Backward Stochastic Partial Differential Equations with Jumps and Application to Optimal Control of Random Jump Fields

Transcription

1 Backward Sochaic Parial Differenial Equaion wih Jump and Applicaion o Opimal Conrol of Random Jump Field Bern Økendal 1,2, Frank Proke 1, Tuheng Zhang 1,3 June 7, 25 Abrac We prove an exience and uniquene reul for a general cla of backward ochaic parial differenial equaion wih jump. Thi i a ype of equaion which appear a adjoin equaion in he maximum principle approach o opimal conrol of yem decribed by ochaic parial differenial equaion driven by Lévy procee. AMS Subjec Claificaion: Primary 6H15 Secondary 93E2, 35R6. 1 Inroducion Le B, and η() = Rn zñ(d, dz); be an m-dimenional Brownian moion and a pure jump Lévy proce, repecively, on a filered probabiliy pace (Ω, F, F, P ). Fix T > and le φ(ω) be an F T -meaurable random variable. Le b : [, T R n R n m R n be a given vecor field. Conider he problem o find hree F -adaped procee p() R n, q() R n m and r(, z) R n m uch ha dp() = b(, p(), q())d + q()db + r(, z)ñ(d, dz), (, T ) (1.1) R n p(t ) = φ a.. (1.2) Thi i a backward ochaic (ordinary) differenial equaion (BSDE). I i called backward becaue i i he erminal value p(t ) = φ ha i given, no he iniial value p(). Sill p() i required o be F -adaped. In general hi i only poible if we alo are free o chooe q() and r(, z) (in an F -adaped way). The heory of BSDE, when η =, i now well developed. See e.g. [EPQ, [MY, [PP1, [PP2 and [YZ and he reference herein. In he jump cae (η ) BSDE have been udied. See [FØS, [NS, [S and he reference herein. There are many applicaion of hi heory. Example include he following: (i) The problem of finding a replicaing porfolio of a given coningen claim in a complee financial marke can be ranformed ino a problem of olving a BSDE. 1 CMA, Deparmen of Mahemaic, Univeriy of Olo, P.O. Box 153 Blindern, N-316 Olo, Norway. okendal@mah.uio.no, proke@mah.uio.no 2 Norwegian School of Economic and Buine Adminiraion, Helleveien 3, N-545 Bergen, Norway 3 Deparmen of Mahemaic, Univeriy of Mancheer, Oxford Road, Mancheer M13 9PL, England, U.K. zhang@mah.man.ac.uk 1

2 (ii) The maximum principle mehod for olving a ochaic conrol problem involve a BSDE for he adjoin procee p(), q(), r(, z). For more informaion abou hee and oher applicaion of BSDE we refer o [EPQ and [YZ and reference herein. The purpoe of hi paper i o udy backward ochaic parial differenial equaion (BSPDE) wih jump. They are defined in a imilar way a BSDE, bu wih he baic equaion being a ochaic parial differenial equaion raher han a ochaic ordinary differenial equaion. More preciely, we will udy a cla of BSPDE which include he following: Find adaped procee Y (, x), Z(, x), Q(, x, z) uch ha dy (, x) = AY (, x)d + b(, x, Y (, x), Z(, x))d + Z(, x)db + Q(, x, z)ñ (d, dz), (, x) (, T ) Rn R n (1.3) Y (T, x) = φ(x, ω) (1.4) Here dy (, x) denoe he Iô differenial wih repec o, while A i a parial differenial operaor wih repec o x and Ñ(d, dz) i he compenaed Poion random meaure aociaed wih a Lévy proce η( ). The funcion b : [, T R n R R R i given and o i he erminal value funcion φ(x) = φ(x, ω). We aume ha φ(x) i F T -meaurable for all x and ha E[ φ(x) 2 dx <, (1.5) R n where E denoe expecaion wih repec o P. We are eeking he procee Y (, x), Z(, x) and Q(, x, z) uch ha (1.3) and (1.4) hold. The procee Y (, x), Z(, x) and Q(, x, z) are required o be F -adaped, i.e., Y (, x), Z(, x) and Q(, x, z) are F -meaurable for all x R n, z R and we alo require ha [ E R n {Y (, x) 2 + Z(, x) 2 + Q(, x, z) 2 ν(dz)}d dx <, (1.6) R n where ν( ) i he Lévy meaure of he underlying Lévy proce. Equaion of hi ype are of inere becaue hey appear a adjoin equaion in a maximum principle approach o opimal conrol of ochaic parial differenial equaion. See Secion 2. Example 1.1 Conider he following BSPDE: dy (, x) = 1 2 Y (, x)d + Z(, x)db + Q(, x, z)ñ(d, dz), (, x) (, T ) Rn R n (1.7) Y (T, x) = φ(x) (1.8) Here Y (, x) = n 2 Y (,x) i=1 x 2 i aifie E[ R φ(x) 2 dx <. n i he Laplacian wih repec o x applied o Y (, x), and φ(x) 2

3 where In hi imple cae, we are able o find he oluion explicily: We fir ue he Iô repreenaion heorem o wrie, for almo all x, φ(x) = h(x) + g(, x, ω)db + k(, x, z, ω)ñ(d, dz) R n h(x) = E[φ(x), g(, x, ) and k(, x, z) are F -meaurable for all, x and { } E[ g 2 (, x, ) + k 2 (, x, z)ν(dz) ddx <. R n R n Le R f(x) = (2π) n x y 2 2 f(y)exp( )dy, > R n 2 be he raniion operaor for Brownian moion defined for all meaurable f : R n R uch ha he inegral converge. Then i i well known ha (R f(x)) = 1 2 (R f(x)) (1.9) Now define Y (, x) = R T ( g(,, ω)db + k(,, z, ω)ñ(d, dz) + h( ))(x) R n = (R T g(,, ω))(x)db + (R T k(,, z, ω))(x)ñ(d, dz) + (R T h)(x) R n (1.1) Then dy (, x) [ = 1 2 (R T g(,, ω))(x)db 1 2 (R T k(,, z, ω))(x)ñ (d, dz) 1 R n 2 R T h( )(x) d + (R T g(,, ω))(x)db + R T (k(,, z, ω))(x)ñ(d, dz) R n = 1 2 Y (, x)d + Z(, x)db + Q(, x, z)ñ(d, dz), R n where and Z(, x) = (R T g(,, ω))(x) (1.11) Q(, x, z) = (R T k(,, z))(x). (1.12) Hence he procee Y (, x), Z(, x) and Q(, x, z) given by (1.1) (1.12) olve he BSPDE (1.7) (1.8). 3

4 In he general cae i i no poible o find explici oluion of a BSPDE. However, in Secion 3 we will prove an exience and uniquene reul for a general cla of uch equaion. We will achieve hi by regarding he BSPDE of ype (1.3) (1.4) a a pecial cae of a backward ochaic evoluion equaion for Hilber pace valued procee. Thi, in urn, i udied by aking finie dimenional projecion and hen aking he limi. Thi i he well known alerkin approximaion mehod which ha been ued by everal auhor in oher connecion. See e.g. [B1, [B2 and [P. We alo refer reader o [PZ for he general heory of ochaic evoluion equaion on Hilber pace. The re of he paper i organized a follow: In Secion 2 we prove a (ufficien) maximum principle for opimal conrol of random jump field, i.e. oluion of SPDE driven by Lévy procee (Theorem 2.1). Thi principle involve a BSPDE of he form (1.3)-(1.4) in he aociaed adjoin procee. In Secion 3 we give he precie framework of our general exience and uniquene reul. The exience and uniquene reul and i proof are given in Secion 4. 2 The ochaic maximum principle In hi ecion we prove a verificaion heorem for opimal conrol of a proce decribed by a ochaic parial differenial equaion (SPDE) driven by a Brownian moion B() and a Poion random meaure Ñ(d, dz). We call uch a proce a (conrolled) random jump field. The verificaion heorem ha he form of a ufficien ochaic maximum principle and he adjoin equaion for hi principle urn ou o be a backward SPDE driven by B( ) and Ñ(, ). Thi par of he paper i an exenion of [Ø o he cae including jump and an exenion of [FØS o SPDE conrol. Le Γ(, x) = Γ (u) (, x); [, T, x R k be he oluion of a (conrolled) ochaic reacion-diffuion equaion of he form dγ(, x) = [(LΓ)(, x) + b(, x, Γ(, x), u(, x)) d + σ(, x, Γ(, x), u(, x))db() + θ(, x, Γ(, x), u(, x), z)ñ (d, dz); (, x) [, T (2.1) R Γ(, x) = ξ(x); x (2.2) Γ(, x) = η(, x); (, x) (, T ) (2.3) Here dγ(, x) = d Γ(, x) i he differenial wih repec o and L = L x i a given parial differenial operaor of order m acing on he variable x R k. We aume ha R k, R l are given open and cloed e, repecively, and ha b : [, T R R, σ : [, T R R, θ : [, T R R R, ξ : R and η : (, T ) R are given meaurable funcion. The proce u : [, T i called an admiible conrol if he equaion (2.1)-(2.3) ha a unique coninuou oluion Γ(, x) = Γ (u) (, x) which aifie [ ( ) E f(, x, Γ(, x), u(, x)) dx d + g(x, Γ(T, x)) dx <, (2.4) 4

5 where f : [, T R R and g : R R are given C 1 funcion. The e of all admiible conrol i denoed by A. If u A we define he performance of u, J(u), by [ ( ) J(u) = E f(, x, Γ(, x), u(, x))dx d + g(x, Γ(T, x))dx. (2.5) We conider he problem o find J R and u A uch ha J := up J(u) = J(u ). (2.6) u A J i called he value of he problem and u A (if i exi) i called an opimal conrol. In he following we le L denoe he adjoin of he operaor L, defined by (L ϕ 1, ϕ 2 ) = (ϕ 1, Lϕ 2 ) ; ϕ 1, ϕ 2 C (), (2.7) where (ϕ, ψ) = (ϕ, ψ) L 2 () = ϕ(x)ψ(x)dx and C () i he e of infiniely many ime differeniable funcion wih compac uppor in. We now formulae a (ufficien) ochaic maximum principle for he problem (2.6): Le R denoe he e of funcion r : [, T R Ω R. Define he Hamilonian by H : [, T R R R R R H(, x, γ, u, p, q, r(, x, )) = f(, x, γ, u) + b(, x, γ, u)p + σ(, x, γ, u)q + R θ(, x, γ, u, z)r(, x, z)ν(dz) (2.8) For each u A conider he following adjoin backward SPDE in he 3 unknown adaped procee p(, x), q(, x) and r(, x, z): dp(, x) = { H γ (, x, Γ(u) (, x), u(, x), p(, x), q(, x), r(, x, )) + L p(, x)}d + q(, x)db() + r(, x, z)ñ(d, dz); < T. (2.9) p(t, x) = g γ (x, Γ(u) (, x)); x (2.1) p(, x) = ; (, x) (, T ). (2.11) The following reul may be regarded a a ynhei of Theorem 2.1 in [Ø and Theorem 2.1 in [FØS: R Theorem 2.1(Sufficien SPDE maximum principle for opimal conrol of reaciondiffuion jump field) 5

6 Le û A wih correponding oluion Γ(, x) of (2.1)-(2.3) and le p(, x), q(, x), r(, x, ) be a oluion of he aociaed adjoin backward SPDE (2.9)-(2.11). Suppoe he following, (2.12)-(2.15), hold: The funcion (2.12) (γ, u) H(γ, u) := H(, x, γ, u, p(, x), q(, x), r(, x, )); (γ, u) R and γ g(x, γ); γ R are concave for all (, x) [, T. For all u A [ E H(, x, Γ(, x), û(, x), p(, x), q(, x), r(, x, )) (2.13) = up H(, x, Γ(, x), v, p(, x), q(, x), r(, x, )) v for all (, x) [, T. ( Γ(, x) Γ(, ) { } 2 x) q(, x) 2 + r(, x, z) 2 ν(dz) ddx < (2.14) R and [ } E p(, x) {σ(, 2 x, Γ(, x), u(, x)) 2 + θ(, x, Γ(, x), u(, x), z) 2 ν(dz) ddx < R (2.15) Then û(, x) i an opimal conrol for he ochaic conrol problem (2.6). Proof. Le u be an arbirary admiible conrol wih correponding oluion Γ(, x) = Γ (u) (, x) of (2.1)-(2.3). Then by (2.5) where Similarly we pu [ J(û) J(u) = E { } f f dxd + {ĝ g} dx, (2.16) f = f(, x, Γ(, x), û(, x)), f = f(, x, Γ(, x), u(, x)) ĝ = g(x, Γ(T, x)) and g = g(x, Γ(T, x)). b = b(, x, Γ(, x), û(, x)), b = b(, x, Γ(, x), u(, x)) σ = σ(, x, Γ(, x), û(, x)), σ = σ(, x, Γ(, x), u(, x)) and θ = θ(, x, Γ(, x), û(, x), z), θ = θ(, x, Γ(, x), u(, x), z). 6

7 Moreover, we e Ĥ = H(, x, Γ(, x), û(, x), p(, x), q(, x), r(, x, )) H = H(, x, Γ(, x), u(, x), p(, x), q(, x), r(, x, )). Combining hi noaion wih (2.8) and (2.16) we ge where and [ I 1 = E Since γ g(x, γ) i concave we have Therefore, if we pu J(û) J(u) = I 1 + I 2, (2.17) { } Ĥ H ( b b) p ( σ σ) q ( θ θ) rν(dz) dxd R (2.18) [ I 2 = E {ĝ g} dx. (2.19) g ĝ g γ (x, Γ(T, x)) (Γ(T, x) Γ(T, x)). Γ(, x) = Γ(, x) Γ(, x) we ge, by he Iô formula (or inegraion by par formula) for jump diffuion ([ØS, Ex. 1.7) I 2 [ g E γ (x, Γ(T, x)) Γ(T, x)dx [ = E p(t, x) Γ(T, x)dx [ ( = E p(, x) Γ(, x) where + + { Γ(, x)d p(, x) + p(, x)d Γ(, x) + (σ σ) q(, x) } d R ) (θ θ) r(, x, z)n(d, dz) dx [ ( { [ ( ) H = E Γ(, x) L x) p(, γ [ + p(, x) L Γ(, x) + (b b) + (σ σ) q(, x) } ) + (θ θ) r(, x, z)ν(dz) d dx, (2.2) R ( ) H = H γ γ (, x, Γ(, x), û(, x), p(, x), q(, x), r(, x, )). 7

8 Combining (2.17)-(2.2) we ge [ ( J(û) J(u) E [ +E ( ) { ΓL p pl Γ} dx d { Ĥ H + ( ) H ) x)} γ Γ(, d dx. (2.21) Since Γ(, x) = p(, x) = for all (, x) (, T ) we ge by an exenion of (2.7) ha { ΓL p pl Γ} dx = for all (, T ). Combining hi wih (2.21) we ge [ ( J(û) J(u) E { Ĥ H + From he concaviy aumpion in (2.12) we deduce ha ( ) H ) x)} γ Γ(, d dx. (2.22) H Ĥ H γ ( Γ, û) (Γ Γ) + H u ( Γ, û) (u û). (2.23) From he maximaliy aumpion in (2.13) we ge ha Combining (2.23) and (2.24) we ge which ubiued in (2.22) give H u ( Γ, û) (u û). (2.24) H Ĥ H γ ( Γ, û) (Γ Γ), J(û) J(u). Since u A wa arbirary hi prove Theorem Framework We now preen he general eing in which we will prove our main exience and uniquene reul for backward SPDE wih jump: Le V, H be wo eparable Hilber pace uch ha V i coninuouly, denely imbedded in H. Idenifying H wih i dual we have V H = H V, (3.1) where V and for he opological dual of V. Le A be a bounded linear operaor from V o V aifying he following coerciviy hypohei: There exi conan α > and λ uch ha 2 Au, u + λ u 2 H α u 2 V for all u V, (3.2) 8

9 where Au, u = Au(u) denoe he acion of Au V on u V. Remark ha A i generally no bounded a an operaor from H ino H. Le K be anoher eparable Hilber pace. Le (Ω, F, P ) be a probabiliy pace. Le {B, } be a cylindrical Brownian moion wih covariance pace K on he probabiliy pace (Ω, F, P ), i.e., for any k K, B, k i a real valued-brownian moion wih E[ B, k 2 = k 2 K. Le (, B()) be a meaurable pace, where i a opological vecor pace. Furher le η() be a Lévy proce on. Denoe by ν(dx) he Lévy meaure of η. Denoe by L 2 (ν) he L 2 -pace of quare inegrable H-valued meaurable funcion aociaed wih ν. Se p() = η() = η() η( ). Then p = (p(), D p ) i a aionary Poion poin proce on wih characeriic meaure ν. See [IW for deail on Poion poin procee. Denoe by N(d, dx) he Poion couning meaure aociaed wih he Lévy proce, i.e., N(, A) = D I p, A(p()). Le Ñ(d, dx) := N(d, dx) dν(dx) be he compenaed Poion meaure. Define F = σ(b, N(, A), A B(), ). Recall ha a linear operaor S from K ino H i called Hilber-Schmid if i=1 Sk i 2 H < for ome orhonormal bai {k i, i 1} of K. L 2 (K, H) will denoe he Hilber pace of Hilber-Schmid operaor from K ino H equipped wih he inner produc S 1, S 2 L2 (K,H) = i=1 S 1k i, S 2 k i H. Le b(, y, z, q, ω) be a meaurable mapping from [, T H L 2 (K, H) L 2 (ν) Ω ino H uch ha b(, y, z, q, ω) i F -adaped,i.e., b(, y, z, q, ) i F -meaurable for all, y, z, q. Suppoe we are given an F T -meaurable, H-valued random variable φ(ω). We are looking for F -adaped procee Y, Z, Q wih value in H, L 2 (K, H) and L 2 (ν) repecively, uch ha he following backward ochaic evoluion equaion hold: dy = AY d + b(, Y, Z, Q )d + Z db + Q (x)ñ(d, dx), (, T ) (3.3) Y T = φ(ω) a.. (3.4) From now on we aume ha he following, (3.5) and (3.6), hold: There exi a conan c < uch ha for all, y 1, z 1, q 1, y 2, z 2, q 2. b(, y 1, z 1, q 1 )(ω) b(, y 2, z 2, q 2 )(ω) H c( y 1 y 2 H + z 1 z 2 L2 (K,H) + q 1 q 2 L 2 (ν)) (3.5) 4 Exience and Uniquene E[ b(,,, ) 2 H d < (3.6) We now ae and prove he main exience and uniquene reul of hi paper. Theorem 4.1 Aume ha E[ φ 2 H <. Then here exi a unique H L 2(K, H) L 2 (ν)-valued progreively meaurable proce (Y, Z, Q ) uch ha (i) (ii) φ = Y + E[ Y 2 H d <, E[ AY d + Z 2 L 2 (K,H) d <, E[ Q 2 L 2 (ν) d <. Z db + Q(x)Ñ(d, dx); T. b(, Y, Z, Q )d + 9

10 A he proof i long, we pli i ino lemma. Lemma 4.2 Aume ha E[ φ 2 H <, and ha b(, y, z, q, ω) = b(, ω) i independen of y, z and q, and E[ b() 2 H d <. Then here exi a unique H L 2(K, H) L 2 (ν)- valued progreively meaurable proce (Y, Z, Q ) uch ha (i) (ii) φ = Y + E[ Y 2 H d <, E[ AY d + Z 2 L 2 (K,H) d <, E[ Q 2 L 2 (ν) d <. Z db + Q(x)Ñ(d, dx); T. b()d + Proof. Exience of oluion. Se D(A) = {v; v V, Av H}. Then D(A) i a dene ubpace of H. Thu we can chooe and fix an orhonormal bai {e 1,...e n,...} of H uch ha e i D(A). Se V n = pan(e 1, e 2,..., e n ). Denoe by P n he projecion operaor from H ino V n. Pu A n = P n A. Then A n i a bounded linear operaor from V n o V n. For he cylindrical Brownian moion B, i i well known ha he following decompoiion hold: B = β i k i (4.1) i=1 where {k 1, k 2,..., k i,...} i an orhonormal bai of K, and β i, i = 1, 2, 3,... are independen andard Brownian moion. Se B n = (β 1,..., βn ). Define F n = σ(b n, N(, A), A B(), ) compleed by he probabiliy meaure P, and pu φ n = E[P n φ FT n and b n () = E[P n b() F n. Conider he following backward ochaic differenial equaion on he finie dimenional pace V n : dy n = A n Y n d + b n()d + Z n dbn + (x)ñ(d, dx) ; < T (4.2) Q n Y n T = φ n (ω) a.. (4.3) A A n i a bounded linear operaor from V n o V n, i follow from he reul of Siu [S ha (4.2) (4.3) admi a unique F n - adaped oluion (Y n, Zn, Qn ), where Y n V n, Z n L 2 (K n, V n ), K n = pan(k 1, k 2,..., k n ) and Q n L 2 (ν). We are going o how ha he equence (Y n, Z n, Q n ) admi a convergen ubequence. Uing Iô formula, we find ha 2E[ E[ Y n 2 H = E[ φ n 2 H 2E[ Y n, P nay n d < Y n, b n () > d E[ Z n 2 L 2 (K d E[ n,v n) d Q n (x) 2 Hν(dx), (4.4) where Z n 2 L 2 (K = n n,v n) i,j=1 (Zn (i, j))2 and for he Hilber-Schmid norm. I follow from (3.2) ha E[ Y n 2 H E[ φ 2 H αe[ Y n 2 V d + λe[ Y n 2 Hd 1

11 +E[ Y n 2 H d + E[ b() 2 H d E[ Z n 2 L 2 (K d E[ n,v n) d Q n (x) 2 H ν(dx). Hence, E[ Y n 2 H + αe[ Y n 2 V d + E[ E[ φ 2 H + (λ + 1)E[ Z n 2 L 2 (K,H) d + E[ Y n 2 H d + E[ d Q n (x) 2 H ν(dx) b() 2 H d where Z n = Z n P n, and P n i he projecion from K ino K n = pan(k 1,..., k n ). In paricular, Se Ȳ n = E[ Y n 2 H E[ φ 2 H + (λ + 1)E[ Y n 2 Hd + E[ Y n 2 Hd. Then (4.5) implie ha d(e(λ+1) Ȳ n) e (λ+1) (E[ φ 2 d H + E[ b() 2 Hd) Hence, E[ Y n 2 Hd C(E[ φ 2 H + E[ b() 2 Hd), where C i an appropriae conan. Thi furher implie ha up{ n up{ n up{ n up{ n E[ Y n 2 Hd} < b() 2 Hd (4.5) E[ Y n 2 V d} < (4.6) E[ Z n 2 L 2 (K,H)d} < (4.7) E[ Q n 2 L 2 (ν)d} < (4.8) For a eparable Hilber pace L, we denoe by M 2 ([, T, L) he Hilber pace of progreively meaurable, quare inegrable, L-valued procee equipped wih he inner produc < a, b > M = E[ < a, b > L d. By he weak compacne of a Hilber pace, i follow from (4.6),(4.7)and (4.8) ha a ubequence {n k, k 1} can be eleced o ha Y n k., k 1 converge weakly o ome limi Y in M 2 ([, T, V ), Z n k., k 1 converge weakly o ome limi Z in M 2 ([, T, L 2 (K, H)) and Q n k., k 1 converge weakly o ome limi Q in M 2 ([, T, L 2 (ν)). Le u prove ha ( a verion of )(Y, Z, Q) i a oluion o he backward ochaic evoluion equaion (3.3) and (3.4). For n i 1, we have ha d Y n, e i = P n AY n, e i d + b n (), e i d + Z n db, e i + Q n (x), e i Ñ(d, dx) = AY n, e i d + b n (), e i d + Z n db, e i + Q n (x), e i Ñ(d, dx) (4.9) 11

12 Le h() be an aboluely coninuou funcion from [, T o R wih h ( ) L 2 ([, T ) and h() =. By he Iô formula, YT n, e i h(t ) = h() AY n, e i d + h() b n (), e i d + h()d Z n db, e i + h() Q n (x), ei Ñ(d, dx) + Replacing n by n k in (4.1) and leing k o obain φ, e i h(t ) = + h() AY, e i d + h()d Z db, e i + h() b(), e i d + Y n, e i h ()d. (4.1) h() Q (x), e i Ñ(d, dx) Y, e i h ()d. (4.11) From (4.1) o (4.11), we have ued he fac ha he linear mapping from M 2 ([, T, L 2 (K, H)) ino L 2 (Ω) defined by (Z) = h()d Z db, e i = j=1 h() Z (k j ), e i dβ j i coninuou and alo he linear mapping F from M 2 ([, T, L 2 (ν)) ino L 2 (Ω) defined by F (Q) = h() Q (x), e i Ñ(d, dx) i coninuou. So, he convergence of (4.1) o (4.11) ake place weakly in L 2 (Ω). Fix (, T ) and chooe, for n 1, 1, + 1 2n, h n () = 1 1 n ( + 1 2n ), 1 2n + 1 2n,, 1 2n Wih h( ) replaced by h n ( ) in (4.11), i follow ha φ, e i = h n () AY, e i d + + h n ()d h n () b(), e i d + Z u db u, e i + 1 n Sending n o infiniy in (4.12) we ge ha φ, e i = + AY, e i d n b(), e i d 1 2n Q (x), e i Ñ(d, dx) + d h() Q (x), e i Ñ(d, dx) Y, e i d. (4.12) Z u db u, e i + Y, e i. 12

13 for almo all [, T ( wih repec o Lebegue meaure). A i i arbirary, hi implie ha φ = AY d + b()d + Z db + Q (x)ñ(d, dx) + Y. (4.13) for almo all [, T ( wih repec o Lebegue meaure). For [, T, define Ŷ = φ AY d b()d Z db Q (x)ñ(d, dx). Then we ee ha (Ŷ, Z, Q ) alo aifie (ii) in he Theorem 4.1 wih Y replaced by Ŷ for all [, T. Hence, (Ŷ, Z, Q ) i a oluion o he equaion (3.3) and (3.4). Uniquene: Le (Y, Z, Q ) and (Ȳ, Z, Q ) be wo oluion of he equaion (3.3). Then A(Y Ȳ)d + Applying Iô formula, we ge (Z Z )db + (Y Ȳ) + (Q (x) Q (x))ñ(d, dx) = = Y Ȳ 2 H + 2 Y Ȳ, dm [ Y Ȳ + Q (x) Q (x) 2 2 H Y Ȳ HÑ(d, dx) A(Y Ȳ), Y Ȳ d + where M = (Z Z )db. By (3.2),we ge ha E[ Y Ȳ 2 H = 2 E[ α λ Z Z 2 L 2 (K,H) d Q (x) Q (x) 2 Hdν(dx) (4.14) E[ A(Y Ȳ), Y Ȳ d E[ Q (x) Q (x) 2 H dν(dx) E[ Y Ȳ 2 V d + λ E[ Y Ȳ 2 H d. E[ Y Ȳ 2 Hd Z Z 2 L 2 (K,H) d By a ronwall ype inequaliy, i follow ha E[ Y Ȳ 2 H =, which prove he uniquene. 13

14 Lemma 4.3 Aume ha E[ φ 2 H <, and ha b(, y, z, q, ω) = b(, z, q, ω) i independen of y. Then here exi a unique H L 2 (K, H) L 2 (ν)-valued progreively meaurable proce (Y, Z, Q ) uch ha (i) (ii) φ = Y + E[ Y 2 H d <, E[ AY d + Z 2 L 2 (K,H) d <, E[ Q 2 L 2 (ν) d <. Z db + Q(x)Ñ(d, dx); T. b(, Z, Q )d + Proof. Se Z =, Q =. Denoe by (Y n, Zn, Qn ) he unique oluion of he backward ochaic evoluion equaion: dy n = AY n d + b(, Zn 1, Q n 1 )d + Z n db + Q n (x)ñ(d, dx) (4.15) YT n = φ(ω). (4.16) The exience of uch a oluion (Y n, Zn, Qn ) ha been proved in Lemma 4.2. Puing M n = Zn db, and by Iô formula we ge ha = Y n+1 T = Y n Y n T 2 H Y n 2 H + 2 b(, Z n, Qn Y n+1 [ Y n+1 Q n+1 In virue of (3.2), for ε >, E[ Y n+1 = 2E[ 2E[ λe[ + εe[ Y n 2 H + E[ A(Y n+1 b(, Z n, Qn Y n+1 b(, Z n, Qn A(Y n+1 Y n ), Y n+1 Y n d ) b(, Zn 1, Q n 1 ), Y n+1 Y n d Y n + Q n+1 Q n 2 H Y n+1 Y n 2 HÑ(d, dx) Q n 2 H dν(dx) Y n, d(m n+1 M n ) + Z n+1 Z n+1 Z n 2 L 2 (K,H) d + E[ Y n ), Y n+1 Y n d ) b(, Zn 1, Q n 1 ), Y n+1 Y n d Y n 2 Hd αe[ Y n+1 Y n 2 V d ) b(, Zn 1 Z n 2 L 2 (K,H) d Q n+1 Q n 2 H dν(dx), Q n 1 ) 2 H d + 1 ε E[ Y n+1 Y n 2 Hd (4.17) 14

15 Chooe ε < 1 4c, where c i he Lipchiz conan in (3.5). I follow from (4.17) ha Y n 2 H + E[ +αe[ Y n+1 Y n 2 V d + E[ E[ Y n+1 (λ + 1 ε )E[ E[ Y n+1 Z n+1 Le β = λ + 1 ε. Then we have from (4.18) ha Z n 2 L 2 (K,H) d Y n 2 Hd E[ Q n+1 Q n 2 H dν(dx) Z n Z n 1 2 L 2 (K,H) d Q n Qn 1 2 H dν(dx) (4.18) d d (eβ E[ + e β E[ + e β E[ + αe β E[ 1 2 eβ E[ eβ E[ Z n+1 Y n+1 Q n+1 Y n+1 Y n 2 Hd Z n 2 L 2 (K,H) d Q n 2 H dν(dx) Y n 2 V d Z n Zn 1 2 L 2 (K,H) d Q n Q n 1 2 Hdν(dx) (4.19) From here, following a imilar proof a in [PP1 we will how ha (Y n, Z n, Q n ) converge o ome limi (Y, Z, Q) in he produc pace of M 2 ([, T, V ),M 2 ([, T, L 2 (K, H)) and M 2 ([, T, L 2 (ν)). Inegraing boh ide in (4.19) we ge ha E[ + Y n e β E[ E[ In paricular, 1 2 Y n 2 H d + Z n+1 E[ [ Q n+1 E[ Z n+1 Q n 2 Hdν(dx)d + α Z n Zn 1 2 L 2 (K,H) deβ d Z n 2 L 2 (K,H) deβ d + E[ Z n Z n 1 2 L 2 (K,H) deβ d + Z n 2 L 2 (K,H) deβ d e β E[ E[ e β E[ e β E[ Y n+1 Q n+1 Y n 2 V de β d Q n Qn 1 2 H dν(dx)d Q n 2 H dν(dx)d (4.2) Q n Q n 1 2 Hdν(dx)d (4.21) 15

16 Thi implie ha E[ Z n+1 Z n 2 L 2 (K,H) deβ d + e β E[ Q n+1 Q n 2 Hdν(dx)d ( 1 2 )n C for ome conan C. Thu, i follow from (4.2) ha Hence, we conclude from (4.18) ha E[ Y n+1 Y n 2 Hd ( 1 2 )n C (4.22) E[ Z n+1 Z n 2 L 2 (K,H) d + E[ Q n+1 Q n 2 H dν(dx)d ( 1 2 )n Cβ {E[ Z n Zn 1 2 L 2 (K,H) d + E[ Uing he above inequaliy repeaedly give Q n Qn 1 2 H dν(dx)} (4.23) E[ Z n+1 Z n 2 L 2 (K,H) d + E[ Q n+1 Q n 2 Hdν(dx) Combining (4.18) and (4.23) we have ha ( 1 2 )n ncβ (4.24) E[ Y n+1 Y n 2 V d ( 1 2 )n (n + 1)Cβ (4.25) I follow now from (4.24) and (4.25) ha he he equence (Y n, Z n, Q n ), n 1 converge in M 2 ([, T, V ) M 2 ([, T, L 2 (K, H)) M 2 ([, T, L 2 (ν)) o ome limi (Y, Z, Q ). Leing n in (4.14), we ee ha (Y, Z, Q ) aifie Y + AY d + b(, Z )d + Z db + Q (x)ñ(d, dx) = φ (4.26) i.e., (Y, Z, Q ) i a oluion o equaion (3.3). Uniquene 16

17 Le (Y, Z, Q ), (Ȳ, Z, Q ) be wo oluion. By Iô formula, a in (4.14) we have E[ Y Ȳ 2 H + E[ Z Z 2 L 2 (K,H) d Conequenly, + E[ = 2E[ 2E[ λe[ E[ E[ By ronwall inequaliy, Q (x) Q (x) 2 H dν(dx) < A(Y Ȳ), Y Ȳ > d b(, Z, Q ) b(, Z, Q ), Y Ȳ d Y Ȳ 2 Hd αe[ Y Ȳ 2 V d Z Z 2 L 2 (K,H) d + ce[ Y Ȳ 2 H d Q (x) Q (x) 2 Hdν(dx) (4.27) E[ Y Ȳ 2 H (λ + c)e[ Y = Ȳ which furher implie Z = Z and Q = Q by (4.27). Proof of Theorem 4.1. Y Ȳ 2 Hd (4.28) Le Y =. Define, for n 1, (Y n+1, Z n+1, Q n+1 ) o be he oluion of he equaion: dy n+1 = AY n+1 d + b(, Y n + Z n+1 db +, Q n+1 )d, Zn+1 Q n+1 (x)ñ(d, dx) (4.29) Y n+1 T = φ (4.3) The exience of (Y n+1, Z n+1, Q n+1 ) i conained in Lemma 4.3. Uing he imilar argumen a in he proof of Lemma 4.3 i can be hown ha (Y n+1, Z n+1, Q n+1 converge o ome limi (Y, Z, Q ), and moreover (Y, Z, Q ) i he unique oluion o equaion (3.3). We omi he deail avoiding he repeaing. Example 4.2 Le H = L 2 (R d ), and e V = H 1 2 (Rd ) = {u L 2 (R d ); u L 2 (R d R d )} Denoe by a(x) = (a ij (x)) a marix-valued funcion on R d aifing he uniform ellipiciy condiion: 1 c I d a(x) ci d for ome conan c (, ). 17 )

18 Le f(x) be a vecor field on R d wih f L p (R d ) for ome p > d. Define Au = div(a(x) u(x)) + f(x) u(x) Then (3.2) i fulfilled for (H, V, A). Thu, for any choice of cylindrical Brownian moion B, any drif coefficien b(, y, z, ω) aifying (3.5) and (3.6) and erminal random variable φ, he main reul in Secion 4 applie. Acknowledgemen Thi work wa iniiaed when he fir and he hird auhor aended a workhop on SPDE in Warwick in May, 21. We would like o hank he organizer Jo Kennedy, Roger Tribe and Jerzy Zabczyk, a well a David Elworhy, for he inviaion and hopialiy. We alo hank he Mahemaic Iniue a Univeriy of Warwick for providing he imulaing reearch environmen. Reference [B1 [B2 A.Benouan: Maximum principle and dynamic programming approache of he opimal conrol of parially oberved diffuion. Sochaic 9 (1983), A.Benouan: Sochaic maximum principle for yem wih parial informaion and applicaion o he eparaion principle. In M.Davi and R.Ellio (edior): Applied Sochaic Analyi. ordon and Breach 1991, pp [EPQ N. El Karoui, S. Peng and M.C. Queuez: Backward ochaic differenial equaion in finance. Mahemaical Finance 7 (1997), [FØS N.C. Framad, B. Økendal and A. Sulem: Sufficien ochaic maximum principle for he opimal conrol of jump diffuion and applicaion o finance. J. Opim. Theory and Appl. 121 (24), and Vol. 124, No. 2 (25) (Erraa). [MY J. Ma and J. Yong: Forward-Backward Sochaic Differenial Equaion and Their Applicaion. Springer LNM 172, Springer-Verlag [NS D. Nualar and W. Schouen: Backward ochaic differenial equaion and Feynman-Kac formula for Lévy procee, wih applicaion in finance. Bernoulli 7 (21), [Ø [ØS [P B. Økendal: Opimal conrol of ochaic parial differenial equaion. Soch. Anal. and Appl. (o appear). B. Økendal, Sulem, A.: Applied Sochaic Conrol of Jump Diffuion. Springer- Verlag 24 (o appear). E. Pardoux: Sochaic parial differenial equaion and filering of diffuion procee. Sochaic 3 (1979),

19 [PP1 E. Pardoux and S. Peng: Adaped oluion of backward ochaic differenial equaion. Syem and Conrol Leer 14 (199), [PP2 E. Pardoux and S. Peng: Backward doubly ochaic differenial equaion and yem of quailinear ochaic parial differenial equaion, Probabiliy Theory and Relaed Field 98 (1994), [PZ [S.D. Prao and J. Zabczyk: Sochaic equaion in infinie dimenion. Cambridge Univeriy Pre, R. Siu: On oluion of backward ochaic differenial equaion wih jump and applicaion. Sochaic Procee and heir Applicaion 66(1997), [YZ J.Yong and.y.zhou: Sochaic Conrol. Springer-Verlag