In nite horizon optimal control of forward-backward stochastic di erential equations with delay.

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1 In nie horizon opimal conrol of forward-backward sochasic di erenial equaions wih delay. Nacira AGAM and Bern ØKSENDAL yz 25 January 213 Absrac Our sysem is governed by a forward -backward sochasic di erenial equaion wih delay. Su cien and necessary maximum principles for parial informaion in in nie horizon are derived. We illusrae our resuls by an applicaion o an opimal consumpion from a cash ow wih delay wih respec o recursive uiliy. Keywords: In nie horizon; Opimal conrol; Sochasic delay equaion; Sochasic di ereniel uiliy; Lévy processes; Maximum principle; Hamilonian; Adjoin processes; Parial informaion. 1 Preliminaries Le (; F; (F ) ; P ) be a complee lered probabiliy space on which a onedimensional sandard Brownian moion B (:) and an independen compensaed Poisson random measure N(d; da) = N(d; da) (da) d are de ned. We sudy he following coupled forward-backward sochasic di erenial equaions (FBSDEs, for shor) conrol sysem wih delay: dx() = b (; X(); X 1 (); X 2 (); u()) d + (; X(); X 1 (); X 2 (); u()) db() + (; X(); X 1 (); X 2 (); u(), a) N(d; da); 2 [; 1) X() = X (); 2 [ ; ] (1:1) Laboraory of Applied Mahemaics, Universiy Med Khider, Po. Box 145, Biskra (7) Algeria. y Cener of Mahemaics for Applicaions (CMA), Universiy of Oslo, Box 153 Blindern, N-316 Oslo, Norway. z The research leading o hese resuls has received funding from he European esearch Council under he European Communiy s Sevenh Framework Programme (FP7/27-213) / EC gran agreemen no [22887]. 1

2 dy () = g (; X(); X 1 (); X 2 (); Y (); Z(); u()) d + Z()dB() + K (; a) N(d; da); 2 [; 1). (1:2) Throughou his paper, we inroduce he following basic assumpions X 1 () = X( ) and X 2 () = e ( r) X(r)dr; >, > are given consans, b : [; 1) U! ; : [; 1) U! ; ; K : [; 1) U! ; g : [; 1) U! ; where he coe ciens b; ; ; f and g are Fréche di ereniable (C 1 ) wih respec o he variables (x; x 1 ; x 2 ; y; z; k(:); u). We denoe by, he se of all funcions k :!. We inerpree he in nie horizon BSDE (1:2) in he sense of Pardoux [8], i.e. for all T < 1, (Y (); Z(); K (; :)) solves he equaion T Y () = Y (T ) + g (s; X(s); X 1 (s); X 2 (s); Y (s); Z(s)) ds T and moreover, E[sup K (s; a) N(ds; da); T, T Z(s)dB(s) (1:3) e Y 2 () + e (Z 2 () + K 2 (s; a) (da))d] < 1 (1:4) for su cienly large consan. See secion 4 in [8] for more deails. Noe ha by he Iô represenaion heorem for Lévy processes ( see Theorem 13:28 in [2]), equaion (1:3) is equivalen o he equaion T Y () = E[Y (T ) + g (s; X(s); X 1 (s); X 2 (s); Y (s); Z(s)) ds j F ]; T. (1:5) Le E F be a given sub lraion, represening he informaion available o he conroller a ime. Le U be a non-empy subse of : We le A E denoe he family of admissible E -adaped conrol processes. The corresponding performance funcional is 2

3 J(u) = E[ f (; X(); X 1 (); X 2 (); Y (); Z(); K (; :) ; u()) d + h(y ())] (1:6) where we assume ha he funcions f and h are Fréche di ereniable (C 1 ) wih respec o he variables (x; x 1 ; x 2 ; y; z; k(:); u) and hey saisfy E[ jf (; X(); X 1 (); X 2 (); Y (); Z(); K (; :) ; u()) j d + jh(y ()j] < 1. (1:7) The opimal conrol problem is o maximise he value funcion E 2 such ha E (X ) = sup u2a E J(u) (1:8) over an admissible conrol domain A E which is assumed o be locally convex. The Hamilonian H : [; 1) L 2 () U L 2 ()! is de ned as H(; x; x 1 ; x 2 ; y; z; k(:); ; p; q; r(:); u) = f(; x; x 1 ; x 2 ; y; z; k; u) + g(; x; x 1 ; x 2 ; y; z; u) + b(; x; x 1 ; x 2 ; u)p + (; x; x 1 ; x 2 ; u)q + (; x; x 1 ; x 2 ; u; a)r(; a)(da). (1:9) We suppose ha he Hamilonian H is Fréche di ereniable (C 1 ) in he variables x; x 1 ; x 2 ; y; z; k. We associae o his problem he following pair of forward- backward SDEs in he adjoin processes (), ( p(); q(); r(; :)): d() (; X(); X 1(); X 2 (); Y (); Z(); K(; :); u(); (); p(); q(); r(; :)) d (; X(); X 1(); X 2 (); Y (); Z(); K(; :); u(); (); p(); q(); r(; :)) db() + r k H(; X(); X 1 (); X 2 (); Y (); Z(); K(; :); u(); (); p(); q(); r(; :)) N(d; da) () = h (Y ()) (1:1) dp() = E(() j F )d + q()db() + r(; a) N(d; da); 2 [; 1) (1:11) where 3

4 () (; X(); X 1(); X 2 (); Y (); Z(); K(; :); u(); (); p(); q(); r(; :)) ( + ; X( + ); X 1 ( + ); X 2 ( + ); Y ( + ); Z( + ); K( + ; :); u( + ); ( + ); p( + ); q( + ); r( + ; :)) e ( + (s; X(s); X 1 (s); X 2 (s); Y (s); Z(s); K(s; :); u(s); (s); p(s); q(s); r(s; :))e s ds). (1:12) The unknown process () is he adjoin process corresponding o he backward sysem (Y (); Z(); K(; a)) and he riple unknown (p(); q(); r(; :)) is he adjoin process corresponding o he forward sysem X(). 2 Su cien maximum principle for parial informaion We will prove in his secion ha under some assumpions he maximizaion of he Hamilonian funcion leads o an opimal conrol. Theorem 2.1 Le u 2 A E wih corresponding soluions X(); X 1 (); X 2 (); Y (); Z(); K(; a); p(); q(); r(; a) and () of equaions (1:1), (1:2), (1:1) and (1:11). Suppose ha: (H 1 ): (Transversaliy condiions) and (H 2 ): (Concaviy) The funcions x! h(x) and lim E[ p(t )( X(T ) X(T ))] lim E [ (T )( Y (T ) Y (T ))]. (x; x 1 ; x 2 ; y; z; k; u)! H(; x; x 1 ; x 2 ; y; z; k(:); p; q; ; r(:); u) are concave, for all 2 [; 1). (H 3 ): (The condiional maximum principle) max E[H(; X(); X 1 (); X 2 (); Y (); Z(); K(; :); v; (); p(); q(); r(; :)) j E ] v2u = E[H(; X(); X 1 (); X 2 (); Y (); Z(); K(; :); u(); (); p(); q(); r(; :)) j E ]. 4

5 (H 4 ): ( Growh condiions) Suppose for all u 2 A E ha he following holds: 2 E[ Y H () + 2 r k H(; a) (da)gd] < 1 (2:1) E[ 2 ()fz 2 () + K 2 (; a)(da)gd] < 1 (2:2) E[ X 2 ()fq 2 () + r2 (; a)(da)gd] < 1 (2:3) E[ p2 ()f 2 () + 2 (; a)(da)gd] < 1 (2:4) where X(); X 1 (); X 2 (); Y (); Z(); K(; a) are he soluions of (1:1), (1:2) corresponding o u, and we are using he () = d dz H(; X(); X 1 (); X 2 (); Y (); z; K(; a); u(); (); p(); q(); r(; :)) j z=z() and similarly wih r k H(; a). Then u() is an opimal conrol for (1:8), i.e. J(u) = sup u2a E J(u). Proof. Assume ha u 2 A E. We wan o prove ha J(u) J(u), i.e.u is an opimal conrol. We pu J(u) J(u) = I 1 + I 2 (2:5) where I 1 = E[ ff(; X(); X 1 (); X 2 (); Y (); Z(); K(; :); u()) f (; X(); X 1 (); X 2 (),Y (), Z(), K (, :), u())g d] and By he de niion of H, we have I 1 = E[ f( H() (() I 2 = E[h( Y ()) h(y ())]. H()) ())q() ()(g() g()) ( b() ( (; a) where we have used he simpli ed noaion b())p() (; a))r(; a)(da)gd] (2:6) H() = H(; X(); X 1 (); X 2 (); y; Z(); K(; :); u(); (); p(); q(); r(; :)) H() = H(; X(); X 1 (); X 2 (); Y (); Z(); K(; :); u(); (); p(); q(); r(; :)) ec. 5

6 Since h is concave, we have h( Y ()) h(y ()) h ( Y ())( Y () Y ()) = ()( Y () Y ()). By Iô s formula and (1:1), we have for all T E[ ()( Y () Y ())] = E[(T )( Y (T ) Y (T )) T ()d( Y T () Y ()) ( Y () Y ())d () T ( Z() ()d T r k H(; a)( K(; a) K(; a))(da)d # Le T! 1, we obain E ()( Y () Y ()) E[ Z 1 f ()(g() Z + r k H(; a)( K(; a) = lim (T E )( Y (T ) g())( Y () K(; a))(da)gd]. Y (T )) Y () + ( Z() () (2:7) Combining (2:6) (2:7), we obain J(u) J(u) lim (T E )( Y (T ) Y (T )) +E[ f( H() H()) ( b() b())p() (() ())q() ( r k (; a) (; a))r(; a)(da) ( Y () H(; a)( K(; a) K(; a))(da)gd]. Y () ( Z() () Since H is concave, we have J(u) +( X 1 () J(u) lim (T E )( Y (T ) X 1 H () + ( X 2 () ( b() b())p() (() ())q() 2 Y (T )) + E 4 Z 1 X 2 H () + (u() Z ( 8 < : ( X() () () 9 3 = (; a) (; a))r(; a)(da) ; d 5. (2:8) 6

7 Applying now, Iô formula o lim = E[ E p(t )( X(T ) p(t )( X(T ) X(T )) X(T )) f( b() b())p() ( X() X())E(() j F ) +(() ())q() + ( (; a) = E[ f( b() b())p() ( X 1 () (; a))r(; a)(da)gd] X 1 ())() +(() ())q() + ( (; a) (; a))r(; a)(da)gd]. (2:9) By he de niion (1:12) of, we have E[ = lim ( X() E[ T + X())()d] (( X( ) X( ))( )d)] T = lim E[ ( ( )( X( ) X( ))d T H () ( X T 1 () X 1 ())d ( H (s) e s ds) e ( ) ( X( ) X( )))d]. (2:1) Subsiuing r = ; we obain H (s)( X 2 (s) = H (s) T = ( = r+ r T + ( s e (s s X 2 (s))ds r) ( H (s)e s ds)e r H (s) e s ds)e ( X(r) X(r))dr ds X(r)) dr ) ( X( ) X( ))d. (2:11) 7

8 Combining (2:8) wih (2:9) (2:11), we deduce ha J(u) Then J(u) J(u) lim E = lim E J(u) lim + E[ [(T )( Y (T ) Y (T ))] lim E[p(T )( X(T ) X(T ))] +E[ (u() ()d] [(T )( Y (T ) Y (T ))] lim E [p(t )( X(T ) X(T ))] +E[ Ef(u() () j E gd]. E [(T )( Y (T ) Y (T ))] lim E[p(T )( X(T ) X(T ))] () j E g(u() By assumpions (H 1 and H 3 ), we conclude J(u) J(u) i.e. for all u 2 A E, u is an opimal conrol. u())d]. 3 Necessary condiions of opimaliy for parial informaion A drawback of he previous secion is ha he concaviy condiion is hard o saisfy in applicaion. Indeed, we derive necessary condiions for an opimal conrol wih parial informaion. We assume he following: (A 1 ) For all u 2 A E and all 2 A E bounded, here exiss > such ha u + s 2 A E for all s 2 (, ). (A 2 ) For all, h and all bounded E -mesurable random variables, he conrol process () de ned by () = 1 [s;1) () (3:1) belongs o A E. (A 3 ) For all bounded 2 A E, we de ne he derivaive processes () := d ds Xu+s () j s= (3:2) () := d ds Y u+s () j s= (3:3) () := d ds Zu+s () j s= (3:4) 8

9 We can see ha (; a) := d ds Ku+s (; a) j s= (3:5) and Noe ha d ds Xu+s 1 () j s= = d ds Xu+s () j s= = ( ) d ds Xu+s 2 () j s= = e ( () = for 2 [ ; ]. r) ()dr. Theorem 3.1 Soppose ha u 2 A E wih corresponding soluions X(); X 1 (); X 2 (); Y (); Z(); K(; a); (); p(); q() and r(; a) of equaions (1:1), (1:2), (1:1) and (1:11). Assume ha (2:1) (2:4) hold. And lim E[ p(t )( X(T ) X(T ))], lim E[ (T )( Y (T ) Y (T ))]. Then he following asserions are equivalen. (i) For all bounded 2 A E, (ii) For all 2 [; 1), d ds J(u + s) j s= H(; X(); X 1 (); X 2 (); Y (); Z(); K(; :); u; (); p(); q(); r(; :)) j E ] u=u() =. Proof. I follows from (1:1) ha d() @x()() + @x ()() + ()( @x(; a)() + (; a)( and d() = ()() ) () ()( ) e ( e ( e ( ) () ) (; a) () r) ()()gd r) ()()gd e ()()gd + ()db() + r) (; a)()g N(d; da), r) (r)dr (; a) N(d; ()() 9

10 where for simpliciy of noaion, we b(; X(); X 1(); X 2 (); u()) ec. Suppose ha asserion (i) holds. Then = d ds J(u + s) j s= = @x ()() + ()( ) () e ( r) @z ()() ()() + r k f(; a) (; a)gd + h ( Y ())()]. (3:6) We know by he de niion of () @x (; a)r(; a)(da) and he same (): By he Iô formula, we ge E[h ( Y ()())] = E[ ()()] = lim E[ (T )(T )] T lim E[ f ()() ()() ()( ) () e ( r) ()() ()()) + () + () + r k H(; a) (; a)(da)gd]. lim E[ (T )( Y (T ) Y (T ))], we obain (3:7) lim E[ (T )Y u+s (T )] lim E[ (T )Y u (T )] (3:8) for all 2 A E and all s 2 ( ; ). Then d ds [ lim E f (T )Y u+s (T )g] s= =. (3:9) = d ds [ lim E f (T )Y u+s (T )g] s= = lim E[ (T ) d ds Y u+s (T )] s= (3:1) = lim E[ (T )(T )], he inerchanging of he limi w.r.. he derivaive operaor holds for uniform limis wih uniform convergence of he derivaive. Inerchanging derivaive and inegraion is jusi ed if d ds [ lim Ef (T ) (T )g] s= z(w) 1

11 for some inegrable funcion z wih (T ) = Y u+s (T ), X u+s (T ). Combining (3:7) (3:1),we ge E[h ( Y ()())] = E[ ()()] = E[ ()() ()( and hence (3:6) becomes () e ( r) ()() ()()) + () + () + r k H(; a) (; a)(da)gd] = d ds J(u + s) j s= = @x ()() + ()( Z ) @z ()() ()() + r k f(; a) (; a) ()( ) ()()) + () e ( e ( r) ()() r) ()() () + () + r k H(; a) (; a)(da)gd]. Using he same agrumens as in (3:8) (3:1) for we obain lim E[p(T )( X(T ) X(T ))], lim E[p(T ) (T )] =. Applying Iô o p(t )(T ), we have = lim E[p(T ) (T )] = E[ + @x()() + ()E (() j F ) d + ()( ) () e @x ()() + ()( + @x(; a)() + (; a)( ) (; a) = d ds J(u + s) j s= ()()d). r) ()()gd ) () e ( e ( r) ()()gd r) a)()g(da)d] Therefore ()()d) =. 11

12 Use () = 1 [s;1) () where (!) is bounded and E -mesurable, s and ge E( Z 1 (s)ds ) = Di ereniaing wih respec o h = we obain ha (s) ) = This holds for all s and all, we conclude ( ) j E ) =. This proves ha asserion (i) implies (ii). To complee he proof, we need o prove he converse implicaion. This is achieved by using ha every bounded 2 A E can be approximaed by linear combinaions of conrols of he form (3:1). 4 An applicaion o opimal consumpion wih respec o recursive uiliy Le X() = X (c) () be a cash ow modelled by 8 < dx() = X( : X() = x > )[b ()d + ()db() + (; a) N(d; da)] c()d; (4:1) where b (), () and (; a) are given bounded F -predicable processes, is a xed delay and (; a) > 1 for all (; a) 2 [; 1). The process c() is our conrol process, inerpreed as our relaive consumpion rae such ha X (c) () > for all. We le A denoe he family of all F -predicable relaive consumpion raes. To every c 2 A we associae a recursive uiliy process Y c () = Y () de ned as he soluion of he in nie horizon BSDE T Y () = E[Y (T ) + g (s; Y (s); c(s)) ds j F ] for all T, (4:2) valid for all deerminisic T < 1. (See Du e & Epsein (1992)). Suppose he soluion Y () of he in nie horizon BSDE (4:2) sais es he condiion (1:4) and le c(s); s be he consumpion rae. We assume ha he funcion g(; y; c) sais es he following condiions: 12

13 g(; y; c) is convex wih respec o y y; c) has an inverse: E [jg(s; Y (s); c(s))j] ds < 1 for all c 2 A (4:4) I(; v; w) = if v v (; g(; y; c)) 1 (v) if v v (; w) where y; ). We sudy he problem o nd c 2 A such ha supy c () = Y c (). (4:5) c2a We call such a process c a recursive uiliy opimal relaive consumpion rae. We see ha he problem (4:5) is a special case of problem (1:8) wih f =, h(y) = y, u = c and J(u) = Y () b(; x; x 1 ; x 2 ; u) = x 1 b () c (; x; x 1 ; x 2 ; u) = x 1 () (; x; x 1 ; x 2 ; u; a) = x 1 (; a) In his case he Hamilonian de ned in (1:9) akes he form H(; x; x 1 ; x 2 ; y; z; k(:); ; p; q; r(:); u) = g(; y; c) + (x 1 b () c) p + x 1 ()q + x 1 (; a)r(a)(da) (4:6) Maximizing H as a funcion of c gives he rs order condiion (; Y (); c()) = E[p() j E ] (4:7) for an opimal c(). The pair of adjoin processes (1:1)-(1:11) is given by and d() = (; Y (); c())d () = 1 (4:8) Z dp() = E(() j F )d + q()db() + 13 r(; a) N(d; da); 2 [; 1) (4:9)

14 where Equaion (4:8) has he soluion () = [b ()p( + ) + ()q( + ) + (; a)r( + ; a)(da)] (4:1) () = exp( which subsiued ino (4:7) gives (s; Y (s); (; Y (); c()) (s; Y (s); c(s))ds) = E[p() j E ] is i equal o his????? or d d (s; Y (s); c(s))ds)) = E[p() j E ]; (s; Y (s); c(s))ds) = 1 + E[p(s) j E s ]ds; Hence, o nd he opimal consumpion rae c i su ces o nd E[p(s) j E s ]; s. We refer o Theorem (5:1) in [1] for he soluion of he ABSDE (4:1): Sep 1 does no depend on p() dp() = E[f ( + )q( + ) + (; a)r( + ; a)(da)g j F ]d Sep2 General +q()db() + r(; a) N(d; da) dp() = E[fb ( + )p( + ) ( + )q( + ) + (; a)r( + ; a)(da)g j F ]d + q()db() + r(; a) N(d; da) Subsiuing he las expression (4:12) o he SDDE (4:1), we rewrie i as a BSDE dx() = [I(; p()) + b ()X( )]d + ()X( )db() + X( ) (; a) N(d; da); (4:13) 14

15 which can be represened as in he sens (4:2) T X() = E[X(T ) + g 1 (s; X(s); c(s)) ds j F ] for all T, (4:14) Togeher wih he ransversaliy condiion (H 1 ) of Theorem 2:1, i.e., lim E[ p(t )( X(T ) X(T ))] (4:15) and lim E[ (T )( Y (T ) Y (T ))]. (4:16) eferences [1] N. Agram, S. Haadem, B. Øksendal, and F. Proske. A maximum principle for in nie horizon delay equaions.arxiv ( 212). [2] G. Di Nunno, B. Øksendal and F. Proske: Malliavin Calculus for Lévy Processes wih Applicaion o Finance. Springer. Correced 2 nd prining 29. [3] D. Du e and LG. Epsein. Sochasic di erenial uiliy. Economerica, volume 6, Issue 2 (1992), [4] I. Elsanosi, B. Øksendal, and A. Sulem. Some solvable sochasic conrol problems wih delay. Sochasic and Sochasic repors, 71:69 89, 2. [5] S. Haadem, B. Øksendal, F.Proske. A maximum principle for jump di usion processes wih in nie horizon.arxiv (212). [6] Q. Meng, Opimal conrol problem of fully coupled forward-backward sochasic sysems wih OlivierPoisson jumps under parial informaion. [7] B. Øksendal, A. Sulem, Maximum principles for opimal conrol of forward backward sochasic di erenial equaions wih jumps, SIAM J. Conrol Opim. 48 (5) (21) [8] B. Øksendal, A. Sulem, T. Zhang. A maximum principle of opimal conrol of sochasic delay equaions and ime-advanced backward sochasique di erenial equaions, Adv. Appl. Prob., 43 (211), [9] E. Pardoux. BSDE s, weak convergence and homogenizaions of semilinear PDE s. In F.H. Clark and.j. Sern, ediors, Nonlinear Analysis, Di erenial Equaions and Conrol, pages Kluwer Academic, Dordrech, [1] O. Menoukeu-Pamen (211): Opimal conrol for sochasic delay sysem under model uncerainy. Submied. 15

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