Singular control of SPDEs and backward stochastic partial diffe. reflection

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1 Singular conrol of SPDEs and backward sochasic parial differenial equaions wih reflecion Universiy of Mancheser Join work wih Bern Øksendal and Agnès Sulem Singular conrol of SPDEs and backward sochasic parial diffe

2 Absrac We sudy singular conrol problems for sochasic parial differenial equaions. We esablish sufficien and necessary maximum principles for an opimal conrol of such sysems. The associaed adjoin processes saisfy a kind of backward sochasic parial differenial equaion (BSPDE) wih reflecion. Exisence and uniqueness of BSPDEs wih reflecion are obained. Singular conrol of SPDEs and backward sochasic parial diffe

3 The conrol problem Le D be a given bounded domain in R d. We consider a general sysem where he sae Y (, x) a ime and a he poin x D R is given by a sochasic parial differenial equaion (SPDE) as follows: dy (, x) = {AY (, x) + b(, x, Y (, x))}d + σ(, x, Y (, x))db() + λ(, x, Y (, x))ξ(d, x) ; (, x) [0, T ] D Y (0, x) = y 0 (x) ; x D Y (, x) = 0 ; (, x) (0, T ) D. (1) Here A is a given linear second order parial differenial operaor. Singular conrol of SPDEs and backward sochasic parial diffe

4 The conrol problem We assume ha he coefficiens and b(, x, y) : [0, T ] D R R, σ(, x, y) : [0, T ] D R R, λ(, x, y) : [0, T ] D R R are C 1 funcions wih respec o y. The se of possible conrols, A, is a given family of adaped processes ξ(, x), which are non-decreasing and lef-coninuous w.r.. for all x, ξ(0, x) = 0. The performance funcional has he form [ J(ξ) = E f (, x, Y (, x))ddx + g(x, Y (T, x))dx + D D 0 0 ] h(, x, Y (, x))ξ(d, x), (2) D Singular conrol of SPDEs and backward sochasic parial diffe

5 The conrol problem where f (, x, y), g(x, y) and h(, x, y) are bounded measurable funcions which are differeniable in he argumen y and coninuous w.r... We wan o maximize J(ξ) over all ξ A, where A is he se of admissible singular conrols. Thus we wan o find ξ A (called an opimal conrol) such ha sup J(ξ) = J(ξ ) ξ A Singular conrol of SPDEs and backward sochasic parial diffe

6 Sufficien maximum principle Define he Hamilonian H by H(, x, y, p, q)(d, ξ(d, x)) = {f (, x, y) + b(, x, y)p + σ(, x, y)q}d + {λ(, x, y)p + h(, x, y)}ξ(d, x). (3) To his Hamilonian we associae he following backward SPDE (BSPDE) in he unknown process (p(, x), q(, x)): { dp(, x) = A p(, x)d + H (, x, Y (, x), p(, x), y q(, x))(d, ξ(d, x))} + q(, x)db() ; (, x) (0, T ) D (4) wih boundary/erminal values p(t, x) = g (x, Y (T, x)) ; x D (5) y p(, x) = 0 ; (, x) (0, T ) D. (6) Here A denoes he adjoin of he operaor A. Singular conrol of SPDEs and backward sochasic parial diffe

7 Sufficien maximum principle Theorem[1. Sufficien maximum principle] Le ˆξ A wih corresponding soluions Ŷ (, x), ˆp(, x), ˆq(, x). Assume ha and Assume ha E[ y h(x, y) is concave (7) (y, ξ) H(, x, y, ˆp(, x), ˆq(, x))(d, ξ(d, x)) D ( is concave. (8) 0 {(Y ξ (, x) Ŷ (, x))2ˆq 2 (, x) + ˆp 2 (, x) (σ(, x, Y ξ (, x)) σ(, x, Ŷ (, x)) 2 }d)dx] <, (9) Singular conrol of SPDEs and backward sochasic parial diffe

8 Sufficien maximum principle for all ξ A. Moreover, assume ha he following maximum condiion holds: {λ(, x, Ŷ (, x))ˆp(, x) + h(, x, Ŷ (, x))}ξ(d, x) {λ(, x, Ŷ (, x))ˆp(, x) + h(, x, Ŷ (, x))}ˆξ(d, x) for all ξ A. (10) Then ˆξ is an opimal singular conrol. Singular conrol of SPDEs and backward sochasic parial diffe

9 Sufficien maximum principle Theorem[2.Sufficien maximum principle II] Suppose he condiions of he above Theorem hold. Suppose ξ A, and ha ξ ogeher wih is corresponding processes Y ξ (, x), p ξ (, x), q ξ (, x) solve he coupled SPDE-RBSPDE sysem consising of he SPDE (1) ogeher wih he refleced backward SPDE (RBSPDE) given by dp ξ (, x) { = A p ξ (, x) + f y (, x, Y ξ (, x)) + b y (, x, Y ξ (, x))p ξ (, x) + σ } y (, x, Y ξ (, x))q ξ (, x) d { λ y (, x, Y ξ (, x))p ξ (, x) + h } y (, x, Y ξ (, x)) ξ(d, x) ; (, x) [0, T ] D Singular conrol of SPDEs and backward sochasic parial diffe

10 Sufficien maximum principle λ(, x, Y ξ (, x))p ξ (, x) + h(, x, Y ξ (, x)) 0 ; for all, x, a.s. {λ(, x, Y ξ (, x))p ξ (, x) + h(, x, Y ξ (, x))}ξ(d, x) = 0 ; for all, x, a.s. p(t, x) = g y (x, Y ξ (T, x)) ; x D p(, x) = 0 ; (, x) (0, T ) D. Then ξ maximizes he performance funcional J(ξ). Singular conrol of SPDEs and backward sochasic parial diffe

11 A necessary maximum principle A weakness of he sufficien maximum principle obained in he previous secion are he raher resricive concaviy condiions, which do no always hold in applicaions. Therefore i is of ineres o obain a maximum principle which does no need hese condiions. Theorem[3.Necessary maximum principle] (i) Suppose ξ A is opimal, i.e. max J(ξ) = ξ A J(ξ ). (11) Le Y, (p, q ) be he corresponding soluion associaed wih ξ. Then λ(, x, Y (, x))p (, x) + h(, x, Y (, x)) 0 (12) for all, x [0, T ] D, a.s. Singular conrol of SPDEs and backward sochasic parial diffe

12 A necessary maximum principle and {λ(, x, Y (, x))p (, x) + h(, x, Y (, x))}ξ (d, x) = 0 (13) for all, x [0, T ] D, a.s. Singular conrol of SPDEs and backward sochasic parial diffe

13 A necessary maximum principle (ii) Conversely, suppose ha here exiss ˆξ A such ha he corresponding soluions Ŷ (, x), (ˆp(, x), ˆq(, x)) of (1) and (4)-(5), respecively, saisfy λ(, x, Ŷ (, x))ˆp(, x)+h(, x, Ŷ (, x)) 0 and for all, x [0, T ] D, a.s. (14) {λ(, x, Ŷ (, x))ˆp(, x) + h(, x, Ŷ (, x))}ˆξ(d, x) = 0 (15) for all, x [0, T ] D, a.s. Then ˆξ is a direcional sub-saionary poin for J( ), in he sense ha 1 lim y 0 + y (J(ˆξ + yζ) J(ˆξ)) 0 for all ζ V(ˆξ). (16) Singular conrol of SPDEs and backward sochasic parial diffe

14 Exisence and uniqueness for BSPDEs wih reflecion Nex, I will presen he exisence and uniqueness resul for refleced backward sochasic parial differenial equaions. For noaional simpliciy, we choose he operaor A o be he Laplacian operaor. However, our mehods work equally well for general second order differenial operaors like A = 1 2 d i,j=1 (a ij (x) ), x i x j where a = (a ij (x)) : D R d d is a measurable, symmeric marix-valued funcion which saisfies he uniform ellipic condiion d λ z 2 a ij (x)z i z j Λ z 2, z R d i,j=1 and x D for some consan λ, Λ > 0 Singular conrol of SPDEs and backward sochasic parial diffe

15 Exisence and uniqueness for BSPDEs wih reflecion Firs we will esablish a comparison heorem for BSPDEs, which is of independen ineres. Consider wo backward SPDEs: du 1 (, x) = u 1 ()d b 1 (, u 1 (, x), Z 1 (, x))d + Z 1 (, x)db, u 1 (T, x) = φ 1 (x) a.s. (17) du 2 (, x) = u 2 ()d b 2 (, u 2 (, x), Z 2 (, x))d + Z 2 (, x)db, u 2 (T, x) = φ 2 (x) a.s. (18) From now on, if u(, x) is a funcion of (, x), we wrie u() for he funcion u(, ). Singular conrol of SPDEs and backward sochasic parial diffe

16 A comparison heorem The following resul is a comparison heorem for backward sochasic parial differenial equaions. Theorem[4. Comparison heorem for BSPDEs] Suppose φ 1 (x) φ 2 (x) and b 1 (, u, z) b 2 (, u, z). Then we have u 1 (, x) u 2 (, x), x D, a.e. for every [0, T ]. Seps of he proof. For n 1, define funcions ψ n (z), f n (x) as follows (see [DP1]). 0 if z 0, ψ n (z) = 2nz if 0 z 1 n, (19) 2 if z > 1 n. { 0 if x 0, f n (x) = x 0 dy y 0 ψ n(z)dz if x > 0. (20) Singular conrol of SPDEs and backward sochasic parial diffe

17 A comparison heorem We have f n(x) = 0 if x 0, nx 2 if x 1 n, 2x 1 n if x > 1 n. (21) Also f n (x) (x + ) 2 as n. For h K := L 2 (D), se F n (h) = f n (h(x))dx. Applying Io s formula we ge F n (u 1 () u 2 ()) = F n (φ 1 φ 2 ) + D F n(u 1 (s) u 2 (s))( (u 1 (s) u 2 (s)))ds Singular conrol of SPDEs and backward sochasic parial diffe

18 A comparison heorem + F n(u 1 (s) u 2 (s))(b 1 (s, u 1 (s), Z 1 (s)) b 2 (s, u 2 (s), Z 2 (s)))ds 1 2 F n(u 1 (s) u 2 (s))(z 1 (s) Z 2 (s))db s F n (u 1 (s) u 2 (s))(z 1 (s) Z 2 (s), Z 1 (s) Z 2 (s))ds =: I 1 n + I 2 n + I 3 n + I 4 n + I 5 n, (22) Singular conrol of SPDEs and backward sochasic parial diffe

19 A comparison heorem Afer carefully analyzing every erm on he righ and afer cancelaion of erms, we can show ha F n (u 1 () u 2 ()) F n (φ 1 φ 2 ) + C D ((u 1 (s, x) u 2 (s, x)) + ) 2 dxds F n(u 1 (s) u 2 (s))(z 1 (s) Z 2 (s))db s (23) Take expecaion and le n o ge E[ ((u 1 (, x) u 2 (, x)) + ) 2 dx] dse[ ((u 1 (s, x) u 2 (s, x)) + ) 2 dx] D (24) Gronwall s inequaliy yields ha E[ ((u 1 (, x) u 2 (, x)) + ) 2 dx] = 0, (25) D which complees he proof of he heorem D Singular conrol of SPDEs and backward sochasic parial diffe

20 Exisence and uniqueness for BSPDEs wih reflecion Le V = W 1,2 0 (D) be he Sobolev space of order one wih he usual norm. Consider he refleced backward sochasic parial differenial equaion: du() = u()d b(, u(, x), Z(, x))d + Z(, x)db η(d, x), (0, T ), (26) u(, x) L(, x), (u(, x) L(, x))η(d, x)dx = 0, 0 D u(t, x) = φ(x) a.s. (27) Singular conrol of SPDEs and backward sochasic parial diffe

21 Exisence and uniqueness for BSPDEs wih reflecion Theorem[5. Exisence and Uniqueness] Assume ha E[ φ 2 K ] <. and ha b(s, u 1, z 1 ) b(s, u 1, z 1 ) C( u 1 u 2 + z 1 z 2 ). Le L(, x) be a measurable funcion which is differeniable in and wicely differeniable in x such ha φ(x) L(T, x) and 0 D L (, x) 2 dxd <, L(, x) 2 dxd <. 0 D Then here exiss a unique K L 2 (D, R m ) K-valued progressively measurable process (u(, x), Z(, x), η(, x)) such ha Singular conrol of SPDEs and backward sochasic parial diffe

22 Exisence and uniqueness for BSPDEs wih reflecion (i) E[ 0 u() 2 V d] <, E[ 0 Z() 2 L 2 (D,R m ) d] <. (ii) η is a K-valued coninuous process, non-negaive and nondecreasing in and η(0, x) = 0. (iii) u(, x) = φ(x) + u(, x)ds + b(s, u(s, x), Z(s, x))ds Z(s, x)db s + η(t, x) η(, x); 0 T, (iv) u(, x) L(, x) a.e. x D, [0, T ]. (v) 0 D (u(, x) L(, x))η(d, x)dx = 0 (28) where u() sands for he K-valued coninuous process u(, ) and (iii) is undersood as an equaion in he dual space V of V. Singular conrol of SPDEs and backward sochasic parial diffe

23 The proof of exisence and uniqueness I will indicae how we prove he heorem. we inroduce he penalized BSPDEs: du n () = u n ()d b(, u n (, x), Z n (, x))d + Z n (, x)db n(u n (, x) L(, x)) d, (0, T ) (29) u n (T, x) = φ(x) a.s. (30) According o [ØPZ], he soluion (u n, Z n ) of he above equaion exiss and is unique. We are going o show ha he sequence (u n, Z n ) has a limi, which will be a soluion of he equaion (28). Singular conrol of SPDEs and backward sochasic parial diffe

24 The proof of exisence and uniqueness Firs we need some a priori esimaes. Lemma[1] Le (u n, Z n ) be he soluion of equaion (29). We have sup n E[sup u n () 2 K ] <, (31) sup E[ n sup E[ n 0 0 u n () 2 V ] <, (32) Z n () 2 L 2 (D,R m )] <. (33) Singular conrol of SPDEs and backward sochasic parial diffe

25 The proof of exisence and uniqueness We also need he following crucial esimaes. Lemma[2] Suppose he condiions in Theorem 5 hold. Then here is a consan C such ha E[ ((u n (, x) L(, x)) ) 2 dxd] C n 2. (34) 0 D Main ideas of he proof. Le f m be defined as in he proof of Theorem 4. Then f m (x) (x + ) 2 and f m(x) 2x + as m. For h K, se G m (h) = f m ( h(x))dx. D The idea is o apply Io s formula o he process u n () L() and for he funcional G m ( ). Singular conrol of SPDEs and backward sochasic parial diffe

26 The proof of exisence and uniqueness Lemma[3]. Le (u n, Z n ) be he soluion of equaion (29). We have lim E[ sup u n () u m () 2 n,m K ] = 0, (35) 0 T lim E[ u n () u m () 2 n,m V d] = 0. (36) 0 lim E[ Z n () Z m () 2 n,m L 2 (D,R m ) d] = 0. (37) 0 Singular conrol of SPDEs and backward sochasic parial diffe

27 The proof of exisence and uniqueness Main ideas of he proof. Applying Iô s formula, i follows ha = 2 u n () u m () 2 K < u n (s) u m (s), (u n (s) u m (s)) > ds < u n (s) u m (s), b(s, u n (s), Z n (s)) b(s, u m (s), Z m (s)) > < u n (s) u m (s), Z n (s) Z m (s) > db s < u n (s) u m (s), n(u n (s) L(s)) m(u m (s) L(s)) > Z n (s) Z m (s) 2 L 2 (D,R m ) ds ( Now we esimae each of he erms on he righ side. Singular conrol of SPDEs and backward sochasic parial diffe

28 The proof of exisence and uniqueness 2 = 2 < u n (s) u m (s), (u n (s) u m (s)) > ds u n (s) u m (s) 2 V ds. (39) By he Lipschiz coninuiy of b and he inequaliy ab εa 2 + C ε b 2, one has 2 C < u n (s) u m (s), b(s, u n (s), Z n (s)) b(s, u m (s), Z m (s)) > d u n (s) u m (s) 2 K ds Z n (s) Z m (s) 2 L 2 (D,R m ) ds. (40 Singular conrol of SPDEs and backward sochasic parial diffe

29 The proof of exisence and uniqueness In view of (34), we can show ha 2E[ 2m(E[ (E[ +2n(E[ (E[ < u n (s) u m (s), n(u n (s) L(s)) m(u m (s) L(s)) > D D C ( 1 n + 1 m ). D ((u n (s, x) L(s, x)) ) 2 dxds]) 1 2 ((u m (s, x) L(s, x)) ) 2 dxds]) 1 2 D ((u n (s, x) L(s, x)) ) 2 dxds]) 1 2 ((u m (s, x) L(s, x)) ) 2 dxds]) 1 2 ( Singular conrol of SPDEs and backward sochasic parial diffe

30 The proof of exisence and uniqueness I follows from (38) and (39) ha E[ u n () u m () 2 K ] E[ Z n (s) Z m (s) 2 L 2 (D,R m ) ds] +E[ C u n (s) u m (s) 2 V ds] E[ u n (s) u m (s) 2 K ]ds + C ( 1 n + 1 ). (42) m Applicaion of he Gronwall inequaliy yields lim n,m {E[ un () u m () 2 K ]+1 2 E[ Z n (s) Z m (s) 2 L 2 (D,R m ) ds]} = 0, (43) Singular conrol of SPDEs and backward sochasic parial diffe

31 The proof of exisence and uniqueness lim E[ u n (s) u m (s) 2 n,m V ds] = 0. (44) By (43) and he Burkholder inequaliy we can furher show ha The proof is complee lim E[ sup u n () u m () 2 n,m K ] = 0. (45) 0 T Singular conrol of SPDEs and backward sochasic parial diffe

32 The proof of exisence and uniqueness Proof of Theorem 5. From Lemma 3 we know ha (u n, Z n ), n 1, forms a Cauchy sequence. Denoe by u(, x), Z(, x) he limi of u n and Z n. Pu η n (, x) = n(u n (, x) L(, x)) Lemma 3.4 implies ha η n (, x) admis a non-negaive weak limi, denoed by η(, x), in he following Hilber space: K = {h; h is a K-valued adaped process such ha E[ wih inner produc < h 1, h 2 > K = E[ 0 h(s) 2 K ds] < } (46) 0 D h 1 (, x)h 2 (, x)ddx]. Singular conrol of SPDEs and backward sochasic parial diffe

33 The proof of exisence and uniqueness Se η(, x) = 0 η(s, x)ds. Then η is a coninuous K-valued process which is increasing in Le n in (29) o obain u(, x) = φ(x) + u(, x)ds + b(s, u(s, x), Z(s, x))ds Z(s, x)db s + η(t, x) η(, x); 0 T. (47) Furhermore we can show ha (u, η) fulfills he required properies. Singular conrol of SPDEs and backward sochasic parial diffe

34 The proof of exisence and uniqueness Uniqueness. Le (u 1, Z 1, η 1 ), (u 2, Z 2, η 2 ) be wo such soluions o equaion (28). By Iô s formula, we have = 2 u 1 () u 2 () 2 K < u 1 (s) u 2 (s), (u 1 (s) u 2 (s)) > ds < u 1 (s) u 2 (s), b(s, u 1 (s), Z 1 (s)) b(s, u 2 (s), Z 2 (s)) > ds < u 1 (s) u 2 (s), Z 1 (s) Z 2 (s) > db s < u 1 (s) u 2 (s), η 1 (ds) η 2 (ds) > Z 1 (s) Z 2 (s) 2 L 2 (D,R m ) ds (48) Singular conrol of SPDEs and backward sochasic parial diffe

35 The proof of exisence and uniqueness Noe ha = 2E[ 2E[ 2E[ +2E[ 2E[ D < u 1 (s) u 2 (s), η 1 (ds) η 2 (ds) >] (u 1 (s, x) L(s, x))η 1 (ds, x)dx] D D D (u 1 (s, x) L(s, x))η 2 (ds, x)dx] (u 2 (s, x) L(s, x))η 2 (ds, x)dx] (u 2 (s, x) L(s, x))η 1 (ds, x)dx] 0 (49) This observaion allows o prove Singular conrol of SPDEs and backward sochasic parial diffe

36 The proof of exisence and uniqueness E[ u 1 () u 2 () 2 K ] E[ Z 1 (s) Z 2 (s) 2 L 2 (D,R m ) ds] C E[ u 1 (s) u 2 (s) 2 K ]ds. (50) Appealing o Gronwall inequaliy, his implies u 1 = u 2, Z 1 = Z 2 which furher gives η 1 = η 2 from he equaion hey saisfy. Singular conrol of SPDEs and backward sochasic parial diffe

37 Link o opimal sopping This par provides a link beween he soluion of a refleced backward sochasic parial differenial equaion and an opimal sopping problem. Le u(, x) be he soluion of he following refleced BSPDE. = u(, x) 1 T φ(x) + u(, x)ds + k(s, x, u(s, x), Z(s, x))ds 2 Z(s, x)db s + η(t, x) η(, x); 0 T, u(, x) L(, x), (u(s, x) L(s, x))η(d, x)dx = 0 a.s. (51) 0 D Singular conrol of SPDEs and backward sochasic parial diffe

38 Link o opimal sopping Le S,T be he se of all sopping imes τ saisfying τ T. For τ S,T, define R (τ, x) = τ P s k(s, x)ds+p τ L(τ, x)χ {τ<t } +P τ φ(x)χ {τ=t }, where k(s, ) = k(s,, u(s, ), Z(s, )) and P denoes he hea semigroup generaed by he Laplacian operaor 1 2. Singular conrol of SPDEs and backward sochasic parial diffe

39 Link o opimal sopping Here, and in he following we will use he simplified noaion P k(s, x) = (P k(s, ))(x) ec. Theorem[6. Opimal sopping] u(, x) is he value funcion of he he opimal sopping problem associaed wih R (τ, x), i.e., u(, x) = esssup τ S,T E[R (τ, x) F ] (52) Singular conrol of SPDEs and backward sochasic parial diffe

40 References C. Donai-Marin, E. Pardoux: Whie noise driven SPDEs wih reflecion. Probab. Theory Rel. Fields 95, 1-24(1993). C. Donai-Marin, E. Pardoux: EDPS réfléchies e calcul de Malliavin. (French)[SPDEs wih reflecion and Malliavin Calculus]. Bull. Sci. Mah. 121(5)(1997), A. Gegou-Pei, E. Pardoux: Equaions Différenielles Sochasiques Rérogrades Réfléchies Dans Un Convexe. Sochasics and Sochasics Repors 57 (1996) U. G. Haussmann, E. Pardoux: Sochasic variaional inequaliies of parabolic ype. Appl. Mah. Opim. 20(1989), Singular conrol of SPDEs and backward sochasic parial diffe

41 References D. Nualar, E. Pardoux: Whie noise driven by quasilinear SPDEs wih reflecion. Probab. Theory Rel. Fields 93,77-89(1992). B. Øksendal, A. Sulem: Singular sochasic conrol and opimal sopping wih parial informaion of jump diffusions. Preprin 2010, Oslo. B. Øksendal, F. Proske and T. Zhang: Backward sochasic parial differenial equaions wih jumps and applicaion o opimal conrol of random jump fields. Sochasics 77:5 (2005) Singular conrol of SPDEs and backward sochasic parial diffe

42 References E. Pardoux and S. Peng: Adaped soluions of backward sochasic differenial equaions. Sysem and Conrol Leers 14 (1990) E. Pardoux and S. Peng: Backward doubly sochasic differnial equaions and sysems of quasilinear sochasic parial differenial equaions. Probab. Theory and Rel. Fields 98 (1994) T.Zhang : Whie noise driven SPDEs wih reflecion: srong Feller properies and Harnack inequaliies. Poenial Analysis 33:2 (2010) Singular conrol of SPDEs and backward sochasic parial diffe