Kalman Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems

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1 J. Mah. Anal. Appl. 34 8) Kalman Bucy filering equaions of forward and backward sochasic sysems and applicaions o recursive opimal conrol problems Guangchen Wang a,b,,zhenwu b a School of Mahemaical Sciences, Shandong Normal Universiy, Jinan 514, PR China b School of Mahemaics and Sysem Sciences, Shandong Universiy, Jinan 51, PR China Received 13 April 6 Available online 3 January 8 Submied by J. Glaz Absrac This paper is concerned wih Kalman Bucy filering problems of a forward and backward sochasic sysem which is a Hamilonian sysem arising from a sochasic opimal conrol problem. There are wo main conribuions worhy poining ou. One is ha we obain he Kalman Bucy filering equaion of a forward and backward sochasic sysem and sudy a kind of sabiliy of he aforemenioned filering equaion. The oher is ha we develop a backward separaion echnique, which is differen o Wonham s separaion heorem, o sudy a parially observed recursive opimal conrol problem. This new echnique can also cover some more general siuaion such as a parially observed linear quadraic non-zero sum differenial game problem is solved by i. We also give a simple formula o esimae he informaion value which is he difference of he opimal cos funcionals beween he parial and he full observable informaion cases. 8 Elsevier Inc. All righs reserved. Keywords: Backward sochasic differenial equaion; Feynman Kac formula; Kalman Bucy filering; Linear quadraic non-zero sum differenial game; Recursive opimal conrol; Sabiliy 1. Inroducion To solve parially observed sochasic opimal conrol problems consiss of wo componens. One is esimaion, he oher is conrol. The esimaion par is relaed o filering problems. The mos successful resul of filering heory was obained for linear sysems by Kalman [5] and Kalman and Bucy [6] in 196 and 1961, respecively. In he case of linear sysems, parially observed opimal conrol problems can be parly reaed by a separaion heorem originally obained by Wonham [17] in This heorem allows us o firs compue filering of saes, and hen o solve fully This work is parially suppored by he Naural Science Foundaion of PR China ) and Shandong Province Z6A1), he Naional Basic Research Program of PR China 973 Program, No. 7CB81494) and he New Cenury Excellen Young Teachers Program of Educaion Minisry, PR China. * Corresponding auhor a: School of Mahemaical Sciences, Shandong Normal Universiy, Jinan 514, PR China. addresses: wgcmahsdu@sohu.com, wguangchen@mail.sdu.edu.cn G. Wang), wuzhen@sdu.edu.cn Z. Wu). -47X/$ see fron maer 8 Elsevier Inc. All righs reserved. doi:1.116/j.jmaa.7.1.7

2 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) observed opimal conrol problems driven by he filering saes. A sysemaic inroducion of linear filering heory and is applicaion o opimal conrol can be found in he books of Lipser and Shiryayev [9] and Bensoussan []. However we noe ha he signal processes in he above filering problems are he soluions of forward sochasic differenial equaions SDEs in shor). Nonlinear backward sochasic differenial equaions BSDEs in shor) have been independenly inroduced by Pardoux and Peng [1] and Duffie and Epsein [3]. For a BSDE coupled wih a forward SDE, Peng [13] gave a probabilisic inerpreaion for a large kind of he second order quasi-linear parial differenial equaion PDE in shor). This resul generalized he well-known Feynman Kac formula o a non-linear case. El Karoui e al. [7] gave some imporan properies of BSDEs and heir applicaions o opimal conrol and financial mahemaics. Peng [14] derived a general maximum principle for a fully observed forward sochasic conrol sysem. I is well known ha an opimal conrol can be represened by an adjoin process which is he soluion of a BSDE. Then he opimal sae equaion and he adjoin equaion consis of a Hamilonian sysem which is a forward and backward sochasic differenial equaion FBSDE in shor). The sudy of coupled FBSDEs sared in las early 9s. In his PhD disseraion, Anonelli [1] obained he firs resul on he solvabiliy of an FBSDE over a small ime duraion. Ma e al. [1] provided explici relaions among he forward and he backward componens of he adaped soluion via a quasi-linear PDE, bu hey required he non-degeneracy held for he forward diffusion and he non-randomness held for he coefficiens. Hu and Peng [4], Peng and Wu [15] go he exisence and uniqueness resul of an FBSDE wih he arbirarily fixed large ime duraion under a monooniciy condiion on he coefficiens, which is resricive in a differen way. We refer he reader o he book of Ma and Yong [11] for a sysemaic inroducion of FBSDEs. In [8] and [16], Li and Tang derived some general maximum principles for parially observed forward sochasic conrol sysems, which covered mos of he resuls of references herein. To ge an observable maximum principle, hey used backward sochasic PDEs o characerize he corresponding Hamilonian sysem. In fac, i is a naural reques o characerize he Hamilonian sysem by filering for FBSDEs. However, here exiss few work dealing wih his opic. In our paper, we will sudy filering problems of a forward and backward sochasic sysem arising from an opimal conrol problem. And hen he heoreical resul is applied o a parially observed recursive opimal conrol problem in Secion 4. To our bes knowledge, hese kinds of resuls have no been found in exising works. In he coming secion, we presen he Kalman Bucy filering equaion corresponding o he aforemenioned forward and backward sochasic sysem. In Secion 3, we sudy a kind of sabiliy of he filering equaion obained in he above secion. We also give an example of a forward and backward sochasic sysem which has a sable explici observable soluion. Duffie and Epsein [3] presened a concep of sochasic differenial recursive uiliy which is an exension of he sandard addiive uiliy wih he insananeous uiliy depending no only on an insananeous consumpion rae c ) bu also on he fuure uiliy. As has been noed by El Karoui e al. [7], he sochasic differenial) recursive uiliy process can be regarded as he soluion of a special BSDE. From BSDEs poin of view, El Karoui e al. [7] gave he formulaion of recursive uiliies and heir properies. Using soluions of BSDEs o describe cos funcionals of conrol sysems, we ge recursive opimal conrol problems. In Secion 4, we sudy a parially observed recursive opimal conrol problem. Using a new echnique which is differen o Wonham s separaion heorem, we obain a unique opimal conrol which is a linear feedback of he sae filering esimaion. We noice ha his new echnique can cover some more general siuaion. For example, i can be used o solve a parially observed linear quadraic non-zero sum differenial game problem, which is more general han he aforemenioned recursive problem. From he financial mahemaics poin of view, Yang and Ma [18] gave a definiion of informaion value. In he las secion, our ask is o esablish a formula, which shows he imporance of more observable informaion o conrollers. How o esimae he informaion value of he recursive opimal conrol problem is also sudied in his secion.. Kalman Bucy filering equaions In his secion, we firs inroduce a forward and backward sochasic sysem arising from a classical opimal conrol problem, and hen derive he Kalman Bucy filering equaion for his kind of sysem. Le Ω, F,F ), P ) be a filered complee probabiliy space equipped wih a naural filraion F = σ {ξ,w 1 s), W s): s }, F = F T, where W 1 ), W )) is a -dimensional sandard Brownian moion defined on he space, and T> is a fixed real number. ξ is a Gaussian random variable, independen of W 1 ), W )), wih he mean m and he variance n.

3 18 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) Throughou he paper, for he sake of convenience, we only consider 1-dimensional sochasic sysem. For he muli-dimensional case, we can ge similar resuls by he same mehod. Suppose ha we have a conrol sysem, whose evoluion is described by he following equaion: { ) dx) = A)X)+ B)v) d + C1 ) dw 1 ) + C ) dw ), 1) X) = ξ, where v ) is defined by { } U ad = v ) v) is an F -adaped process valued in R and saisfies E v 4 ) d < +. Every elemen in U ad is called an admissible conrol. We suppose ha X ) has an effec on he wealh of a conroller, however he conroller canno influence he sysem, and acs o proec his advanages by v ) U ad. The payoff corresponding o v ) U ad is recursive, which means ha he cos funcional is given by J v ) ) = Y) = EY), ) where Y ), Z 1 ), Z )) is a soluion of he BSDE dy) = a)x ) + b)y)+ f 1 )Z 1 ) + f )Z ) + c)v ) ) d Z 1 ) dw 1 ) Z ) dw ), 3) YT)= X T ). We need he following hypohesis: H1) a ), c ) ε>, A ), B ), C 1 ), C ), f 1 ) and f ) are uniformly bounded deerminisic funcions wih respec o [,T]. Since he drif erm in 3) conains Z 1 ), Z )), i brings us some rouble o express he cos funcional ). To simplify i, we define a probabiliy measure Q on he space Ω, F) by { T dq T dp = exp f 1 ) dw 1 ) + f ) dw ) 1 } f 1 ) + f )) d. From H1), according o Girsanov s heorem, i follows ha U ), V )) defined by U)= W 1 ) f 1 s) ds and V)= W ) f s) ds is a -dimensional sandard Brownian moion defined on he space Ω, F,F ), Q). I is easy o prove ha U ), V )) and ξ remain muually independen and ξ keeps he same probabiliy law as before on Ω, F,F ), Q). Then we can rewrie 1) and 3) as follows: { dx) = A)X)+ B)v)+ C1 )f 1 ) + C )f ) ) d + C 1 ) du) + C ) dv ), 4) X) = ξ, { dy) = a)x ) + b)y)+ c)v ) ) d Z 1 ) du) Z ) dv ), YT)= X 5) T ). By he definiion of U ad, we know ha if v ) U ad hen E Q v4 ) d < +. In his case, E Q X 4 )<+, i.e., E Q Y T ) < +. So here exiss a unique soluion for 4) and 5), respecively. Therefore he corresponding cos funcional is rewrien as

4 J v ) ) [ T = E Q G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) ] e bs)ds a)x ) + c)v ) ) T d + e b)d X T ), 6) where E Q denoes he mahemaical expecaion on he space Ω, F,F ), Q). Minimizing 6) subjec o v ) U ad and 4) formulaes a fully observed opimal conrol problem. For simpliciy, we denoe his problem by Problem FO). Any u ) U ad saisfying J u ) ) = min J v ) ) v ) U ad is called an opimal conrol. The corresponding sae rajecory and he cos funcional are called an opimal sae rajecory and an opimal cos funcional denoed by x ) and Ju )), respecively. Since he drif erm in 4) conains he deerminisic funcion C 1 )f 1 ) + C )f ), he classical echnique of compleing squares canno be used direcly o solve Problem FO). However, Peng s maximum principle see Peng [14]) is sill an alernaive ool. From he maximum principle, i is easy o check ha u) = 1 B)c 1 )e bs)ds y), a.e., a.s. 7) is an opimal conrol of Problem FO). Here he adjoin process y ) saisfies he following Hamilonian sysem which is an FBSDE dx) = A)x) 1 B )c 1 )e ) bs)ds y) + C 1 )f 1 ) + C )f ) d + C 1 ) du) + C ) dv ), dy) = a)e bs)ds x) + A)y) ) 8) d z 1 ) du) z ) dv ), T x) = ξ, yt)= e bs)ds xt ). From a resul in Peng and Wu [15], we know ha 8) admis a unique soluion x ), y ), z 1 ), z )) and he FBSDE has a pracical background in opimal conrol. In he following, we will discuss he filering problem for he forward and backward sochasic sysem 8). For simpliciy, we keep same noaions as before. Suppose ha he sae variable x ), y ), z 1 ), z )) canno be observed direcly, however we can observe a noisy process Z ) relaed o x ), whose dynamic is described by he equaion { ) dz) = D)x)+ F)Z) d + H)dW ), 9) Z) =, in oher way, { dz) = D)x)+ F)Z)+ f )H ) ) d + H ) dv ), 1) Z) =. We inroduce he following hypohesis: H) D ), F ), H ) ε> and H 1 ) are uniformly bounded deerminisic funcions wih respec o. Obviously, if H) holds, hen here exiss a unique soluion for 9) as well as 1). Remark.1. The linear combinaions of y), z 1 ), z )) can be considered in he drif erm of he observaion equaion 9). For his case, we can sill deal wih i by same echniques, so we only consider he observaion equaion as above. Our filering problem is o find explici expressions for he bes esimaion in he sense of square error) wih respec o he observaions Z ) up o ime, denoed by ˆx),ŷ),ẑ 1 ), ẑ )), for he sae x), y), z 1 ), z )), i.e., we wan o find he explici expressions for [ ] [ ] [ ˆx) = E Q x) Z, ŷ) = EQ y) Z, ẑ1 ) = E Q z1 ) ] [ Z, ẑ ) = E Q z ) ] Z 11) and heir square error esimaion. Here Z = σ {Zs); s }.

5 184 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) Our mehod is firs o look for he relaions of x ), y ), z )) hen o compue ˆx ), ŷ ), ẑ 1 ), ẑ )) by classical filering heory for forward SDEs. From he general non-linear Feynman Kac formula see Peng [13]), if we se y) = u, x)), hen z 1 ) and z ) can be wrien as z 1 ) = C 1 ) x u,x) ), where u, x) is a classical soluion of he following PDE: Here u, x) + Lu, x) + a)e bs)ds x + A)u, x) =, T ut, x) = e bs)ds x. Lu, x) = 1 C 1 ) + C )) + z ) = C ) x u,x) ), 1) u, x) x A)x 1 B )c 1 )e bs)ds u, x) + C 1 )f 1 ) + C )f ) ) u, x). x By he erminalcondiionof 13), we se u, x) = Π)x + π), where Π ) and π ) saisfy respecively Π)+ A)Π) 1 B )c 1 )e bs)ds Π ) + a)e bs)ds =, ΠT)= e bs)ds 14) and π)+ A) 1 B )c 1 )e ) bs)ds Π) π)+ C 1 )f 1 ) + C )f ) ) Π)=, 15) πt) =. From he classical Riccai differenial equaion heory, we know ha here exiss a unique soluion for 14) and 15), respecively. By 1) and 14), we ge y) = Π)x)+ π), z 1 ) = C 1 )Π), z ) = C )Π), 16) where x ) saisfies [ dx) = A) 1 B )c 1 )Π)e )x) bs)ds + C 1 )f 1 ) + C )f ) 1 B )c 1 )π)e ] bs)ds d + C 1 ) du) + C ) dv ), x) = ξ. 17) Here Π ) and π ) come from 14) and 15), respecively. From 1), 16) and 17), i is easy o see ha x ) is Gaussian, hen Z ) is Gaussian, so is x ), y ), Z )) valued in R 3. Therefore here exiss a recursive filering formula for ˆx ), ŷ ), ẑ 1 ), ẑ )). In fac, here we applied he muual independence of ξ and U ), V )). Obviously, ẑ 1 ) = C 1 )Π), ẑ ) = C )Π). 18) Then we only need o compue ˆx) and ŷ) defined by 11). Le P)= E Q x) ˆx)) be he square error of he esimaion ˆx). From he fac ha x) ˆx)) Z and x) ˆx) is Gaussian, we know ha x) ˆx) is independen of Z.So P)= E Q x) ˆx) ) = EQ [ x) ˆx) ) Z ]. 19) 13)

6 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) Thanks o Theorem 8.1 in Lipser and Shiryayev [9], we obain he following equaions for ˆx) and P): [ d ˆx) = A) 1 B )c 1 )Π)e ) bs)ds ˆx) + C 1 )f 1 ) + C )f ) 1 B )c 1 )π)e ] bs)ds d + C ) + D)H 1 )P ) ) d W), ˆx) = m, P) A) 1 B )c 1 )Π)e ) bs)ds P)+ C ) + D)H 1 )P ) ) C1 ) C ) =, P) = n, where he process W)= H 1 ) dz) D)ˆxs) F)Z) f )H ) ) d = V)+ ) 1) Ds)H 1 s) xs) ˆxs) ) ds ) is an observable 1-dimensional sandard Brownian moion defined on Ω, Z,Z ), Q), which is he so-called innovaion process. Taking condiional expecaions on boh sides of 16), we ge ŷ) = Π)ˆx) + π), where ˆx ) is he soluion of ). So we have Theorem.. Le H1) and H) hold. Then he filering esimaion ˆx ), ŷ ), ẑ 1 ), ẑ )) of he sae x ), y ), z 1 ), z )), which is he soluion of 8), are given by ), 3) and 18). This resul will be used o sudy a parially observed recursive opimal conrol problem in Secion Sabiliy of Kalman Bucy filering equaions In his secion, we will sudy a kind of sabiliy of he filering equaions ) and 3) wih respec o heir iniial values and prove ha ) and 3) are sable under our framework. Furhermore, we also give a worked-ou example of a forward and backward sochasic sysem which has a sable explici observable filering soluion. We firs give Definiion 3.1. For any T, assume ha ˆx 1 ) and ˆx ) are wo iniial values, and ha ˆx 1 ) and ˆx ) are he corresponding filering esimaion values. The filering equaion ) is called sable, if for any ε>, here exiss δ = δε) > such ha when E Q ˆx 1 ) ˆx )) <δ, we always have sup ˆx1 ) ˆx ) ) <ε. E Q T In pracice, we hope when he iniial filering esimaion value changes lile, here is also lile difference of he filering esimaion a any ime, i.e., we can ge a sable filering esimaion. Oherwise, he filering resul has lile pracical sense. For our filering equaion ), we can give a more general resul, he coninuous dependence of soluions wih respec o parameers, which implies our desired sable resul. Le us now consider he following equaion depending on a parameer δ R: { d ˆx δ ) = g δ ) ˆx δ ) + h δ )D δ )x δ ) + r δ ) ) d + h δ )H δ ) dv ), ˆx δ ) = m δ, 4) 3)

7 186 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) where h δ ) = H δ ) ) C δ )H δ ) + D δ )P δ ) ), g δ ) = A δ ) 1 r δ ) = C δ 1 f δ 1 ) + Cδ f δ ) 1 B δ ) ) c δ ) ) 1 Π δ )e bδ s) ds h δ )D δ ), B δ ) ) c δ ) ) 1 π δ )e bδ s) ds, and Π δ ), π δ ), x δ ) and P δ ) are soluions of 14), 15), 17) and 1) respecively where all coefficiens depend on he parameer δ. In fac, 4) can be obained from ) and ). Obviously, if all variables of 4) do no depend on he parameer δ, Eq. 4) can be regarded as he filering equaion ) corresponding o he sysem 8) and 1). We assume he following hypohesis: H3) a δ ), b δ ), c δ ), f δ 1 ), f δ ), Aδ ), B δ ), C δ 1 ), Cδ ), Dδ ), H δ ) and [H δ )] 1 are coninuous wih respec o δ and uniformly bounded wih respec o and δ. Then we have Theorem 3.. Le H3) hold. Then he soluion ˆx δ ) of 4) is coninuous abou he parameer δ R. Proof. For noaional convenience, we se x) =ˆx δ 1 ) ˆx δ ), ˇx) = x δ 1 ) x δ ), ḡ) = g δ 1 ) g δ ), r) = r δ 1 ) r δ ), hd)) = h δ 1 )D δ 1 ) h δ )D δ ), So, we have x) = + hh )) = h δ 1 )H δ 1 ) h δ )H δ ). [ g δ 1 s) xs) +ḡs)ˆx δ 1 s) + h δ 1 s)d δ 1 s) ˇxs) + hh )s)x δ s) + rs) ] ds hh )s) dv s) + x). Hölder s inequaliy implies ha and x ) g δ 1 s) ) ds hd) s) ds x s) ds + 7 sup T x ) = 7C x s) ds + 7 ḡ s) ds + 7 hd) s) ds ḡ s) ds ˆx δ s) ) ds + 7 x δ s) ) ds + 7T r s) ds + 7 x δ s) ) T ds + 7T ˆx δ s) ) T ds + 7C h δ 1 s)d δ 1 s) ) ds hh )s) dv s)) + 7 x ), ˇx s) ds r s) ds + 7 sup T ˇx s) ds hh )s) dv s)) + 7 x ).

8 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) From B D G inequaliy, we have where sup E Q T ψt δ 1,δ ) = 7 x ) 7C E Q + 7 ḡ s) ds E Q sup x ) d + ψt δ 1,δ ), 5) s hd) s) ds E Q ˆx δ s) ) T ds + 7C E Q x δ s) ) T ds + 7T sup ˇx s) ds r s r s) ds + 7E Q x ). By H1) and H), we know ha ψ T δ 1,δ ) converges o in L T P ) as δ 1 δ. Here we have already applied he coninuous dependence propery of he soluion of SDE on he parameer δ. From Gronwall s inequaliy and 5), we ge sup E Q T The proof is compleed. x ) ψ T δ 1,δ )e 7C T. I is easy o check ha he sabiliy of he filering equaion ) is a paricular case of he coninuous dependence of he soluion on he parameer. Therefore we have Corollary 3.3. Le H3) hold. Then he filering equaion ) is sable in he sense of Definiion 3.1. Example 3.4. We se B ) = C ) and all he oher coefficiens in 8) and 1) be non-zero consans. By Theorem., we ge he corresponding Riccai equaion { P) AP ) + D H P ) C1 =, P) =, which has a soluion P)= λ 1 λ λ 3 exp{ λ λ 1 )D } H, 1 λ 3 exp{ λ λ 1 )D } H where λ 1 = D H AH A H + C1 ) D, λ = D H AH + A H + C1 ) D, λ 3 = n λ 1 n λ. The filering equaion { ) d ˆx) = A ˆx) + C1 f 1 d + DH 1 P)d W), ˆx) = m has a soluion ˆx) = m + C 1 f 1 A 1 e A 1 ) + DH 1 Ps)e A s) d Ws).

9 188 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) Therefore ŷ) = Π)ˆx) + π), ẑ) = C 1 Π), where Π)= ) a A + e AT ) a A, π) = C 1 f 1 A 1 [ a A + ) e AT ) e AT )) a A e AT ) 1 )]. Obviously, he soluion ˆx ), ŷ ), ẑ )) of our filering problem is sable in he sense of Definiion A parially observed recursive opimal conrol problem The objecive of his secion is o sudy a parially observed recursive opimal conrol problem, which has a close connecion wih resuls in Secion. We use he same noaions as hose in Secion. Le us consider he following sae and observaion equaions: dx 1 ) = A)X 1 ) + C 1 )f 1 ) + C )f ) ) d + C 1 ) du) + C ) dv ), d Z 1 ) = D)X 1 ) + F) Z 1 ) + f )H ) ) d + H ) dv ), 6) X 1 ) = ξ, Z 1 ) =, { Ẋ ) = A)X ) + B)v), X ) =, Z ) = D)X ) + F) Z ), Z ) =, 7) where v ) U ad and all coefficiens saisfy H1) and H). For any v ) U ad, i is easy o check ha X 1 ) + X ) and Z 1 )+ Z ) are he unique soluion of 4) and 1), respecively, i.e., X ) = X 1 )+X ), Z ) = Z 1 )+ Z ). Se Z = σ { Z 1 s); s }. We presen he following: Definiion 4.1. A conrol variable v ) is called admissible, if v) is an R-valued sochasic process adaped o Z and Z and saisfying E v4 ) d < +. The se of admissible conrols is denoed by Ū ad. Remark 4.. From Definiion 4.1, we claim ha if v ) Ū ad hen Z = Z, T. In fac, i is clear ha Z Z, T. On he oher hand, if v ) Ū ad, from 7) we know ha X ) is Z -adaped, so is Z ). Thus Z) = Z 1 ) + Z ) is Z -adaped. Tha is o say, Z Z, T. Definiion 4.1 implies us o deermine he conrol by he observable process. Bu he observable process does no depend on he conrol. Oherwise, here is an immediae difficuly when he observable process depends on he conrol. I is he main reason ha he sae and he observaion equaions are decoupled. I follows from Definiion 4.1 and Remark 4. ha [ ] [ ˆX) = E Q X) Z = EQ X1 ) ] Z + X ) = ˆX 1 ) + X ). Since 6) is similar o 1) and 17), from Theorem., we easily ge he following resul. Proposiion 4.3. For any v ) Ū ad,leh1) and H) hold. Then he sae variable X ), which is he soluion of 4), has a filering esimaion d ˆX) = A) ˆX) + B)v)+ C 1 )f 1 ) + C )f ) ) d + C ) + D)H 1 )Δ) ) dū), 8) ˆX) = m,

10 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) where he observable 1-dimensional sandard Brownian moion Ū ) is defined as Ū)= V)+ Ds)H 1 s) Xs) ˆXs) ) ds and Δ ) = E Q X ) ˆX )) saisfies { Δ) A)Δ)+ C ) + D)H 1 )Δ) ) C 1 ) C ) =, Δ) = n. 9) Remark 4.4. Obviously, he soluion Δ ) of 9) does no depend on he admissible conrol v ) Ū ad. This is very imporan o solve he following Problem PO). Our problem is o seek a suiable v ) Ū ad o minimize he cos funcional Jv )) defined by 6) subjec o 4) and 1). If an admissible conrol ū ) Ū ad saisfies J ū ) ) = min J v ) ), v ) Ūad hen ū ) is called an opimal conrol, and he corresponding sae rajecory deermined by 4) is denoed by x ). For simpliciy, we denoe he above problem by Problem PO). This is a parially observed recursive opimal conrol problem. A classical solving mehod is o combine Wonham s separaion heorem wih a direc consrucion mehod inroduced in Bensoussan []. However, under our framework, we will inroduce a new echnique o solve i in hree seps. In deails, we firs regard Problem PO) as Problem FO) for a momen and seek is opimal soluion, nex we conjecure a candidae opimal conrol ū ) of Problem PO). To ge an explici observable opimal conrol, we apply he filering esimaion of BSDEs o characerize he adjoin process ȳ ). This is differen o Li and Tang [8], Tang [16], in which backward sochasic PDEs was used o describe adjoin processes. A he las sep, we verify ha ū ) defined by 36) is indeed an opimal conrol. In conras wih Wonham s separaion heorem, our mehod can be regarded as a backward separaion echnique. Follow our new echnique, i is much more convenien, direc and valid o solve Problem PO) han using he mehod inroduced in Bensoussan []. I needs o poin ou ha his idea is inspired by Li and Tang [8] and Tang [16], in which some heoreical resuls of maximum principles were derived, however hey did no illusrae how o use heir heoreical resuls o ge an explici observable opimal conrol of a parially observed opimal conrol problem. Moreover, o apply Girsanov s heorem, which is necessary o obain a maximum principle, Li and Tang need a crucial assumpion, i.e., he drif erm in heir observaion equaion is uniformly bounded wih respec o he sae x ) and he conrol v ). Alhough in our seing, i sill does no conain he conrol v ), bu linear wih respec o X ), Z )), which parly generalizes he resuls of Li and Tang. This is an anoher main difference o heirs. Sep 1. Opimal soluion of Problem FO). Recalling he opimal conrol defined by 7), we claim ha i is also unique. In fac, le u 1 ) and u ) be opimal, and he corresponding rajecories be X 1 ) and X ). Since 4) is a linear sysem, X 1 )+X ) and X 1 ) X ) are he rajecories under he conrols u 1 )+u ) and u 1 ) u ). Se Ju 1 )) = Ju )) = α, where α is a consan. From Parallelogram law, i follows ha α = J u 1 ) ) + J u ) ) ) ) ) u1 ) + u ) u1 ) u ) u1 ) + u ) = J + J = J { T + 1 E Q e bs)ds[ a) X 1 ) X ) ) + c) u1 ) u ) ) ] T d + e b)d X 1 T ) X T ) ) } α + 1 E Q e bs)ds c) u 1 ) u ) ) d,

11 19 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) which implies E Q e bs)ds c) u 1 ) u ) ) d. Thus u 1 ) u ), i.e., Problem FO) exiss a unique opimal conrol. Sep. Conjecure. Obviously, Ū ad U ad, i.e., he opimal conrol ū ) of Problem PO) is an elemen of U ad. For Problem PO), we canno fully observe he sae variable X ), and we also canno observe he adjoin process ȳ ), bu we can observe he noisy process Z ) relaed o X ). Our inuiion is o replace ȳ ) by is filering esimaion ˆȳ ). Inroduce an observable conrol variable ū) = 1 B)c 1 )e bs)ds ˆȳ), where x ), ȳ ), z 1 ), z )) saisfies he FBSDE d x) = A) x) 1 B )c 1 )e ) bs)ds ˆȳ) + C 1 )f 1 ) + C )f ) d + C 1 ) du) + C ) dv ), dȳ) = a)e bs)ds x) + A)ȳ) ) d z 1 ) du) z ) dv ), T x) = ξ, ȳt ) = e bs)ds xt ). The above equaion 31) is similar o 8), excep ha he drif erm of he forward SDE in 31) conains he observable process ˆȳ ). For mahemaically rigorous, we assume E ˆȳ Q 4 ) d < +. Then he forward SDE in 31) admis a unique soluion. So is he BSDE here. Tha is o say, for a given suiable ˆȳ ), here exiss a unique soluion o he FBSDE 31). On he oher hand, he following formulas 3) and 34) show ha he aforemenioned assumpion abou ˆȳ ) is indeed reasonable. Since E ˆȳ Q 4 ) d < +, i is clear ha ū ) defined by 3) is admissible. From sep one, we conjecure ha ū ) is a candidae opimal conrol of Problem PO). To prove he conjecure is rue in sep 3, nex we will give a more explici form of ū ) by compuing ˆ x ), ˆȳ )). Alhough in 31), he drif erm of he forward SDE conains ˆȳ ). Forunaely, ˆȳ ) is observable. So i does no bring difficuly for us o compue ˆ x ), ˆȳ )). From Proposiion 4.3, we easily derive ha d ˆ x) = + C ) + D)H 1 )Δ) ) dū), ˆx) = m. A) ˆ x) 1 B )c 1 )e bs)ds ˆȳ) + C 1 )f 1 ) + C )f ) Solving 31) by usual echniques for BSDEs, we ge T T ˆȳ) = e bs)ds+ As) ds [ ] E Q xt ) Z + ) d 3) 31) 3) s as)e br)dr+ s Ar) dr [ ] E Q xs) Z ds. 33) We now claim ha ˆȳ) = Π)ˆ x) + π), 34) where Π ), π ) and ˆ x ) are he soluions of 14), 15) and 3). In fac, if we le Ψ be he fundamenal soluion of ) A) Λ) Ψ = Ψ Λ) A) combining 31) wih 3), hen we have

12 where ) ˆ xs) = Ψs,) xs) s + ) s ˆ x) + x) G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) ) 1 Ψs,r) λr) dr 1 ) C r) + Dr)H 1 r)δr) Ūr) Ψs,r) d Ur), C 1 r) C r) Vr) Λ) = 1 B )c 1 )Π)e bs)ds, λ) = C 1 )f 1 ) + C )f ) 1 B )c 1 )π)e bμ)dμ. I is easy o check ha ) [ ] ) E Q xs) Z 1 = 1 Ψs,) 1 s s = e Ar) Λr)) dr ˆ x) + Subsiuing 35) ino 33), we have ˆȳ) = Π)ˆ x) + π) s ˆ x) + 1 ) Ψs,r) 1 1 ) λr) dr s e r Aμ) Λμ)) dμ λr) dr. 35) wih T Π)= e bs)ds+ As) Λs)) ds + s as)e br)dr+ s Ar) Λr)) dr ds, T π)= e bs)ds+ As) ds + s s as)e br)dr+ s Ar) dr e r Aμ) Λμ)) dμ λr) dr s e r Aμ) Λμ)) dμ λr)dr ds. From he exisence and uniqueness of soluions o 14) and 15), i is easy o verify ha Π ) and π ) saisfy 14) and 15), i.e., Π ) Π ), π ) π ). Tha is o say, he claim 34) is rue. On he oher hand, combining 3) and 34), we verify E ˆ x Q 4 ) d < +. So does ˆȳ ), i.e., he aforemenioned assumpion abou ˆȳ ) is reasonable. Thus he solvabiliy of 31) is furhermore confirmed. Sep 3. Proof of opimizaion. In his sep, we will prove ha ū) = 1 B)c 1 )e bs)ds Π)ˆ x) + π) ) 36) is a unique opimal conrol of Problem PO), where ˆ x ) saisfies 3) wih ˆȳ ) displaced by 34). Since ˆX ) X ) ˆX )), he cos funcional 6) can be rewrien as J v ) ) = J v ) ) + e bs)ds T a)δ)d + e bs)ds ΔT ) 37)

13 19 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) wih J v ) ) [ T = E Q e bs)ds a) ˆX ) + c)v ) ) ] T d + e bs)ds ˆX T ), 38) where ˆX ) and Δ ) saisfy 8) and 9), respecively. For any v ) Ū ad, we easily derive ha J v ) ) J ū ) ) { T = E Q e bs)ds[ a) ˆX) ˆ x) ) + c) v) ū) ) ] d + e bs)ds ˆXT ) ˆ xt ) ) } + Θ 39) wih { T Θ = E Q e bs)ds[ a)ˆ x) ˆX) ˆ x) ) + c)ū) v) ū) )] d + e bs)ds ˆ xt ) ˆXT ) ˆ xt ) )}. 4) Since all he erms depending on Δ ) have disappeared and he firs erm a he righ-hand side of 39) is non-negaive, we know ha J v ) ) J ū ) ) Θ. We claim ha Θ, which implies ha ū ) defined by 36) is opimal. In fac, noing 14), 15), 8), 3) and 34), i follows from Iô s formula ha d ˆȳ) = A) ˆȳ) + a)e bs)ds ˆ x) ) d Π) C ) + D)H 1 )Δ) ) dū), E Q [ ˆȳT ) ˆXT ) ˆ xt ) )] = E Q + E Q ˆX) ˆ x) ) d ˆȳ) [ A) ˆX) ˆ x) ) + B) v) ū) )] ˆȳ)d. Noing 34), 36) and ˆȳT ) = e bs)ds ˆ xt ), subsiuing he above wo formulas ino 4), we ge Θ = E Q e bs)ds c)ū) + B)ˆȳ) ) v) ū) ) d. Thus 41) implies ha ū ) defined by 36) is an opimal conrol. Applying Parallelogram law similar o sep one, we can also prove he uniqueness of ū ). Now we only need o compue Jū )). Subsiuing ū ) ino 38), we ge where J ū ) ) = Σ B )c 1 )π )e bs)ds d, 4) 41)

14 [ T Σ = E Q + 1 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) a)e bs)ds B )c 1 )Π )e ) bs)ds ˆ x ) d ] B )c 1 )Π)π)e bs)ds T ˆ x)d + e bs)ds ˆ x T ). Applying Iô s formula o 1 Π ) ˆ x )+π ) ˆ x ), inegraing on he inerval [,T], aking expecaions and comparing i wih 4), we derive J ū ) ) = Therefore we have [ 1 Π) C ) + D)H 1 )Δ) ) + C1 )f 1 ) + C )f ) ) π) 1 4 B )c 1 )π )e ] bs)ds d + a)δ)e bs)ds d + ΔT )e bs)ds + 1 m Π) + m π). 43) Theorem 4.5. Le H1) and H) hold. Then he unique opimal conrol and he cos funcional of Problem PO) are given by 36) and 43), respecively. Remark 4.6. If we se b ) = f 1 ) = f ), our Theorem 4.5 reduces o he classical resuls obained in Lipser and Shiryayev [9] and Bensoussan [], in which Wonham s separaion heorem is used o ge an explici observable opimal conrol. Remark 4.7. We find ha he filering esimaion ˆ x ), ˆȳ )) of x ), ȳ )), which is he soluion of 31), play an imporan role in looking for an opimal conrol ū ) of Problem PO). Alhough in 31), he drif erm of he forward SDE conains ˆȳ ). Forunaely, ˆȳ ) is observable. So i does no bring any difficuly for us o compue ˆ x ), ˆȳ )). This is also a moivaion for us o sudy he filering problems of a Hamilonian sysem which is an FBSDE in Secion. In fac, he above backward separaion echnique can cover a general siuaion han Problem PO). For example, we can formulae he following parially observed linear quadraic non-zero sum differenial game problem. Following he backward echnique and applying Theorem., we obain an explici observable Nash equilibrium poin of he game problem. Example 4.8. For simpliciy, le us only consider he case of wo players. The 1-dimensional sae and observaion equaions are as follows: { dx) = Ax) + B1 v 1 ) + B v ) ) d + C 1 dw 1 ) + C dw ), 44) x) = ξ, { dz) = D)x)d + F)dW ), Z) =. 45) Here and below, for convenience, we le M i, N i >, Q i, B i, C i i = 1, ) and A be consans, D ) and F ) be bounded deerminisic in [,T], F 1 ) be also bounded. ξ is an F -measurable Gaussian random variable, independen of W 1 ), W )), wih he mean m and he variance n.

15 194 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) The cos funcionals of wo players are denoed by J i v1 ), v ) ) [ T ] = 1 E Mi x ) + N i vi )) d + Q i x T ). 46) Our problem is o find a pair of u 1 ), u )) such ha J 1 u1 ), u ) ) = min v 1 ) Ũad J 1 v1 ), u ) ), J u1 ), u ) ) = min J u1 ), v ) ), v ) Ũad where { Ũ ad = v i ) v i ) is an R-valued sochasic process adaped o Z and Z and saisfies E vi }. ) d < + Then u 1 ), u )) is called a Nash equilibrium poin of he game problem 46) subjec o 44) and 45). Since he drif erm of he sae equaion 44) conains wo admissible conrols corresponding o wo players, i is more general han Problem PO). Forunaely, boh of hem are observable. And he sae and observaion equaions 44) and 45) are similar o 4) and 1). Thus i does no bring any rouble o esimae he sae x ) by filering. The solving mehod is similar o Problem PO), so we omi some of he similar proofs and only give key resuls here. Inroduce he following Riccai differenial equaions: { Σ 1 ) + AΣ 1 ) N 1 B Σ 1)Σ ) N 1 1 B 1 Σ 1 ) + M 1 =, 47) Σ 1 T ) = Q 1, { Σ ) + AΣ ) N 1 1 B 1 Σ 1)Σ ) N 1 B Σ ) + M =, 48) Σ T ) = Q. The above wo equaions are coupled ogeher. To prove ha here exis soluions o hem, we need an addiional assumpion N 1 1 B 1 = N 1 B. Inroduce he following equaions: Σ)+ AΣ) N 1 1 B 1 Σ ) + M 1 + M =, ΣT)= Q 1 + Q, 49) Σ 1 ) + A N 1 B Σ)) Σ 1 ) + M 1 =, Σ 1 T ) = Q 1, 5) Σ ) + A N 1 1 B 1 Σ)) Σ ) + M =, Σ T ) = Q. 51) I is clear ha 49) admis a unique soluion. Thus 5) and 51) also exis a unique soluion, respecively. Le Σ ) = Σ 1 ) + Σ ). We can verify ha Σ ) saisfies 49), i.e., Σ ) = Σ ). Subsiuing Σ ) = Σ 1 ) + Σ ) ino 5) and 51), we easily know ha 47) and 48) exis a unique soluion, respecively. Following he backward echnique and applying Theorem., we ge an explici observable Nash equilibrium poin ũ i ) = Ni 1 B i Σ i ) ˆ x), 5) where ˆ x ) is he soluion of { d ˆ x) = A N 1 1 B 1 Σ 1) N 1 B Σ ) ) ˆ x)d + C + D)F 1 )Δ) ) d W), 53) ˆ x) = m. The square error Δ ) = Ex ) ˆx )) and he innovaion process W ) saisfy respecively { Δ) AΔ) + C + D)F 1 )Δ) ) C 1 ) C ) =, Δ) = n, W)= W ) + Ds)F 1 s) xs) ˆxs) ) ds.

16 G. Wang, Z. Wu / J. Mah. Anal. Appl. 34 8) Remark 4.9. We noe ha he diffusion erms in 44) do no conain he sae or he conrol variable. Oherwise, we canno find an explici observable Nash equilibrium poin, even in he case of full observaion. To our bes knowledge, his is sill an open quesion. 5. Compuing he informaion value In his secion, our ask is o compue he difference of he opimal recursive cos funcionals of Problem PO) and Problem FO) ΔJ = J ū ) ) J u ) ), 54) which is called he informaion value. Subsiuing u ) given in 7) ino he cos funcional 6), by usual echniques, we have J u ) ) = [ 1 C 1 ) + C )) Π)+ C 1 )f 1 ) + C )f ) ) π) 1 4 B )c 1 )π )e ] bs)ds d + 1 m + n ) Π) + m π). From 43) and 54) we easily ge he following informaion value formula ΔJ = 1 + Therefore we have [ C ) + D)H )Δ) D)Δ)+ C )H ) ) C 1 )Π)] d a)δ)e bs)ds d + ΔT )e bs)ds 1 n Π). 55) Theorem 5.1. Le H1) and H) hold. Then he informaion value ΔJ defined by 54) can be wrien as 55). Remark 5.. The informaion value formula shows he following fac: o minimize he cos funcional, i is very imporan for conrollers o collec more informaion. ΔJ does no depend on m, which is he mean of he Gaussian random variable ξ, and increases in Δ ), he square error of he filering esimaion. Obviously, hese resuls coincide wih our inuiion. Remark 5.3. We noe ha here are no sae and conrol variables in he diffusion coefficiens of he FBSDE 8) and he conrol sysem 1). For ha case, we canno ge he explici filering esimaion for his kind of forward and backward sochasic sysem, and he opimal conrol for he parially observed recursive opimal conrol problem. To our bes knowledge, i is sill an open problem. For parially observed forward and backward sochasic sysems, here exiss few heoreical resul up o now. We hope ha we could furhermore develop his kind of heory and find more applicaions in our fuure work. Acknowledgmens The auhors would like o hank he anonymous referee for he careful reading and helpful commens and suggesions ha led o an improved version of his paper. The auhors also hank Dr. Mingyu Xu for many helpful advices. References [1] F. Anonelli, Backward-forward sochasic differenial equaions, Ann. Appl. Probab ) [] A. Bensoussan, Sochasic Conrol of Parially Observable Sysems, Cambridge Universiy Press, 199. [3] D. Duffie, L. Epsein, Sochasic differenial uiliy, Economerica 6 199)

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