime-inconien naure. BSVIE urn ou o be a proper ool for decribing hem. To hi end, we need o eablih he well-poedne of QBSVIE. In hi paper, we have achieved hi. Conequenly, he heory of equilibrium recurive uiliy and equilibrium dynamic rik meaure are uccefully eablihed. Reference 1 N. Agram, Dynamic rik meaure for BSVIE wih jump and emimaringale iue, arxiv: v1 mah.oc 3 Mar