Solvability of Indefinite Stochastic Riccati Equations and Linear Quadratic Optimal Control Problems

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1 *Manuscrpt 2nd revson) Clck here to vew lnked References Solvablty of Indefnte Stochastc Rccat Equatons and Lnear Quadratc Optmal Control Problems Janhu Huang a Zhyong Yu b a Department of Appled Mathematcs, The Hong Kong Polytechnc Unversty, Hong Kong, Chna b School of Mathematcs, Shandong Unversty, Jnan 251, Chna January 7, 214 Abstract A new approach to study the ndefnte stochastc lnear quadratc LQ) optmal control problems, whch we called the equvalent cost functonal method, s ntroduced by Yu 15 n the setup of Hamltonan system. On the other hand, another mportant ssue along ths research drecton, s the possble state feedback representaton of optmal control and the solvablty of assocated ndefnte stochastc Rccat equatons. As the response, ths paper contnues to develop the equvalent cost functonal method by extendng t to the Rccat equaton setup. Our analyss s featured by ts ntroducton of some equvalent cost functonals whch enable us to have the brdge between the ndefnte and postvedefnte stochastc LQ problems. Wth such brdge, some solvablty relaton between the ndefnte and postve-defnte Rccat equatons s further characterzed. It s remarkable the solvablty of the former s rather complcated than the latter hence our relaton provdes some alternatve but useful vewpont. Consequently, the correspondng ndefnte lnear quadratc problem s dscussed for whch the unque optmal control s derved n terms of state feedback va the soluton of Rccat equaton. In addton, some example s studed usng our theoretcal results. Key words: Stochastc Rccat Equaton, Stochastc Lnear Quadratc Optmal Control, Backward Stochastc Dfferental Equaton BSDE). AMS subject classfcaton: 93E2, 49N1, 6H1. 1 Introducton In ths paper, we consder the followng stochastc lnear quadratc for short, LQ hereafter) optmal control problem: Mnmze Ju) = 1 T } { Qx 2 E T, x T + R t x t, x t + 2 S t u t, x t + N t u t, u t dt 1) dx t = A t x t + B t u t ) dt + C t x t + D ) tu t dw t, Subject to 2) x = h, Ths work s supported by the Natonal Natural Scence Foundaton of Chna ) and the Natural Scence Foundaton of Shandong Provnce, Chna ZR21AQ4). The authors acknowledge the support of RGC Earmarked grant 599. Emal: majhuang@polyu.edu.hk Correspondng author. Emal: yuzhyong@sdu.edu.cn 1

2 where W t = Wt 1,..., Wt d ) s a d-dmensonal Brownan moton and the control varable u t takes ts value n some Eucldean space. Note that n most lterature of LQ problem, the matrx-valued process S t s set to be zero thus no cross-term S t u t, x t nvolves. For sake of convenence, we call the LQ problem NC for short of no cross-term) f S t. Correspondngly, a stochastc Rccat equaton SRE) s ntroduced for solvng the stochastc LQ problem. In partcular, the SRE assocated to our stochastc LQ problem s a matrx-valued nonlnear backward stochastc dfferental equaton BSDE) wth the followng form: dk t = GA t, B t, C t, D t ; R t, N t, S t ; K t, L t )dt L tdwt, 3) K T = Q, where we denote by L = L 1, L 2,..., L d ) and the coeffcent G s GA, B, C, D; R, N, S; K, L) := KA + A K + L C + C ) L ) + C ) KC + R S + KB + S + KB + C ) KD N + C ) KD. 1 D ) KD Note that a BSDE s an Itô-type stochastc dfferental equaton SDE) n whch the termnal nstead ntal condton s specfed. The BSDE admts an adapted soluton par K, L) under rather mld condtons. Unlke the forward SDE, the addtonal soluton component L s ntroduced here to make the BSDE have adapted solutons to the fltraton generated by underlyng Brownan moton and termnal condton. Borrowng the term from mathematcal fnance, the second component L maybe nterpreted as the rsk-adjustment factor. In ths sense, the soluton structure of BSDEs s very dfferent to that of forward SDEs. For more detals about BSDEs, the nterested readers may refer the orgnal paper by Pardoux and Peng 8 n the general nonlnear case, and the monograph by Yong and Zhou 14, Chapter 7. In prncple, f the SRE 3) admts a soluton, then the correspondng stochastc LQ problem s solvable, and an optmal control can be represented explctly as a lnear state feedback by the soluton of SRE 3). It s well-known that for determnstc NC-LQ problem namely, S t, C t, D t and all the other coeffcents are determnstc matrces), the control weght N t n cost functonal should be nonnegatve defnte to make the LQ problem well-posed e.g., see Anderson and Moore 1 for more detals). If N t s nonnegatve but beng degenerate on a set wth postve Lebesgue measure, the related LQ problem may have no optmal control see an example n Chen and Yong 4). Keep ths n mnd, n the early research stage to stochastc NC-LQ problems, the postve defnteness condton for N t was always assumed and as a result, the derved stochastc LQ theory s very smlar to that of determnstc one. Among them, we would lke to menton the followng results whch are more related to our present work. In case all the coeffcents A, B, C, D, Q, R N are determnstc and N s unformly postve, t follows L = and SRE 3) s reduced to a nonlnear matrx-valued ordnary dfferental equaton wthout randomness. The solvablty for ths case s obtaned by Wonham 13. For the general random coeffcents, the solvablty of SRE 3) wth unformly postve N and vanshed S s a long standng open problem whch s proposed by Bsmut 2 and Peng 1 respectvely. Tang 12 obtaned one exstence and unqueness result whch completely solved ths open problem. 2 4)

3 However, t was observed by Peng 9, Chen, L and Zhou 3 that there actually have some essental dfference between the determnstc and stochastc LQ problems. In fact, there do exst some non-trval examples n whch the control weghts N t are allowed to be ndefnte but the correspondng LQ problems are stll well-posed. The study of those ndefnte stochastc LQ problems possesses not only ts theoretcal merts but also broad-range applcatons. Some examples nclude the mean-varance portfolo selecton problems studed by Lm and Zhou 7 where the control weght N s equal to zero; the polluton control model gven by Chen, L and Zhou 3 where N s negatve defnte. The solvablty of ndefnte stochastc LQ problem s closely lnked to the solvablty of ndefnte SREs. There are many works focusng on ths ssue and we menton a few as follows. If the control weght N s ndefnte, Chen, L and Zhou 3, Theorem 4.6 gave a necessary and suffcent condton for the exstence and unqueness of SRE when C =, S = and all the other matrces are determnstc. Chen and Zhou 5 extended the above work by lettng C but stll n the determnstc framework. A suffcent condton for the solvablty of SRE s proved there. In case of random coeffcents, Hu and Zhou 6 establshed the unqueness of solutons of SRE n generalty, but the exstence result s proved only for some specal cases. Recently, Qan and Zhou 11 mproved the results of 6 by usng some technques of BSDEs. However, all these works are stll far away to a complete soluton of the solvablty problem of ndefnte SRE. In the paper of Yu 15, a new approach whch we named the equvalent cost functonal method, was ntroduced to deal wth the ndefnte LQ problems n the context of Hamltonan systems. The present paper explores the solvablty of ndefnte SRE see Eq.3)) by applyng the equvalent cost functonal method. The man dea s the ntroducton of so-called equvalent functonal whch can be nterpreted n the followng rough sense. Two dfferent cost functonals are sad to be equvalent f they are used to evaluate the performance of same controlled system but always lead to the same evaluaton result. In other words, two functonals are equvalent f they always lead to the same decson makng for a gven controlled system. Such equvalence has some nterestng mplcatons: the orgnal stochastc LQ problem wth ndefnte control weght can be transformed nto some postve defnte stochastc LQ problem, provded we can fnd some equvalent but postve cost functonals. In fact, as we wll demonstrate later, ths s always the case. Relyng on such transformaton, we can nvestgate the solvablty property of ndefnte SREs by vrtue of the solvablty of postve SREs. Furthermore, we can get some results of ndefnte stochastc LQ problems based on the exstng results of postve-defnte LQ problems. Instead of handlng the more dffcult ndefnte SRE drectly whch s the man method n the lterature, see for example 3, 5, 6, 7, 11), our method provdes some alternatve but more effectve way va the brdge of equvalent cost functonals, whch makes our analyss more smpler n many cases see the example at the paper end). As mentoned by above ntroducton, t s always the NC-LQ problems S = ) to be nvestgated n most exstng lterature and there exst consderable related results. On the other hand, to dscuss the equvalent cost functonals, t s more preferable to study the LQ problems wth nonzero cross terms namely, S ) but the related studes seems relatvely few. Among them, Yong and Zhou 14 dd some research n smlar framework but ther results can not be drectly appled to our setup here. In ths paper, as a prelmnary result, we dscuss the non-zero cross term LQ problems wth postve control weght whch we called the standard LQ problem). Dfferent from Yong and Zhou 14, we ntroduce an nvertble lnear transform see Lemma 3.2) whch lnks the standard LQ problems wth the correspondng NC-LQ problems. Based on t, by such transform and the exstng results of NC-LQ problems n the lterature, we obtan the desred result of our standard LQ problems. Ths result see Theorem 3.5 below) s new n lterature and can be vewed as another contrbuton of our paper. The rest of ths paper s structured as follows. In Secton 2, we ntroduce some notatons and formulate the stochastc LQ problem. Secton 3 ams to study the LQ problem wth unformly 3

4 postve control weght N and nonzero cross tem S. We reduce the general standard stochastc LQ problem to a stochastc NC-LQ problem by a lnear transformaton. As the man result, Secton 4 s devoted to study the ndefnte SRE and ndefnte LQ problem wth the help of equvalent cost functonals. Some example s also dscussed. Secton 5 concludes our paper. 2 Notatons and problem formulaton Let R n be n-dmensonal Eucldean space wth Eucldean norm and Eucldean nner product,. Let R n m be the Hlbert space consstng of all n m) matrces wth nner product A, B := tr{ab }, for any A, B R n m, 5) where the superscrpt denotes the transpose of vectors or matrces. The norm A of A nduced by the nner product s A = traa. In partcular, we denote by S n the set of all n n) symmetrc matrces, S n + the set of all n n) nonnegatve defnte matrces, and Ŝn + the set of all n n) postve defnte matrces. Let T > be a fxed tme horzon and Ω, F, F, P) be a complete fltered probablty space on whch a d-dmensonal standard Brownan moton {W t := W 1 t,..., W d t ), t T } s defned, F = {F t, t T } s the natural fltraton of W augmented by all P-null sets. Let M denote a Eucldean space or a set of matrces except Ŝ n +. Now we ntroduce the followng notatons. L Ω, F T, P; M), the set of all M-valued F T -measurable bounded random varables; L F, T ; M), the set of all M-valued F-adapted bounded processes; L 2 F, T ; M), the set of all M-valued F-adapted processes u such that E T u t 2 dt < ; 6) SF 2, T ; M), the set of all M-valued F-adapted processes x whch have contnuous paths such that E sup x t 2 < ; 7) t,t L F, T ; Ŝn +), the set of all Ŝ n +-valued F-adapted bounded processes N. Moreover, N s unformly postve,.e., there exsts a constant δ > such that N t v, v δ v 2 for any v R n, a.s. a.e. In ths paper, we consder the followng lnear controlled stochastc dfferental equaton SDE, for short): dx t = A t x t + B t u t ) dt + C t x t + D ) tu t dw t, t, T, 8) x = h, where the ntal condton h R n, A, C L F, T ; Rn n ), B, D L F, T ; Rn m ), = 1, 2,..., d. Clearly, for any u L 2 F, T ; Rm ), there exsts a unque strong soluton x SF 2, T ; Rn ) to equaton 8). u L 2 F, T ; Rm ) s called an admssble control, x s called the correspondng admssble state process, and x, u) s called an admssble par. In addton, we are gven a quadratc cost functonal: Ju) := 1 T } { Qx 2 E T, x T + R t x t, x t + 2 S t u t, x t + N t u t, u t dt, 9) 4

5 where R L F, T ; Sn ), S L, T ; R n m ), N L F, T ; Sm ) and Q L Ω, F T, P; S n ). Suppose the controller wants to mnmze the cost functonal J ) by selectng an approprate admssble control u. Our problem s to fnd an admssble control ū L 2 F, T ; Rm ) such that Jū) = nf Ju). 1) u L 2 F,T ;Rm ) We call the above problem the stochastc lnear quadratc optmal control problem. For smplcty, we call t LQ problem 8)-9). If we fnd an admssble control ū satsfyng 1), then we call t an optmal control of LQ problem 8)-9). The correspondng state process x and x, ū) are called an optmal state process and an optmal par, respectvely. In the theory of LQ problem, t s very natural and appealng to connect the LQ problem 8)- 9) wth the stochastc Rccat equaton SRE, for short) 3) for the possble feedback regulator desgn. To conclude ths secton, we gve the defnton of soluton of SRE 3) whch s smlar to that of Tang 12. Defnton 2.1. A soluton of SRE 3) s a par of adapted matrx processes K, L) such that ) T ) N + d D ) KD s a.s.a.e. postve; moreover T ) K, L) satsfes the backward SRE 3). L s 2 ds <, a.s.; 11) GA t, B t, C t, D t ; R t, N t, S t ; K t, L t ) dt <, a.s.; 12) 3 LQ problem wth postve control weght cost In ths secton, we study the LQ problem 8)-9) under the followng standard assumptons. ) R S Assumpton STN). Q L Ω, F T, P; S+), n S L N F, T ; Sn+m + ) and N L F, T ; Ŝm + ). Remark 3.1. From ) the basc operatons of nonnegatve matrces, we know f N L F, T ; Ŝm + ), R S then S L N F Ω, F T, P; S+ n+m ) s equvalent to R SN 1 S L F Ω, F T, P; S+). n By contrast to the NC-LQ problems, an addtonal tem 2 S t u t, x t appears n the cost functonal 9) n our settng. Next we wll reduce our LQ problem to the NC-LQ problem as below: where Mnmze Jũ) := 1 Subject to d x t = { 2 E x = h, Q x T, x T + T Ãt x t + B t ũ t ) dt + R } t x t, x t + N t ũ t, ũ t dt C t x t + D tũ t ) dw t, t, T, R t := R t S t N 1 t S t, Ã t := A t B t N 1 t S t, C t := C t D tn 1 t S t, = 1,..., d. 15) It s worth notng that, n vew of Remark 3.1, Assumpton STN) s exactly the usual postve assumpton when we study the NC-LQ problem13)-14). 5 13) 14)

6 Lemma 3.2. Let Assumpton STN) hold true. For any two process pars x, u) and x, ũ), ntroduce the followng nvertble lnear transformaton: for t, T, ) ) ) xt I O xt = ũ t Nt 1 St 16) I u t or ) ) ) xt I O xt = u t Nt 1 St. 17) I ũ t Then the followng two statements are equvalent: ) x, u) s an admssble optmal) par of the LQ problem 8)-9); ) x, ũ) s an admssble optmal) par of the NC-LQ problem 13)-14). Moreover, we have Ju) = Jũ). Proof. The admssble part. Through drect calculaton, t s easy to verfy the statement ) s equvalent to the statement ), and Ju) = Jũ). The optmal part. Let x, u) be an optmal par of the LQ problem 8)-9), then the correspondng x, ũ) s an admssble par of the NC-LQ problem 13)-14). For any admssble par x 1, ũ 1 ) of NC-LQ problem 13)-14), denote x 1, u 1 ) the correspondng admssble par of the LQ problem 8)-9). Then t follows that Jũ 1 ) = Ju 1 ) Ju) = Jũ). 18) From the arbtrarness of ũ 1, we know x, ũ) s an optmal par of the NC-LQ problem 13)- 14). The above lemma tells us that there exsts some equvalent relatonshp between these two LQ problems. We would lke to analyze ther relatonshp n terms of sutable SREs. Lemma 3.3. Under Assumpton STN), the SREs correspondng to LQ problem 8)-9) and NC-LQ problem 13)-14) are the same one,.e. where GA, B, C, D; R, N, S; K, L) = GÃ, B, C, D; R, N, O; K, L), 19) R := R SN 1 S, Ã := A BN 1 S, C := C D N 1 S, = 1,..., d. 2) Proof. For smplcty of notatons, we denote by 1 V := N + D ) KD S + KB + Ṽ := N + 1 D ) KD KB + We calculate Ṽ N 1 S 1 = N + D ) KD KB + = V. N + 1 D ) KD N + C ) KD, 21) C ) KD. 22) C D N 1 S ) KD D ) KD N 1 S 6 23)

7 Now, from the defntons 4), 2), 21), 22) and the relatonshp between V and Ṽ 23), we have GA, B, C, D; R, N, S; K, L) and = KA + A K + + S + KB + L C + C ) L ) + C ) KC + R C ) KD V, GÃ, B, C, D; R, N, O; K, L) = K A BN 1 S ) + A BN 1 S ) K 24) L C D N 1 S ) + C D N 1 S ) L ) C D N 1 S ) K C D N 1 S ) + R SN 1 S ) KB + The dfference between 24) and 25) s gven as follows: C D N 1 S ) KD V + N 1 S ). GÃ, B, C, D; R, N, O; K, L) GA, B, C, D; R, N, S; K, L) = , 25) 26) where 1 := KBN 1 S 2 := SN 1 B K 3 := 4 := 5 := S + KB + S + L D N 1 S C ) KD N 1 S SN 1 S, 27) SN 1 D ) L SN 1 D ) K C D N 1 S ), 28) C ) KD V + N 1 S ), 29) SN 1 D ) KD V + N 1 S ), 3) S + KB + C ) KD V. 31) We notce that, here we wrte the last tem of 25) as We have = 1 = S + KB + S + KB + C ) KD N 1 S, 32) C ) KD N 1 S. 33) 7

8 Therefore, =. Moreover 2 = SN 1 KB + 4 = SN 1 N + Then, from 23) and 2), we have = SN 1 N + C D N 1 S ) KD, 34) D ) KD V + N 1 S ). 35) D ) KD Ṽ + KB + By 22), we get =. Consequently, we obtan the desred concluson: and we fnsh the proof. C ) KD. 36) GÃ, B, C, D; R, N, O; K, L) = GA, B, C, D; R, N, S; K, L), 37) The NC-LQ problem 13)-14) has been well studed by varous lterature. In partcular, Tang 12 gave the solvablty of the assocated SRE under Assumpton STN). Now, for the convenence of readers, we state the followng results. Lemma 3.4 Tang 12). Under Assumpton STN), the SRE dk t = GÃt, B t, C t, D t ; R t, N t, O; K t, L t )dt K T = Q L tdwt, 38) admts a unque adapted soluton K, L) L F, T ; Sn +) L 2 F, T ; Sn ) d ). Moreover, the NC-LQ problem13)-14) admts a unque optmal par x, ũ) whch s determned by 1 ũ t = N t + Dt) K t Dt K t B t + L tdt + C t) K t Dt x t, d x t = x = h, Ãt x t + B t ũ t ) dt + C t x t + D tũ t ) dw t, t, T, and the value of optmal cost s gven by Jũ) = 1/2) K h, h. We note that the SRE 38) s separated off from the control and state, so ts soluton K, L) s also ndependent of them. One can frst solves K, L), and then apples 39) to construct the optmal feedback control. For the purpose of practcal applcatons, one may use the numercal smulaton of K, L). Due to the SRE s a specal BSDE, the nterested readers may refer the lterature on numercal smulaton of BSDEs, for example Zhao, Chen and Peng 16. Wth the help of above prelmnary results, we are ready to state the man result of ths secton. 39) 8

9 Theorem 3.5. Under Assumpton STN), the SRE 3) admts a unque adapted soluton K, L) L F, T ; Sn +) L 2 F, T ; Sn ) d ). Consequently, N + d D ) KD L F, T ; Ŝm + ). Moreover, the LQ problem 8)-9) admts a unque optmal par x, u) gven by 1 u t = N t + Dt) K t Dt S t + K t B t + L tdt + Ct) K t Dt x t, dx t = A t x t + B t u t ) dt + C t x t + Dtu ) 4) t dw t, t, T, x = h, and the value of optmal cost s Ju) = 1/2) K h, h. Proof. We need only verfy the representaton of optmal control u, and other statements n the theorem follow drectly from the above lemmas. By the lnear transformaton ntroduced n Lemma 3.2, and the defnton of optmal control ũ for the NC-LQ problem 13)-14) n Lemma 3.4, we have u = ũ N 1 S x 1 = N + D ) KD KB + = N + N + 1 D ) KD N + D ) KD N 1 S x 1 D ) KD S + KB + Ths completes the proof. C D N 1 S ) KD x C ) KD x. When all the coeffcents A, B, C, D, R, S, N, Q are determnstc, SRE 3) turns out to be some matrx-valued ordnary dfferental equaton wthout randomness: dk t = GA t, B t, C t, D t ; R t, S t, N t ; K t, ), dt 42) K T = Q. Wonham 13 proved the exstence and unqueness for 42). Moreover the soluton K s nonnegatve. Corollary 3.6. Let Assumpton STN) hold true and suppose all coeffcents A, B, C, D, R, S, N, Q are determnstc, then the Rccat equaton 42) admts a unque soluton K. Moreover K s nonnegatve. Consequently, N + d D ) KD s unformly postve. Furthermore, the LQ problem 8)-9) admts a unque optmal par x, u) determned by 1 u t = N t + Dt) K t Dt S t + K t B t + Ct) K t Dt x t, dx t = A t x t + B t u t ) dt + C t x t + Dtu ) 43) t dw t, t, T, x = h, and the value of optmal cost functonal s Ju) = 1/2) K h, h. 41) 9

10 4 LQ problem wth ndefnte control weght cost In case the Assumpton STN) doesn t hold true for nstance, the control weght N s ndefnte), t s stll possble for the LQ problem 8)-9) to be well-posed and the optmal par exsts. In ths secton, we wll apply the equvalent cost functonal method proposed n Yu 15 to deal wth the LQ problem 8)-9) wthout Assumpton STN). Let us frst recall the defnton of equvalent cost functonal ntroduced n 15. Defnton 4.1. For a gven controlled system, f there exst two cost functonals J ) and J ) satsfyng: for any admssble controls u 1 and u 2, Ju 1 ) < Ju 2 ) f and only f Ju 1 ) < Ju 2 ), then we say J ) s equvalent to J ). Remark 4.2. It s not dffcult to see the followng two statements are equvalent: 1) the cost functonal J ) s equvalent to J ); 2) for any admssble controls u 1 and u 2, ) Ju 1 ) < Ju 2 ) f and only f Ju 1 ) < Ju 2 ); ) Ju 1 ) = Ju 2 ) f and only f Ju 1 ) = Ju 2 ); ) Ju 1 ) > Ju 2 ) f and only f Ju 1 ) > Ju 2 ). Obvously, when we use two equvalent cost functonals J ) and J ) to evaluate the same controlled system, we wll get the same results. Especally, for LQ problem wth cost functonal J ) and LQ problem wth J ), the exstence and unqueness of ther optmal pars are equvalent, and the optmal pars concde. The dea to deal wth LQ problem J ) wthout Assumpton STN) s as follows: f we can fnd a cost functonal J ) whch s equvalent to orgnal J ) but the matrces n J ) satsfy Assumpton STN), then we can study LQ problem J ) through the LQ problem J ). In ths sense, the equvalent cost functonals provde some new and more flexble approach to deal wth LQ problem J ) wthout Assumpton STN). Let us denote by { Θ := H L F, T ; t t Sn ) H t = H + Γ s ds + Λ sdws for all t, T, } where Γ, Λ L F, T ; Sn ), = 1,..., d. For any H Θ, we apply Itô s formula to H t x t, x t on the nterval, T : 44) H h, h = E H T x T, x T { + E + 2 T HB + HA + A H + Λ D + Λ C + C ) Λ ) + C ) HC + Γ x t, x t C ) HD u t, x t + D ) HD u t, u t }dt. 45) It s worth notng the constant H h, h s represented by any admssble par x, u) n a quadratc form. Therefore we can add H h, h /2 to orgnal cost functonal J ) to get some equvalent one, denoted by J H ). Through ths, we may transform J ) wthout Assumpton 1

11 STN) to J H ) satsfyng Assumpton STN). Especally, when H s nonnegatve, then nonnegatve tem d D ) HD u, u n 45) may transform a nonpostve control weght N n J ) to a postve one n J H ). The detals can be llustrated as follows. For each H Θ, we defne J H u) := Ju) 1 2 H h, h = 1 2 E { Q H x T, x T + T R H t x t, x t + 2 S H t u t, x t + N H t u t, u t dt}, 46) where R H = R + HA + A H + S H = S + HB + N H = N + Λ D + Λ C + C ) Λ ) + C ) HC + Γ, 47) C ) HD, 48) D ) HD, 49) Q H = Q H T. Snce J H ) and J ) dffer by only a constant H h, h /2, so they are equvalent. In other words, we get a famly of equvalent cost functonals {J H ), H Θ}. Through a straghtforward calculaton, we have Lemma 4.3. For any H Θ, the exstence and unqueness of SREs assocated wth LQ problem 8), 46) and LQ problem 8)-9) are equvalent. Moreover, f K H, L H ) s a soluton of the SRE assocated wth LQ problem 8), 46), then 5) K t = K H t + H t, L t = L H, t + Λ t, = 1,..., d, t, T 51) s a soluton of the SRE assocated wth LQ problem 8)-9). Here, the LQ problem 8), 46) means the LQ problem wth state gven by 8) and cost functonal gven by 46). The above lemma reveals an nterestng fact that there s a nce structure between the set of SREs and the set of Θ: the sum of an SRE and a gven process n Θ s stll an SRE. Theorem 4.4. If there exsts some H Θ such that R H, S H, N H, Q H ) satsfes Assumpton STN), then the SRE assocated wth LQ problem 8)-9) admts a unque soluton K, L) L F, T ; Sn ) L 2 F, T ; Sn ) d ). In addton, N + d D ) KD L F, T ; Ŝm + ). Moreover, the LQ problem 8)-9) admts a unque optmal par x, u) determned by 1 u t = N t + Dt) K t Dt S t + K t B t + L tdt + Ct) K t Dt x t, dx t = A t x t + B t u t ) dt + C t x t + Dtu ) 52) t dw t, t, T, x = h, and the value of optmal cost functonal s gven by Ju) = 1/2) K h, h. 11

12 Proof. For H Θ, by Theorem 3.5, the SRE assocated wth LQ problem 8), 46) admts a unque soluton K H, L H ) L F, T ; Sn +) L 2 F, T ; Sn ) d ). By Lemma 4.3, the SRE assocated wth LQ problem 8)-9) admts a unque soluton K, L) L F, T ; Sn ) L 2 F, T ; Sn ) d ). In addton, N + D ) KD = N H + D ) K H D L F, T ; Ŝm + ). 53) By Theorem 3.5 agan, the LQ problem 8), 46) admts a unque optmal par x, u) determned by 1 u t = Nt H + Dt) Kt H Dt St H + Kt H B t + L H, t Dt + Ct) Kt H Dt x t, dx t = A t x t + B t u t ) dt + C t x t + Dtu ) t dw t, t, T, x = h. 54) Snce J H ) s equvalent to J ), then x, u) s also the unque optmal par of LQ problem8)-9). A drect calculaton leads to another representaton of u as 1 u t = N t + Dt) K t Dt S t + K t B t + L tdt + Ct) K t Dt x t. 55) At the last step, let us calculate the optmal cost functonal for LQ problem 8)-9): Ju) = J H u) H h, h = 1 2 KH h, h H h, h = 1 2 K h, h, 56) where the second equaton s also from Theorem 3.5. The proof s completed. Theorem 4.5. For LQ problem 8)-9), f the assocated SRE admts a soluton K, L) L F, T ; Sn ) L F, T ; Sn ) d ) and N + d D ) KD L F, T ; Ŝm + ), then there exsts an equvalent cost functonal J H ), H Θ such that R H, S H, N H, Q H ) satsfes Assumpton STN). Proof. For the frst component K of the soluton of SRE, we consder the equvalent cost functonal J K ). It s easy to verfy Q K =, 57) N K = N + D ) KD, 58) S K = S + KB + C ) KD, 59) R K = S K N K) 1 S K ), 6) and R K, S K, N K, Q K ) satsfes Assumpton STN). When all the coeffcents A, B, C, D, R, S, N, Q are determnstc, we consder the followng set nstead of Θ, { ˆΘ := H L, T ; S n t ) H t = H + Γ s ds for all t, T, } 61) where Γ L, T ; S n ). 12

13 The same argument leads to Theorem 4.6. Suppose all the coeffcents A, B, C, D, R, S, N, Q are determnstc, then for the LQ problem 8)-9), the followng two statements are equvalent. ) There exsts an equvalent cost functonal J H ), H ˆΘ such that R H, S H, N H, Q H ) satsfes Assumpton STN); ) The Rccat equaton 42) admts a unque soluton K. Moreover N + d D ) KD s unformly postve. Example 4.7. Let the dmenson of state process x s one. Consder the followng LQ problem: Mnmze Ju) = 1 2 E x 2 T, 62) ) dx t = r t x t + θt u t dt + e u t dwt, Subject to 63) x = h, where r, θ are F-adapted bounded processes, and {e ; = 1, 2,..., d} s an orthogonal bass of R d. Obvously, Assumpton STN) does not hold true for J ). Now we ntroduce an equvalent cost functonal J H ) satsfyng Assumpton STN). Because r and θ are bounded, there exsts a constant K > such that 2r t θ t θ t K for any t, T. We defne H t = e KT t 2r s θ s θs)ds, t, T. 64) It s easy to check that H Θ wth e 2KT H t 1, Γ t = 2r t θ t θ t )H t and Λ t, = 1, 2..., d, for any t, T. We calculate Q H = 1 H T, N H t = H t I d, S H t = H t θ t, R H t = 2H t r t + Γ t, 65) where I d denotes the d d dentty matrx. Then we verfy Assumpton STN) holds true for J H ) thus LQ problem 62)-63) s solvable. In partcular, the correspondng SRE ) dk t = 2r t θt θ t K t 2θt L t L t L t dt L K tdwt, t 66) K T = 1 admts a unque soluton K, L) wth K beng unformly bounded and unformly postve, and L beng square ntegrable. The unque optmal control u s of the followng feedback form: u t = θ t + L t x t. 67) K t At the end of ths example, we note that the SRE 66) s just the one studed by Lm and Zhou 7 wheren they consdered the mean-varance portfolo selecton problem arsng from mathematcal fnance. Here we use the equvalent cost functonal method to revst the solvablty of 66), and the proof here s new and rather quck than the one used n 7. 13

14 5 Concluson The solvablty of ndefnte stochastc Rccat equatons has mportant values for both theoretcal analyss and real applcatons. Nevertheless, the study of ths ssue s very complcated hence only very few cases have been treated to date and the assumptons mposed there are strong. In ths paper, we propose a dfferent method to study ths ssue by the ntroducton of some equvalent cost functonal. Ths connects the ndefnte Rccat equatons to postve-defnte Rccat equatons. Through ths connecton, we obtan more general solvablty of ndefnte Rccat equatons under weak assumptons. Our method also provdes some alternatve and useful vewpont to consder other mportant optmzaton problems: for example, the lnear quadratc mean-feld control or dfferental game wth ndefnte control weghts). Ths wll be dscussed n our future works. References 1 B.D.O. Anderson and J.B. Moore, Optmal control-lnear quadratc methods, Prentce- Hall, New York, J. M. Bsmut, Controle des systems lnears quadratques: applcatons de l ntegrale stochastque, n Sémnare de Probabltés XII, Lecture Notes n Math. 649, C. Dellachere, P.A. Meyer, and M. Wel, eds., Sprnger-Verlag, Berln 1978), pp S. Chen, X. L and X. Zhou, Stochastc lnear quadratc regulators wth ndefnte control weght costs, SIAM J. Control Optm., ), pp S. Chen and J. Yong, Stochastc lnear quadratc optmal control problems, Appl. Math. Optm, 4321), pp S. Chen and Z. Zhou, Stochastc lnaer quadratc regulators wth ndefnte control weght costs. II, SIAM J. Control Optm., 392), pp Y. Hu and X. Zhou, Indefnte stochastc Rccat equatons, SIAM J. Control Optm., 4223), pp A.E.B. Lm and X.Y. Zhou, Mean-varance portfolo selecton wth random parameters n a complete market, Math. Oper. Res., 2722), pp E. Pardoux and S. Peng, Adapted soluton of a backward stochastc dfferental equaton, Systems Control Lett., 14199), pp S. Peng, New development n stochastc maxmum prncple and related backward stochastc dfferental equatons, n proceedngs of 31st CDC Conference, Tucson S. Peng, Open problems on backward stochastc dfferental equatons, n Control of Dstrbuted Parameter and Stochastc Systems Hangzhou, 1998), S. Chen, et al., eds., Kluwer Academc Publshers, Boston, pp , Z. Qan and X. Zhou, Exstence of solutons to a class of ndefnte stochastc Rccat equatons, SIAM J. Control Optm., 51213), pp S. Tang, General lnear quadratc optmal stochastc control problems wth random coeffcents: lnear stochastc Hamlton systems and backward stochastc Rccat equatons, SIAM J. Control Optm., 4223), pp

15 13 W.M. Wonham, On a matrx Rccat equaton of stochastc control, SIAM J. Control, 61968), pp J. Yong and Z. Zhou, Stochastc controls: Hamltonan systems and HJB equatons, Sprnger-Verlag, New York, Z. Yu, Equvalent cost functonals and stochastc lnear quadratc optmal control problems, ESAIM Control Optm. Calc. Var., 19213), pp W. Zhao, L. Chen and S. Peng, A new knd of accurate numercal method for backward stochastc dfferental equatons, SIAM J. Sc. Comput. 2826), pp

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