Marina L. Kleptsyna 1, Alain Le Breton 2 and Michel Viot 2

Transcription

1 ESAIM: PS January 28, Vol. 2, p DOI:.5/ps:2746 ESAIM: Probabiliy and Saisis SEPARATION PRINCIPLE IN THE FRACTIONAL GAUSSIAN LINEAR-QUADRATIC REGULATOR PROBLEM WITH PARTIAL OBSERVATION Marina L. Klepsyna, Alain Le Breon 2 and Mihel Vio 2 Absra. In his paper we solve he basi raional analogue o he lassial linear-quadrai Gaussian regulaor problem in oninuous-ime wih parial observaion. For a onrolled linear sysem boh he sae and observaion proesses are driven by raional Brownian moions, we desribe expliily he opimal onrol poliy whih minimizes a quadrai perormane rierion. Aually, we show ha a separaion priniple holds, i.e., he opimal onrol separaes ino wo sages based on opimal ilering o he unobservable sae and opimal onrol o he ilered sae. Boh inie and ininie ime horizon problems are invesigaed. Mahemais Subje Classiiaion. 93E, 93E2. 6G5, 6G44. Reeived July 2, 26. Revised January 8, 27. Inroduion Several onribuions in he lieraure have been already devoed o he exension o he lassial heory o oninuous-ime sohasi sysems driven by Brownian moions o analogues in whih he driving proesses are raional Brownian moions (Bm s or shor). The raabiliy o he basi problems in prediion, parameer esimaion, ilering and onrol is now raher well undersood (see, e.g., [,6,5,6], and reerenes herein). Neverheless, as ar as we know, i is no ye demonsraed ha opimal onrol problems an also be handled or raional sohasi sysems whih are only parially observable. So, our aim here is o illusrae he aual solvabiliy o suh onrol problems by exhibiing an explii soluion or he ase o he simples linear-quadrai model. We deal wih he raional analogue o he so-alled linear-quadrai Gaussian regulaor problem in one dimension. The real-valued proesses X =(X, ) and Y =(Y, ), represening he sae dynami and he available observaion reord, respeively, are governed by he ollowing linear sysem o sohasi Keywords and phrases. Fraional Brownian moion, linear sysem, opimal onrol, opimal ilering, quadrai payo, separaion priniple. Laboraoire de Saisique e Proessus, Universié du Maine, av. Olivier Messiaen, 7285 Le Mans edex 9, Frane; Marina.Klepsyna@univ-lemans.r 2 LaboraoiredeModélisaion e Calul, Universié J. Fourier, BP 53, 384 Grenoble edex 9, Frane; Alain.Le-Breon@imag.r EDP Sienes, SMAI 28 Arile published by EDP Sienes and available a hp:// or hp://dx.doi.org/.5/ps:2746

2 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 95 dierenial equaions, whih shall be as usual inerpreed as inegral equaions: { dx = a()x d + b()u d + σ()dv H,,X = x, dy = A()X d + B()dW H,,Y =. () Here V =(V H, ) and W =(W H, ) are independen normalized Bm s wih Hurs parameer H in [/2, ) and x a ixed iniial ondiion. The oeiiens a, b, σ, A and B are assumed o be bounded and smooh enough (deerminisi) unions o R + ino R, wihb nonvanishing and B bounded. We suppose ha a eah ime one may hoose he inpu u in view o he pas observaions {Y s, s } in order o drive he orresponding sae, X = X u say, whih hene also as upon he observaion, Y = Y u say. Then, given a os union whih evaluaes he perormane o he onrol aions, he lassial problem o onrolling he sysem dynamis on some ime inerval so as o minimize his os ours. Here, in he inie ime horizon ase, given some ixed T>, we onsider he quadrai payo J T deined or a onrol poliy u =(u, [,T]) by J T (u) =E { q T X 2 T + [q()x 2 + r()u2 ]d}, (2) q T is a posiive onsan and q =(q(), [,T]) and r =(r(), [,T]) are ixed (deerminisi) posiive oninuous unions. I is well-known ha when H = /2 and hene he noises in () are Brownian moions, hen (see, e.g., [3,3,7]), he soluion ū o he orresponding problem, alled he opimal onrol, is provided or all [,T]by ū = b() r() ρ()π ( X); X = Xū ; Ȳ = Y ū, (3) π ( X) is he ondiional mean o X given {Ȳs, s } and ρ =(ρ(), [,T]) is he unique nonnegaive soluion o he bakward Riai dierenial equaion ρ() = 2a()ρ() q()+ b2 () r() ρ2 (); ρ(t )=q T. (4) In (3), he opimal iler π ( X) is generaed by he ollowing so-alled Kalman-Buy sysem on [,T]: dπ ( X) = [a()π ( X)+b()ū ]d + A() B 2 () γ()[dȳ A()π ( X)d],π ( X) =x γ() = 2a()γ()+σ 2 () A2 () B 2 () γ2 (), γ() =, (5) he soluion γ() o he las orward Riai equaion is nohing bu he variane IE( X π ( X)) 2 o he ilering error. Moreover, he minimal os J T (ū) isgivenby J T (ū) =ρ()x 2 + A 2 () T B 2 () ρ()γ2 ()d + q T γ(t )+ q()γ()d. (6) This resul is known as he separaion or erainy-equivalene priniple. I means ha, opimally, he proessing devie whih akes he observaion reord {Ȳs, s } and onvers i ino a onrol value ū separaes ino wo sages: ompuaion o he saisi π ( X) and ompuaion o he onrol value ū as a union o his saisi. The main eaure is ha hese operaions are independen in he sense ha he Kalman-Buy iler does no depend in any way on he oeiiens q T,q,r deining he onrol problem, as he onrol union

3 96 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT does no depend on he noise parameers σ, B, i.e., he onroller behaves as i π ( X) were he aual sae X, whih explains he erm erainy-equivalene. Our irs goal here is o show ha aually when he sysem () is driven by Bm s wih some H (/2, ) insead o Brownian moions, an explii soluion o he opimal onrol problem under he perormane rierion (2) is sill available in erms o a kind o separaion priniple. One o he ruial poins is ha, as in he ase H =/2, again or H (/2, ) he original problem an be redued o he opimal onrol problem wih omplee observaion o he ilered sae π (X). This an be raher easily noied hanks o he obvious orhogonaliy relaion E(X π (X))u = or any suiable variable u whih is measurable wih respe o {Y s, s }. Then, o solve he opimal onrol problem onerning π (X), we adap he approah led in [] o address he ase o omplee observaion in a linear-quadrai regulaor problem wih a raional Brownian perurbaion. Aually, we shall deal also wih he ininie ime horizon problem. In his ase, we assume ha he oeiiens a, b, σ, A and B in () are ixed onsans and we hoose an average quadrai payo per uni ime J whih is deined or a onrol poliy u =(u, ) by J (u) = sup T + T [qx 2 + ru2 ]d, (7) q and r are posiive onsans. Here, i is well-known ha when H =/2, hen (see, e.g., [3]) o ge an opimal onrol ū, one has jus, in he soluion o he inie ime horizon problem, o subsiue or he union ρ, given by (4), he nonnegaive soluion ρ o he algebrai Riai equaion 2a ρ q + b2 r ρ2 =,i.e., In oher words, ū is provided or all by ρ = r b 2 [a + δ ]; δ = a 2 + b2 q. (8) r ū = b r ρπ ( X); X = Xū ; Ȳ = Y ū, (9) he opimal iler π ( X) is sill generaed by (5). So, again a separaion priniple holds. Moreover he opimal os J (ū) isgivenby J (ū) = A2 B 2 ρ γ2 + q γ a.s., () γ is he nonnegaive soluion o he algebrai Riai equaion 2aγ + σ 2 A2 B γ 2 =,i.e., 2 γ = B2 A 2 [a + δ ]; δ = a 2 + A2 B 2 σ2. () We shall also exend hese resuls o he ase H (/2, ). Again, he main idea o he approah is ha he original problem an be redued o he opimal onrol problem wih omplee observaion o he ilered sae π (X). Bu, onrarily o he inie ime horizon ase, here one o he diiul poins is o obain an orhogonaliy ondiion in he sense ha he i as T ends o ininiy o a ime average T T (X π (X))u d is equal o zero almos surely or any suiable proess u. The veriiaion o ha ondiion and he analysis o several oher ruial ergodi ype properies require he preise sudy o he asympoi behavior o various proesses whih have ompliae sruures. Then, o solve he opimal onrol problem onerning π (X), we adap he approah led in [2] o address he ase o omplee observaion in a linear-quadrai regulaor problem wih a raional Brownian perurbaion. The paper is organized as ollows. A irs in Seion, we ix some noaions and preinaries. In pariular, we assoiae o he original problems auxiliary ilering and onrol problems onerning Volerra ype inegral

4 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 97 dynamis driven by appropriae Gaussian maringales orresponding o he raional noises. Then, in Seions 3 and 4, he inie ime and he ininie ime horizon onrol problems are solved, respeively. Seion 5 is devoed o a omplemenary analysis o he seond ase or whih anoher opimal onrol deined in erms o a simpler and more explii linear eedbak is desribed. Finally, an Appendix in Seion 5 is dediaed o auxiliary developmens: we derive some ehnial resuls and we invesigae ergodi ype properies o some proesses... Terminology and noaions. Preinaries In wha ollows all random variables and proesses are deined on a given sohasi basis (Ω, F, (F ), P). Moreover he naural ilraion o a proess is undersood as he P-ompleion o he ilraion generaed by his proess. Here, or some H [/2, ), B H = (B H, ) is a normalized raional Brownian moion wih Hurs parameer H means ha B H is a Gaussian proess wih oninuous pahs suh ha B H =,EBH =and EB H s B H = 2 [s2h + 2H s 2H ], s,. (2) O ourse he Bm redues o he sandard Brownian moion when H =/2. For H /2, he Bm is ouside he world o semimaringales bu a heory o sohasi inegraion w.r. o Bm has been developed (see, e.g., [4] or [5]). Aually he ase o deerminisi inegrands, whih is suiien or he purpose o he presen paper, is easy o handle (see, e.g., [5]). Fundamenal maringale assoiaed o B H. There are simple inegral ransormaions whih hange he Bm o maringales (see [9, 5, 6]). In pariular, deining or <s< k H (, s) =κ H s 2 H ( s) 2 H ; κ H =2HΓ( 3 (H 2 H)Γ + ), (3) 2 w H = λ H 2 2H ; λ H = 2HΓ(3 2H)Γ(H + 2 ) Γ( 3 2 H), (4) B = k H (, s)dbs H, (5) hen he proess B is a Gaussian maringale, alled in [5] he undamenal maringale, whose variane union B is nohing bu he union w H. Aually, he naural ilraion o B oinides wih he naural ilraion (B H )obh. In pariular, we have he dire onsequene o he resuls o [9] ha, given a suiably regular deerminisi union g =(g(), ), he ollowing represenaion holds: g(s)db H s = K g H (, s)db s, (6) or H (/2, ) he union K g H is given by K g H (, s) =H(2H ) g(r)r H 2 (r s) H 3 2 dr, s, (7) s and or H =/2 he onvenion K g /2 (,.) g or all is used. Admissible onrols. LeU be he lass o (F )-adaped proesses u =(u ) suh ha he sohasi dierenial sysem () has a unique srong soluion (X u,y u ) whih saisies J(u) < +,, aording o he seing, he os J(u) isevaluaedby(2)or(7),wihx = X u. Aually, as menioned in Seion, or onrol purpose

5 98 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT we are ineresed in poliies suh ha, or eah, u depends only on he pas observaions {Y s, s }. So, we inrodue he lass o admissible onrols as he lass U ad o hose u s in U whih are (Y u )-adaped proesses (Y u ) is he naural ilraion o he orresponding observaion proess Y = Y u. For u U ad, he riple (u, X u,y u ) is alled an admissible riple and i ū U ad is suh ha J(ū) =in{j(u),u U ad }, hen i is alled an opimal onrol and (ū, X,Ȳ ), X = Xū, Ȳ = Y ū, is alled an opimal riple and he quaniy J(ū) is alled he opimal os..2. Auxiliary ilering and onrol problems O ourse, aking ino aoun he resuls realled in Seion or he ase H =/2, our guess is ha again, in he raional world, he opimal onroller behaves as i he ilered sae were he aual sae. Aually, as a onsequene o he orhogonaliy relaion E(X π (X))u =oru s belonging o U ad, one ges ha J T (u) given by (2) an be deomposed as J T (u) = E { q T π 2 (X)+ [q()π 2 T (X)+r()u2 ]d} +q T E(X T π T (X)) 2 + q()e(x π (X)) 2 d. Hene, sine in a he varianes E(X π (X)) 2 o he ilering errors do no depend on he speii onrol u, i appears ha opimizing J T (u) is equivalen o minimizing he irs expeaion in he righ hand side above, whih expeaion depends only on he ilered sae π (X). Now, i is lear ha he soluion o he ilering problem in model () and he soluion o he onrol problem wih omplee observaion in he orresponding model or π (X) will be ruial in our analysis. So, we give some preinary resuls abou hese wo problems, adaping o he presen onex some o our previous works. Filering problem. Aually, he ilering problem in model (), wihou he addiional erm b()u in he sae dynami, has been solved in [9]. Bu, in he presen seing, sine only u s belonging o U ad are involved, i is raher immediae o exend he resul in he ollowing erms. Wih k H given by (3), we inrodue he observaion undamenal semimaringale Z whih is deined rom Y by: I an be represened as Z = Z = k H (, s)b (s)dy s. (8) Q(s)dw H s + W, W is he Gaussian maringale assoiaed o W H hrough (5) and Q() = d dw H k H (, s) A(s) B(s) X sds, (9) wih a derivaive undersood in he sense o absolue oninuiy. The naural ilraions (Z )and(y )oz and Y oinide and moreover he innovaion ype proess ν =(ν, [,T]) an be deined as ollows. Using, or any proess ξ =(ξ ; ) suh ha E ξ < +, he noaion π (ξ) or he ondiional expeaion E(ξ /Y ) o ξ given he σ-ield Y (or equivalenly Z ), he proess ν is given by: ν = Z π s (Q)dw H s, (2)

6 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 99 w H, Z and Q are given by (4), (8) and (9), respeively. Aually, ν does no depend on he admissible onrol u and moreover i is a oninuous Gaussian (Y )-maringale, wih he variane union ν = w H,whih allows a represenaion o he iler π (X). We inrodue he amily o 2 2 marix-valued deerminisi unions Γ (,.) =(Γ (, s), s ) whih saisies he Riai-Volerra ype sysem Γ (, s) = + s s [A (, r)γ (s, r)+γ (, r)a (s, r)]dr C(, r)c (s, r)dw H r s (2) Γ (, r)e 2 Γ (s, r)dwh r, he 2 2 maries A (, r), E 2 and veors C(, r) inr 2 are given by A (, r) =a(r) ; E p(, r) 2 = ; C(, r) = K σ H (, r), q(, r) wih p(, r) = d dw H k H (, v) A(v) d dv; q(, r) = r B(v) dw H Then, he iler π (X) isgovernedby r k H (, v)kh(v, σ r) A(v) dv. (22) B(v) π (X) =x + a(s)π s (X)ds + b(s)u s ds + Γ 2 (, s)dν s, (23) Γ 2 (, s) ishe(, 2)-enry o he marix Γ (, s). Noie ha he probabilisi inerpreaion o Γ (, s) is given in [9] (see also he beginning o seion 6.2 in he Appendix below) and in pariular, or s =, he diagonal enries Γ ii (, s) oγ (, s) are nohing bu he varianes o he ilering errors or X and Q, respeively, rom {Y s, s }, i.e., Γ (, ) =E(X π (X)) 2 ; Γ 22 (, ) =E(Q π (Q)) 2. O ourse, sine he deiniion (2) o he innovaion ν involves he iler π (Q), o generae he iler π (X) rom (23), one needs aually a omplemenary equaion o orm a losed sysem or he pair (π (X),π (Q)). We shall provide suh an equaion below (see also [8] or a global sysem o ilering equaions in he ase o onsan oeiiens wihou onrol). Conrol problem. Now, rom he disussion above, i beomes naural o analyze he onrol problem wih omplee observaion o he sae in a Volerra ype dynami inspired o (23). Preisely, we onsider a sae proess Π = (Π, ) generaed by Π = x + a(s)π s ds + b(s)u s ds + Γ 2 (, s)dm s, (24) M is a Gaussian (F )-maringale wih he variane union M = w H. Here, in he onrol problem whih is relevan regarding our original problem, he lass o admissible onrols u is he whole lass U. O ourse, aording o he seing, he payo J(u) o minimize is evaluaed by (2) or (7) wih Π = Π u in plae o X, respeively. We reognize ha aually he jus saed onrol problems are quie similar o hose whih have been solved in [] and [2]. More preisely, o ge here an opimal onrol ū, we have only o subsiue (, s) in he seings herein. Thereore, in he inie horizon ase, we inrodue he amily o 2 2 marix-valued unions Γ (., s) = (Γ (, s), [s, T ]) suh ha Γ (., s) is he unique nonnegaive symmeri soluion o he bakward Riai Γ 2 (, s) ork H

7 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT dierenial equaion in he variable on [s, T ] Γ (, s) = A (, s)γ (, s) Γ (, s)a (, s) q()d(, s)d (, s) + b2 () r() Γ (, s)e Γ (, s); Γ (T,s)=q T D(T,s)D (T,s), (25) he 2 2 maries A (, s), E and veors D(, s) inr 2 are given by Γ 2 A (, s) =a() (, s) ; E = ( ; D(, s) = Γ 2 ). (, s) Aually, he (, )-enry Γ (, s) ohemarixγ (, s) does no depend on he variable s and i is nohing bu he soluion ρ() o he Riai equaion (4). Again, we shall denoe by Γ ij (, s) he(i, j)-enry o Γ (, s). Now, parallelling he developmens in [], we ge an opimal onrol ū in U suh ha he opimal pair (ū, Π), Π =Πū, isgovernedon[,t] by he sysem ū = b() r() {ρ() Π + [Γ 2 (, s) ρ()γ2 (, s)]dm s} ; Π =Πū, (26) and moreover he opimal os is J T ( v) =ρ()x 2 + Γ 22 (, )dwh. (27) Similarly, in he ininie horizon ime ase, we shall be able o ake benei o he approah led in [2] in order o analyze he auxiliary onrol problem wih omplee observaion assoiaed wih he original onrol problem. 2. Finie ime horizon onrol problem A irs we sae our main resul: Theorem 2.. Le k H (, s), p(, s) and q(, s) be he kernels deined in (3) and (22), respeively. Le also Γ, Γ be he soluions o (2), (25), respeively, and ρ be he soluion o (4). In he onrol problem min u U ad J T (u) subje o (), wih J T deined by (2), an opimal onrol ū in U ad and he orresponding opimal riple (ū, on [,T] by he sysem X,Ȳ ) are governed ū = b() r() {ρ()π ( X)+ ν = and he pair (π ( X),π ( Q)) is generaed by π ( X) =x + π ( Q) =p(, )x + [Γ 2 (, s) ρ()γ2 (, s)]dν s}; X = Xū ; Ȳ = Y ū, (28) k H (, s)b (s)dȳs a(s)π s ( X)ds a(s)p(, s)π s ( X)ds + b(s)ū s ds + π s ( Q)dw H s, (29) b(s)p(, s)ū s ds + Γ 2 (, s)dν s, (3) Γ 22 (, s)dν s. (3)

8 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS Moreover he opimal os is J T (ū) =ρ()x 2 + Γ 22 (, )dwh + q T Γ (T,T)+ q()γ (, )d. (32) Remark 2.2. (a) InheaseH =/2, or all s, k H (, s) =,p(, s) =q(, s) =A()/B(), i is easy o hek ha or all s T, he maries Γ (, s) andγ (, s) redue o: ( A(s) B(s) Γ (, s) =γ(s) ; Γ (, s) =ρ() A(s) B(s) A 2 (s) B 2 (s) A(s) B(s) γ(s) A(s) B(s) γ(s) A 2 (s) B 2 (s) γ2 (s) ρ and γ are he soluions o he Riai equaions given in (4) and (5), respeively. Hene, i is readily seen ha aually he saemen in Theorem 2. redues globally o he well-known resul realled in Seion. (b) Inroduing v = [Γ 2 (, s) ρ()γ 2 (, s)]dν s, one an wrie he opimal onrol ū as ū = b() r() {ρ()π ( X)+ v }. I is worh menioning ha aually he addiional erm v whih appears in he ase H>/2 (and equals zero when H =/2) an be inerpreed in erms o he prediors a ime o he noise omponen Vτ H, τ T based on he observed opimal dynamis (Ȳ s,s ) upoime. Preisely, one an rewrie v = ), φ(τ,)ρ(τ)σ(τ) τ E(V τ H /F ū )dτ, τ T, (33) τ φ(τ,) =exp{ [a(u) b2 (u) ρ(u)]du}, (34) r(u) or, equivalenly, v = E( φ(τ,)ρ(τ)σ(τ)dvτ H /Yū ). This will be made lear aer he proo o Theorem 2.. Proo o Theorem 2.. Clearly, due o is deiniion hrough a losed sysem in erms o he only proess Ȳ, he onrol poliy ū given by (28) belongs o U ad. Moreover, omparing (3) wih (23), we see ha π ( X) is nohing bu he iler o X based on he observaion o Ȳ. Aually, rom he resuls in [9], i an be seen ha π ( Q) given (3) is also he iler o he proess Q whih is he analogue or X o Q deined rom X by (9). This explains why one may also subsiue (29) or (2) in he represenaion o he innovaion ν whih does no depend on he admissible onrol. Le us onsider he proess (p, T ) deined by p = ρ()π ( X)+ [Γ 2 (, s) ρ()γ2 (, s)]dν s, (35) whih in pariular allows he represenaion ū = (b()/r())p. Parallelling he proo in [], i an be shown ha i saisies he ollowing bakward sohasi dierenial equaion: dp = a()p d q()π ( X)d +Γ 2 (, )dν, [,T]; p T = q T π T ( X). (36)

9 2 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT Now we show ha ū minimizes J T over U ad. Given an arbirary u U ad, we evaluae he dierene J T (u) J T (ū) beween he orresponding os and he os or he announed andidae ū o opimaliy. O ourse, we an wrie J T (u) J T (ū) =E {q T [X 2T X 2T ]+ {q()[x 2 X } 2 ]+r()[u 2 ū 2 ]}d. Using he equaliy y 2 ȳ 2 =(y ȳ) 2 +2ȳ(y ȳ) and exploiing he propery ū = (b/r)p, i is readily seen ha J T (u) J T (ū) =E( )+2E( 2 ), = q T [X T X T ] 2 + {q()[x X ] 2 + r()[u ū ] 2 }d, 2 = q T XT [X T X T ]+ Aually, he las inegral an be wrien as {q() X [X X ] b()p [u ū ]}d. {(X X )[q() X + a()p ] p [a()(x X )+b()(u ū )]}d, and moreover, due o (), (23) and (3), X X = π (X) π ( X).Heneweanrewrie 2 in he orm 2 = q T [ X T π T ( X)][π T (X) π T ( X)] + q()[ X π ( X)][π (X) π ( X)]d +q T π T ( X)[π T (X) π T ( X)] + p [a()(π (X) π ( X)) + b()(u ū )]d. [π (X) π ( X)][q()π ( X)+ap ]d Now, aking ino aoun equaion (36), we see ha he dierene o he las wo inegrals an be wrien as (π (X) π ( X))dp p d(π (X) π ( X)) + (π (X) π ( X))Γ 2 (, )dν. Thereore, insering his ino he expression above o 2 and aking he expeaion, sine E[ X π ( X)][π (X) π ( X)] = and he sohasi inegral par gives also zero, we ge ha { E( 2 )=E q T π T ( X)[π T (X) π T ( X)] (π (X) π ( X))dp p d(π (X) π ( X)) }. Now, inegraing by pars, sine p T = q T π T ( X) andπ (X) π ( X) =,iomeshae( 2 ) = and inally J T (u) J T (ū) =E( ). This o ourse means ha ū minimizes J T over U ad. Now we ompue he opimal os J T (ū). Sine and E[π ( X)( X π ( X))] =, we an wrie { J T (ū) = E q T π 2 ( X)+ T X 2 = π2 ( X)+[ X π ( X)] 2 +2π ( X)[ X π ( X)], +q T E[ X T π T ( X)] 2 + [q()π 2 ( X)+r()ū } 2 ]d q()e[ X π ( X)] 2 d.

10 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 3 Comparing (28), (3) wih (24), (26), we reognize ha he irs quaniy in he righ-hand side above is nohing bu he opimal os (27) in he auxiliary onrol problem disussed in Seion 2. Thereore, sine moreover E( X π ( X)) 2 =Γ (, ), we see ha he expression (32) holds or J (ū). T Jusiiaion o Remark 2.2.b Paralleling he disussion in [], i ollows ha φ(τ,) is given by (34) and Bu or any τ we have v = φ(τ,)ρ(τ) τ E(V τ /Yū )dτ, V τ = τ E(V τ /Yū )= Γ 2 (τ,r)dν r. Γ 2 (τ,r)dν r, and hene also τ E(V τ /Yū )= τ Γ2 (τ,r)dν r. Thereore, he equaliy (33) will be valid i we prove Bu or τ>he ollowing represenaion holds: and so we jus need o show ha τ Γ2 (τ,r)dν r = σ(τ) τ E(V τ H /Yū ); τ>. E(V H τ /Yū )= τ Γ2 (τ,r) =σ(τ) τ To prove his equaliy, or r<<τ, we inrodue he quaniy G(τ,r) = [ ] EVτ H ν r dν r, ν r ν r EV H τ ν r. (37) ν r EX τ ν r, X sands or X u wih u. Sine Eπ τ (X )ν r = EX τ ν r, rom equaions (23) and () (wih u ), we an derive he ollowing wo equaions or G(τ,r): G(τ,r) =a(τ)g(τ,r)+ τ τ Γ2 (τ,r), G(τ,r) =a(τ)g(τ,r)+σ(τ) τ τ ν r EV τ H ν r. This gives ha (37) holds and hene also (33).

11 4 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT 3. Ininie ime horizon onrol problem Firs soluion Here we assume ha he oeiiens a, b, σ, A and B in () are ixed onsans and ha q and r are posiive onsans. Then, in equaion (2) or Γ (, s) = ((Γ ij (, s))), he oeiiens A and C ake he pariular orm ( ) KH (, r) A (, r) =a A ; C(, r) =σ B p A, (38) (, r) B q (, r) wih p (, r) = d dw H k H (, v)dv ; q (, r) = d r dw H k H (, v)k H (v, r)dv, (39) r k H is given by (3) and K H denoes K g H deined by (7) when g, i.e., K H (, s) =H(2H ) s r H 2 (r s) H 3 2 dr, s. Taking benei o he approah led in [2] or he ininie ime horizon onrol problem wih omplee observaion, i is naural o inrodue he ollowing amily o auxiliary unions (γ 2 (., s),s ). For any ixed s, we deine he union γ 2 (., s) =(γ2(, s), s) by γ 2 (, s) =δ eδ e δ τ Γ 2 (τ,s)dτ, (4) δ is given by (8). Now, we an sae our main resul: Theorem 3.. Le k H (, s), p (, s), q (, s) and Γ (, s) be he kernels deined in (3), (39) and (2) wih A and C given by (38). Le also he onsans ρ, γ be deined by (8), () and he union γ 2 be given by (4). In he onrol problem min J (u) subje o (), u U ad wih J deined by (7), an opimal onrol ū in U ad and he orresponding opimal riple (ū, X,Ȳ ) are governed by he sysem ū = b r ρ{π ( X)+ and he pair (π ( X),π ( Q)) is generaed by π ( X) =x + π ( Q) = A B {p (, )x + a Moreover he opimal os is [γ 2 (, s) Γ 2 (, s)]dν s } ; X = Xū ; Ȳ = Y ū, (4) ν = B k H (, s)dȳs aπ s ( X)ds + p (, s)π s ( X)ds + b bū s ds + π s ( Q)dw H s, (42) Γ 2 (, s)dν s, (43) p (, s)ū s ds} + Γ 22 (, s)dν s. (44) J (ū) =ζ (H)+qγ (H), (45)

12 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 5 and ζ (H) = A2 Γ(2H +)sinπh ρ γ2 B2 (δ δ ) 2 (δ + δ ) [ δ a { [ δ a (δ + a) δ H 2 ]} δ a δ 2H +(δ δ )(δ a) δ 2H +Γ(2H +) qσ2 2 ( sin [ a2 πh)δ2 γ (H) = σ2 Γ(2H +) 2δ 2H [ δ 2 δ2 δ 2H ] 2 δ a δ H 2 ] δ 2H, (46) + δ + a ] sin πh, (47) δ a wih δ, δ given by (8), (). Remark 3.2. (a) From he proo below, i will appear ha he erm γ (H), given by (47) whih is involved in he represenaion (45) o he opimal os J (ū), is nohing bu he i as ends o ininiy o he variane E( X π ( X)) 2 o he ilering error. Aually, in he saemen above, or δ = δ,hequaniyζ (H) mus be inerpreed as he i o he righ hand side o (46) as δ ends o δ. This i is nohing bu { A 2 B ρ γ 2 Γ(2H+) sin πh (δ 2 2δ 2H+2 + a) [ δ + ( H 2) (δ a) ] } 2 + δ (δ a)[δ +(2H )(δ a)] +Γ(2H +) Hqσ2 ( sin πh)(δ 2 (48) a2 ). 2δ 2H+2 (b) InheaseH =/2, or all s, k H (, s) =,p (, s) =q (, s) =, i is easy o hek ha or all s T,hemarixΓ (, s) and he quaniy γ 2 (, s) redue o: ( A ) Γ (, s) = B γ(s); γ 2 (, s) = A B γ(s), A B A 2 B 2 γ is he soluion o he Riai equaion given in (5). Hene, i is readily seen ha aually he saemen in Theorem 3. redues globally o he well-known resul realled in Seion. () Inroduing v = [γ 2 (, s) Γ 2 (, s)]dν s, one an wrie he opimal onrol ū as ū = b r ρ[π ( X)+ v ]. I is worh menioning ha aually he addiional erm v whih appears in he ase H>/2 (and equals zero when H =/2) an be inerpreed in erms o he prediors a ime o he noise omponen Vτ H,τ based on he observed opimal dynamis (Ȳs,s ) upoime. Preisely, one an rewrie or, equivalenly, v = σ v = σe( e δ (τ ) τ E(V τ H /Yū )dτ, τ, (49) e δ (τ ) dv H τ /Yū ). This an be derived rom he disussion in [2] by means o argumens similar o hose whih have been used above o prove (33).

13 6 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT Proo o Theorem 3.. Conerning he admissibiliy o ū and he inerpreaion o he dieren erms involved in he sysem, one may repea exaly he argumens a he beginning o he proo o Theorem 2.. Le us onsider he proess (p, ) deined by p = ρ{π ( X)+ [γ 2 (, s) Γ2 (, s)]dν s}, (5) whih in pariular allows he represenaion ū = (b/r)p. Parallelling he proo in [2], i an be shown ha i saisies he ollowing sohasi dierenial equaion: dp = ap d qπ ( X)d + ργ 2 (, )dν, ; p = ρx. (5) Now we ompue he os J (ū) orresponding o he onrol ū. Given an arbirary u U ad,weusehenoaion X = X u.foru =ū, weanwrie j T (u) = [qx 2 + ru 2 ]d, j T (ū) = +q [qπ 2 ( X)+rū 2 ]d [ X π ( X)] 2 d +2q π ( X)[ X π ( X)]d. (52) From (4), (43) and (5), we reognize ha he pair (ū,π ( X)) is governed by a dynami similar o ha o he opimal pair obained in [2] or an ininie horizon ime problem under raional Brownian perurbaion and omplee observaion. Therein, he union K H (, s) plays exaly he same role as Γ 2 (, s) here. Then, we may parallel he developmens (see seion 6.3 in he Appendix below) o ge he i a.s. T + T [qπ 2 ( X)+rū 2 ]d = ζ (H), (53) ζ (H) is given by (46). Conerning he seond erm in (52), o ourse again we have E( X π ( X)) 2 = Γ (, ). Moreover, he resul obained in [8] or he iing behavior o his variane o he ilering error in an auonomous linear model driven by raional Brownian noises ells ha + Γ (, ) =γ (H), γ (H) is given by (47). Aually, in he Appendix (. Se. 6.3), we prove ha he proess X π ( X) possesses he ollowing ergodi ype propery: Moreover we shall show ha a.s. So, inally, he i T + holds or J (ū). T j T T + T [ X π ( X)] 2 d = γ (H), a.s. (54) T π ( T + T X)[ X π ( X)]d =. (55) (ū) exiss a.s. and insering (53) (55) ino (52), we see ha he expressio (45)

14 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 7 Nowweshowhaū minimizes J over U ad. Proeeding along seps quie similar o hose ollowed in he proo o Theorem 2., we an evaluae he dierene j T (u) j T (ū) as j T (u) j T (ū) = (T )+2 2 (T ), (T )= {q[x X ] 2 + r[u ū ] 2 }d, and 2 (T ) = q[π (X) π ( X)][ X π ( X)]d + ρ p d[π (X) π ( X)] [π (X) π ( [π (X) π ( X)]dp. 2 X)]γ (, )dν Noie ha, inegraing by pars in he las line above, sine π (X) π ( X) =, we an rewrie 2 (T )as 2 (T ) = + ρ q[π (X) π ( X)][ X π ( X)]d p T [π T (X) π T ( X)] [π (X) π ( X)]γ 2 (, )dν. Sine J (u) = sup T + T j T (u) a.s.and (T ), o ourse we have J (u) J (ū) + in T + T 2(T ) a.s. Hene, in order o prove ha ū minimizes J over U ad, i is suiien o show ha T + T 2(T )=a.s. To his end, i suies o show ha, i u U ad is suh ha J (u) < +, hen he ollowing is hold a.s. and T + T T + T [π (X) π ( X)][ X π ( X)]d =, (56) T + T p [π (X) π ( X)] =, (57) T T T [π (X) π ( These hree properies will be proved in he Appendix (. Se. 6.3). X)]γ 2 (, )dν =. (58) 4. Ininie ime horizon onrol problem Seond soluion Here he seing is he same as in Seion 4. A irs, i is worh emphasizing ha he analysis herein has provided no only an opimal onrol bu also he explii value o he minimal os. Hene or any oher onrol possibly andidae or opimaliy, he only hing o hek is ha i ahieves he minimal os. On his basis, now we propose a simpler and more explii onrol whih is also opimal. Wih respe o he irs soluion, i has he advanage ha he involved eedbak is more learly expressed in erms o he observaion proess Y. Moreover, i is oheren wih he asympoially opimal iler proposed in [] or linear sysems wihou onrol and wih he seond soluion proposed in [2] or he ininie ime horizon opimal onrol problem under omplee observaion.

15 8 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT In he lassial ase H = 2 he noises are sandard Brownian moions and hene he sysem o ilering equaions redues o he usual Kalman-Buy sysem, he asympoi variane o he ilering error is γ given by (). In ha ase, subsiuing he onsan γ or he union γ() in he equaions (5) and (9), one ges he simpler onroller u = b r ρπ ; X = X u ; Y = Y u, dπ = [aπ + bu ]d + A (59) B 2 γ[dy Aπ d], π =, ρ is sill given by (8). Observe ha, aking ino aoun (8) and (), aually π is governed by Hene π an be represened as dπ = [a + δ + δ ]π d + A B 2 γdy ; π =. (6) π = A B 2 γ e [a+δ +δ ]( s) dy s. (6) I an be heked ha, as a iler o X rom he observaions {Ys,s }, π is asympoially opimal in he sense ha, as goes o ininiy, he variane E(X π ) 2 o he orresponding ilering error onverges o he same i γ as E(X π (X )) 2. Moreover, i an also be heked ha he riple (u,x,y ) is aually opimal in he ininie ime horizon onrol problem, i.e., J (u )=J (ū) given by (). Observe ha, in his ase, he opimaliy is ahieved in he lass o onrols whih an be represened as φ( s)dy s. Below, we show ha his sill holds or H> 2 and we ideniy in his lass a onrol or whih he os value is J (ū) given by (45). So, we sar wih a onrol u = u φ in he orm u = Then, rom (), i is readily seen ha we have also φ( s)dy s. u = X = U V ( s)dv H s + X V ( s)dv H s + U W ( s)dw H s, X W ( s)dw H s, (62) X V (τ) = ax V (τ)+bu V (τ),τ ; X V () = σ, X W (τ) = ax W (τ)+bu W (τ),τ ; X W () =, U V (τ) = A U W (τ) = A τ τ φ(v)x V (τ v)dv, τ, φ(v)x W (τ v)dv + Bφ(τ),τ. (63) For simpliiy, here we deal only wih he ase x =.

16 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 9 Aually, i an be reognized ha X W (Bb/Aσ)U V. Hene, or he sysem (62)-(63), we an subsiue he ollowing: u = U V ( s)dvs H + U W ( s)dws H, X = X V ( s)dvs H + Bb (64) U Aσ V ( s)dws H, and X V (τ) = ax V (τ)+bu V (τ),τ ; X V () = σ, U V (τ) = au V (τ)+ Aσ (65) B U (τ),τ ; U () =, W V τ U W (τ) = Bb φ(v)u σ V (τ v)dv + Bφ(τ),τ. We observe ha in he sysem deined by he las wo equaions above whih link φ, U V and U W, o ix a union φ is equivalen o ix a union U W. So, we may orge abou he las equaion and rom now on, insead o a union φ, we look or an appropriae union U W. O ourse, we are ineresed in hose unions U W suh ha, or he proess (u, X) given by (64) (65), he is T + T T u2 d and T + T T X2 d exis and are inie almos surely. Our guess is ha he minimum or J an be obained by hoosing U W in suh a way ha hese is are nohing bu T + Eu 2 T and T + EXT 2 respeively and moreover he minimum value o Ĵ (U W ) = T + E[qX2 T + ru 2 T ], is ahieved. Aually, or a sohasi inegral we an evaluae S = T + ES2 T = H(2H ) Exploiing he represenaion s r 2H 2 = i is easy o hek ha we an rewrie g( s)db H s, B(H 2, 2 2H) s r T + ES2 T = 2HΓ( 3 2 H) Γ(H + 2 )Γ(2 2H) g(s)g(r) s r 2H 2 dsdr. (τ s) H 3 2 (τ r) H 3 2 dτ, g 2 (s)ds, (66) g(s) = d s g(r)(s r) H 2 dr. (67) ds Hene, due o (64) and he independene beween V H and W H, we an rewrie he quaniy Ĵ (U )as W J (ŨW )= 2HΓ( 3 2 H) {q Γ(H + 2 )Γ(2 2H) X 2V (s)+[q B2 b 2 } A 2 σ 2 + r]ũ 2 (s)+rũ 2 (s) ds, (68) V W

17 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT he riple ( X V, ŨV, ŨW ) orresponds o (X, U, U ) by (67). Aually, i an be readily seen rom (65) V V W ha he dynamis whih link X V, ŨV, ŨW are nohing bu X V (τ) = a X (τ)+bũv (τ)+σ(h V 2 )τ H 3 2,τ ; XV () =, (69) Ũ V (τ) = aũv (τ)+aσ B ŨW (τ),τ ; Ũ V () =. Now we solve he ininie ime horizon deerminisi onrol problem. Le us deine he 2 2 maries A, Q and he veor B in R 2 by: A = min J (ŨW ) subje o (69). (7) Ũ W a b ; Q = q a q B2 b 2 A 2 σ 2 + r ; B = and inrodue also he R 2 -valued unions Z V (τ) andk(τ) as Then, we an rewrie (69) and (68) as ) XV (τ) Z V (τ) =( ; K(τ) = Ũ V (τ) Aσ, B ( σ(h 2 )τ ) H 3 2. Z V (τ) =A Z V (τ)+bũw (τ)+k(τ), τ ; ZV () =, (7) and J )= 2HΓ( 3 2 H) (ŨW Γ(H + { 2 )Γ(2 2H) Z (s)q Z V V (s)+rũ 2 (s)}ds. (72) W Hene, o solve he problem (7), i is naural o inrodue he ollowing Hamilonian sysem in he unions Z and p : Z (τ) = A Z (τ) r BB p (τ)+k(τ), Z () = ; Z (τ) =, τ + p (τ) = Q Z (τ) A p (τ); τ + p (τ) =. Observe ha, i ( Z, p ) is a soluion o (73), hen Z =( X, Ũ ) is nohing bu he soluion o (7) orresponding o he onrol V V Ũ (τ) = r B p (τ). W Aually, sandard alulaions permi o show ha i ( Z, p ) is a soluion o (73), hen he pair ( Z, Ũ )is W an opimal pair in he onrol problem (7). So, inally, o solve his problem we have jus o ideniy a soluion o (73). Le us look or a soluion in suh a way ha or some ixed 2 2 marix Γ and R 2 -valued union λ, he union p an be represened as p (τ) =Γ[ Z (τ)+ λ (τ)]. (74) Hene, i is readily seen ha one may ake Γ as he nonnegaive soluion o he algebrai Riai equaion A Γ+ΓA r ΓBB Γ+Q =, (73)

18 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS and λ as he soluion o he dierenial equaion Γ λ (τ) =[r ΓBB A ]Γ λ (τ) ΓK(τ) ; τ + I an be heked ha he maries Γ and Λ = r ΓBB A are given by Γ= B2 r A 2 σ 2 b 2 λ (τ) =. (75) (a + δ )(a + δ )(δ + δ ) b(a + δ )(a + δ ) b(a + δ )(a + δ ) b 2, (76) (2a + δ + δ ) and The eigenvalues o Λ are δ Λ=r ΓBB A a b = (a + δ )(a + δ ). (77) b a+ δ + δ and δ.iδ δ, hen i an be deomposed as δ Λ=P P δ, a + δ a + δ P = b b Similarly, i δ = δ, hen Λ an be deomposed as O ourse, we have or For δ δ,hisgives and in pariular P + = ; P = b (a + δ ). b(δ δ ) b a+ δ δ Λ=P b [b2 +(a + δ ) 2 ] + P δ +, a + δ b ; P b a+ δ + = a + δ b b 2 +(a + δ ) 2. b a+ δ Γ λ () = Γ λ () =e Λ e Λs ΓK(s)ds, Γ λ () = e Λτ ΓK( + τ)dτ. ( ρ γ (δ 2 a2 ) ϕ () (δ 2 ) a2 ) ϕ () σ(δ δ ) b[(δ a) ϕ () (δ a) ϕ ()], (78) ϕ () =e δ e δ s ds H 2 ; ϕ () =e δ e δ s ds H 2, (79) Γ λ () = Γ(H + 2 ) σ δ ( ) ρ γ (δ 2 a2 )δ 2 H (δ 2 a2 )δ 2 H δ b[(δ a)δ 2 H (δ a)δ 2 H ]. (8) Conerning he os, saring rom (73) and evaluaing by inegraion by pars he variaion o p () Z () over [, + ), i is readily seen ha J (Ũ )= 2HΓ( 3 2 H) W Γ(H + 2 )Γ(2 2H) K (τ)γ p (τ)dτ.

19 2 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT Consequenly, evaluaing by inegraion by pars he variaion o λ () Γ Z () over[, + ), we ge ha J (Ũ W ) an also be represened as J (Ũ W )= 2HΓ( 3 2 H) Γ(H + 2 )Γ(2 2H) { 2 K (τ)γ λ (τ)dτ r λ (τ) ΓBB Γ λ (τ)dτ }. Then, evaluaing he variaion o λ () Γ λ () over[, + ), i omes ha also J (Ũ W )= 2HΓ( 3 2 H) Γ(H + 2 )Γ(2 2H) { λ () Γ λ () + λ (τ) Q λ (τ)dτ }. Finally, an alernaive expression or J (Ũ ) an be derived. We inrodue he nonnegaive soluion R o he W algebrai Lyapunov equaion Λ R + RΛ =r BB. Then, evaluaing by inegraion by pars he variaion o λ () ΓRΓ λ () over[, + ), i omes ha we an also wrie J (Ũ W )= 2HΓ( 3 2 H) Γ(H + 2 )Γ(2 2H) { λ () ΓRΓ λ () + 2 K (τ)[i ΓR]Γ λ (τ)dτ }. (8) Aually, we ge R = A2 σ 2 b 2 ab 2B 2 r δ δ (δ + δ ) ab a 2. + δ δ Now, rom (78)-(8), we an ompue he value J (Ũ ). A irs, rom (78), i omes ha he seond erm W wihin brakes in he righ hand side o (8) gives qσ 2 Γ(H + 2 )Γ(2 2H)Γ(2H) 2Γ( 3 2 H) (δ 2 a2 )δ 2H (δ 2 a2 )δ 2H δ 2 δ2 Moreover, rom (8), i omes ha he irs erm wihin brakes in he righ hand side o (8) gives Γ 2 (H + 2 )A2 γ 2 ρ 2B 2 δ + a δ δ (δ δ ) 2 (δ + δ ) { [δ 2 H δ (δ a) δ 2 H δ (δ a)] 2 +δ δ [δ 2 H (δ a) δ 2 H (δ a)] 2}. Finally, rom hese expressions, i an be heked ha J (Ũ ) an be represened as W J (Ũ W )=ζ (H)+qγ (H), ζ (H) andγ (H) are given by (46) and (47), respeively. In oher words, J (Ũ ) is nohing bu he W opimal os J (ū) given by (45) or our original ininie horizon sohasi onrol problem. The ase δ = δ an be reaed similarly o obain he same onlusion, wih he iing value (48) insead o (46) or he erm ζ (H). Now, aking ino aoun he a ha he onneion (67) an be invered by g(s) = d B(H + 2, 3 2 H) ds s g(r)(s r) 2 H dr, (82)

20 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 3 we may reormulae our iniial guess by elling ha he riple (U V, X V, U W ), obained rom (Ũ V, X V, Ũ W ) hrough (82), is a andidae o deine hrough (64) an opimal pair (u,x ) in he original ininie horizon sohasi onrol problem. Aually he proo ha his is rue is inluded below in he proo o he ollowing saemen. Theorem 4.. Le he pairs o onsans ( ρ, δ ) and ( γ,δ ) be deined by (8) and () respeively. In he onrol problem min J (u) u U ad subje o (), wih J deined by (7), an opimal onrol u in U ad and he orresponding riple (u,x,y ) are governed by he sysem u = b r ρ[π + Aσ2 v v ]; X = X u ; B δ δ Y = Y u, (83) he proess π is deined by π =ˆπ + A B γv, (84) wih dˆπ = aˆπ d + bu d + A B 2 γ[dy Aˆπ d]; ˆπ = x, (85) and he proesses v and v saisy he inegral equaions [ ] δ v = δ vs ds + 2 H Γ( 3 2 H)( s) 2 H B [dys sds], (86) v = [ ] δ vs ds + δ 2 H Γ( 3 2 H)( s) 2 H B [dys Aˆπ sds]. (87) Remark 4.2. (a) Aually, in he saemen above, or δ = δ, one has v v.theerm(v v )/(δ δ ) mus be hanged ino a proess v whih saisies he equaion δ v = [vs + δ v s]ds + 2 H 2 H Γ( 3 2 H)( s) 2 H B [dys Aˆπ s ds]. (b) InheaseH =/2, one has v v and hene i is readily seen ha he saemen redues globally o he well known resul realled a he beginning o Seion 5. () AsaileroX rom he observaions {Ys,s }, π deined by (84) is asympoially opimal in he sense ha, as goes o ininiy, he variane E(X π ) 2 o he orresponding ilering error onverges o he same i γ (H) given by (47) as E(X π (X )) 2 or he opimal iler π (X ). This an be seen by reworking he developmens in [] onerning an auonomous linear sysem wih u, whih generaes he pair (X,Y ). The resul herein an be reormulaed as ollows: deining he proesses and dπ = aπ d + A B 2 γ[dy Aπ d]; π =, [ ] v = δ vs ds + δ 2 H Γ( 3 2 H)( s) 2 H B [dys Aπ s ds], hen π + A B γv is asympoially opimal as a iler o X rom he observaions {Ys,s }. Now, observing ha aually X ˆπ X π, Y. Aˆπds Y. Aπ ds and hene also v v, i ollows ha π =ˆπ + A B γv is also asympoially opimal as a iler o X rom he observaions {Ys,s }.

21 4 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT Proo o Theorem 4.. The saemen or arbirary values o σ and B an be easily derived rom he saemen or σ = B =. So, or simpliiy, here we deal only wih his pariular ase. Moreover, sine he ase δ = δ an be reaed similarly, we assume also ha δ δ. From he disussion above, we ge he andidae pair (u,x ): u = U H ( s)dv V s A p 2 r ( s)dw s H, X = X H ( s)dv V s + b A U V ( s)dw H s, (88), or Γ and Λ deined by (76) and (77), he union Z =(X V, U V ) is he soluion o Z = Λ Z r BB Γλ ; Z () = (, ), (89) and p 2 is he seond omponen o he veor p =Γ(Z + λ )Γλ orresponds hrough (82) o Γ λ given by (78)-(79). Aually, applying (82) o (78) (79), we ge ha Γλ is given by: Γλ () = ϕ () = δ 2 H Γ( 3 2 H) δ ρ γ ( (δ 2 a2 )ϕ () (δ 2 ) a2 )ϕ (), (9) δ b[(δ a)ϕ () (δ a)ϕ ()] e δ ( s) ds 2 H ; ϕ () = δ 2 H Γ( 3 2 H) e δ ( s) ds 2 H. (9) A irs, we prove ha he os J (u ) orresponding o he pair (u,x ) deined by (88) (9) is he opimal os J (ū) given by (45), i.e. T + T [qx 2 Sine by onsruion o he pair (u,x ) we have already guaraneed ha i is suiien o show ha T + T [X 2 + ru 2 ]d = J (ū) a.s. (92) + E[qX 2 + ru 2 ]=J (ū), EX 2 d] = a.s. ; T + T [u 2 Eu 2 ]d = a.s. (93) To esimae he ovarianes EX X +τ and Eu u +τ, we shall ake benei o he ollowing exension o he equaliy (66) or a sohasi inegral S = g( s)dbh s : or any τ ES S +τ = + 2HΓ( 3 2 H) Γ(H + 2 )Γ(2 2H) g(s) g(s + τ)ds, (94) g is deined by (67). Due o (73) (74) and (78) (79), i an be seen ha here exiss a onsan 2 C> suh ha or all s Z (s) C( s H 3 2 ); p (s) C( s H 3 2 ). 2 We use here and hroughou he proo C o denoe an unspeiied posiive onsan, no always he same.

22 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 5 So he g s orresponding o he g s involved in he represenaion (88) o he pair (u,x )inermsov H and W H saisy also suh an inequaliy. Hene, rom (94), we ge ha or all andτ EX X+τ C( τ 2H 2 ); Eu u +τ C( τ 2H 2 ). Consequenly, we an apply asserion (ii) in Proposiion 5. o he Appendix o he oninuous and enered Gaussian proesses X and u wih β =2 2H >andk =. I gives ha he is in (93) hold, whih implies also he validiy o (92). Now we urn o he represenaion (83) (87) o he opimal onrol u. Taking ino aoun he orm (4) o ū in erms o he opimal iler π( X) in he irs soluion o he problem, i is naural o guess ha here a similar deomposiion may hold or u in erms o an asympoially opimal iler or X. In Remark 4.2(), we have already seen ha he proess π =ˆπ + A γv, given by (84), (85) and (87), is a andidae o be suh a iler. Sine i plays a key role in he deiniion o he erms ˆπ and v appearing in he expression o π,i seems reasonable o look or a omplee represenaion o u in erms o he proess Y. Aˆπds. So, as a irs sep, we look or u in he orm u = g ( s)[dy s Aˆπ s ds]. (95) Then i an be heked ha equaion (85) and he irs equaion in (88) imply ha ˆπ = Ψ ( s)[dy s Aˆπ sds], (96) wih a union Ψ saisying Ψ = aψ + bg ; Ψ () = Aγ, (97) g is given by g () =A γu A V r p 2(). Sine p 2 is he seond omponen o he veor p =Γ(Z + λ ) Γ is given by (76) and Z =(X, U ), V V we ge ha g () =ĝ() A r (Γλ ) 2 (), (98) (Γλ ) 2 is he seond omponen o he veor Γλ given by (9) and Now, rom equaion (89), i omes ha ĝ() = a + δ A [ ] a + δ X V b ()+U (). V whih an be rewrien as ĝ = δ ĝ + A r (a + δ )(Γλ ) 2 ; ĝ = aĝ (a + δ )g ; ĝ() = a + δ b ĝ() = a + δ b a + δ A, a + δ A (99) Sine a+δ b = b r ρ and a+δ A = A γ, a omparison o equaions (97) and (99) gives ha ĝ = b r ρψ and hene, due o (98), we have g () = b r ρψ A r (Γλ ) 2 ().

23 6 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT Insering his equaliy ino (95) and aking ino aoun (96), i ollows ha u = b r ρˆπ A r (Γλ ) 2 ( s)[dy s Aˆπ s ds]. Bu rom (9) we have (Γλ ) 2 () = b ρ γ { (δ a)ϕ δ δ () (δ a)ϕ () }, ϕ and ϕ are given by (9). I an be heked ha aually he proesses v () = δ 2 H Γ( 3 2 H)ϕ ( τ)[dy τ Aˆπ τ dτ], and δ 2 H v () = Γ( 3 2 H)ϕ ( τ)[dy τ Aˆπ τ dτ], are nohing bu he soluions o equaions (86) and (87) respeively. Thereore, we an wrie u = b r ρ{ˆπ + A γ δ δ [(δ a)v (δ a)v ]}. Finally, using (84), i is readily seen ha u an be rewrien as in (83). 5. Appendix Auxiliary resuls 5.. Some suiien ondiion on momens or ergodi ype properies o proesses The ollowing suiien ondiion or ergodi ype properies is he key o several seps in our developmens. I has already been used in he proo o Theorem 4. and i will be again repeaedly used below. Noie ha hroughou he saemens and he proos, below we use C o denoe an unspeiied posiive onsan, no always he same. Proposiion 5.. Le ξ =(ξ, ) be a enered oninuous proess. Suppose ha here exiss some onsans C> and β> suh ha he ondiion Eξ ξ +τ C( τ β ), () holds or all and τ. Then he proess (ξ, ) possesses he ollowing ergodi ype properies: (i) T ξ d = a.s. T + T (ii) i moreover ξ is Gaussian, hen or any ineger k, T + T [ξ 2k Eξ 2k ]d = a.s. Proo. For <β<, asserion (i) is aually an immediae onsequene o he las saemen wrien on page 95 in [2] sine i he above ondiion () is saisied hen he ondiion 2α <β< numbered (5.5.3)

24 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 7 herein is obviously ulilled wih α =. Forβ, i is lear ha he ondiion () is sronger han or <β< and so he onlusion remains sill valid. Now, i he enered proess ξ is Gaussian, or any k, we have E[ξ 2k Eξ 2k ][ξ+τ 2k Eξ+τ 2k ] C[Eξ ξ +τ ] 2k, wih some posiive onsan C, and hene also, i ξ saisies (), hen E[ξ 2k Eξ 2k ][ξ+τ 2k Eξ+τ 2k ] C( τ 2kβ ). Consequenly, asserion (ii) ollows by a simple appliaion o he irs saemen in he Proposiion o he enered proess (ξ 2k Eξ 2k, ) wih β hanged ino 2kβ Abou some ehnial resuls used in he proo o Theorem 3. Here we are onerned wih a sysem wih onsan oeiiens. Again, wihou any loss o generaliy and or simpliiy, we assume ha σ = B =andx =. Moreover, sine he ase δ = δ an be reaed similarly, we assume also ha δ δ. For he pair (X, Y ) soluion o he sysem (), we inrodue he ilering error proess as = X π (X),. () Reall ha his proess, whih is enered, does no depend on he speii onrol u involved in he sysem. Hene, o analyze he seond order sruure and he asympoi behavior o, we may deal only wih he ase u. Noie ha, rom he asympoial poin o view, rom [8], we know already ha or ending o ininiy he variane union Γ (, ) =E 2 behavesasollows: + E 2 = γ (H), (2) γ (H) is he onsan given by (47). Bu, in order o derive ergodi ype properies o on he basis o Proposiion 5., we need a muh deeper analysis o he behavior o he ovariane union E s. Preparing or ha, we give a key represenaion o his ovariane union in erms o ovarianes o some oher proesses. Hereaer, he 2 2 maries A H,B H,J and he veor b H in R 2 are deined by: s 2H A H (s) = s 2H ; B H (s) = s 2H s 2H, and J = ; b H (s) = s 2H. Following [8], we inrodue he auxiliary proess ζ =(ζ, ) as he soluion o he equaion ζ = a 2 A H (s)ζ s ds + b H (s)dv s, in whih V denoes he undamenal maringale assoiaed o V H hrough (5). In erms o ζ, he proess Q deined by (9) an be represened as Q() = Aλ H 2(2 2H) b H ()Jζ, (3)

25 8 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT and, or all ixed, we deine anoher auxiliary proess X on [,] as he soluion o he equaion X s = a 2 K H (, u)b H (u)jζ udu + s K H (, u)dv u ; s. From [8], we know ha he ovariane union γ ζζ deined by is he soluion o he Riai equaion γ ζζ (s) =E(ζ s π s (ζ))(ζ s π s (ζ)), s, s γ ζζ (s) = a [A H (u)γ 2 ζζ (u)+γ ζζ (u)a H (u)]du + 2 2H s B H (u)du A2 λ H λ H 4(2 2H) s γ ζζ (u)b H (u)γ ζζ (u)du. (4) Reworking he developmens in [8], one an show ha he ovarianes saisy or all s and he equaion γ Xζ (, s) =E(X s π s (X ))(ζ s π s (ζ)), s,, γ Xζ (, s) = s + {a 2 A H(u) A2 λ H 4(2 2H) γ (u)b ζζ H(u) } γ Xζ (, u)du K H (, u) { a 2 γ ζζ 2H (u)+2 J } b H (u)du. λ H (5) Finally, in erms o he previous ovarianes, one an derive he ollowing represenaion o he ovariane E s : or all s E s = s + s K H (s, r)k H (, r)dwr H + a 2 { a 2 K H(, r)b H(r) A2 λ H 4(2 2H) γ (, r)b Xζ H(r) K H (s, r)b H (r)γ (, r)dr Xζ } γ Xζ (s, r)dr. (6) Now, on he basis o his represenaion and Proposiion 5., we are able o prove he ollowing saemen whih provides ergodi ype properies o he proess. Lemma 5.2. Le be he proess deined by () or he soluion pair (X, Y ) o he sysem (). Then he ollowing asserions hold: (i) here exiss a onsan C> suh ha or all and τ (ii) he proess possesses he ergodi ype propery γ (H) is given by (47), E +τ C( τ 2H 2 ), (7) T 2 T + T d = γ (H) a.s., (8)

26 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 9 (iii) i v =(v, ) is a (Y )-adaped oninuous proess suh ha almos surely sup T T v2 d<+, T + hen T v d = a.s. (9) T + T Proo. Conerning asserion (i), sine he omplee proo is quie long and very ehnial, here we jus give some hins abou he main seps. 3 A irs, in he represenaion (6), expliie expressions o he involved soluions o equaions (4) (5) an be obained. Then, rom hese expressions, reining he mehods used in [8], preise iing properies o he various erms inside he inegrals in he righ hand side o (6) an be analyzed. Finally, i an be shown ha or any ixed τ, E +τ = γ (τ), + he i γ (τ) is inie and explii, wih o ourse in pariular γ () = γ (H) given by (47). Moreover, a omplemenary sudy gives ha τ 2 2H γ τ + (τ) =γ, again he i γ is inie and an be wrien expliily. Hene, i is readily seen ha he asserion (i) holds. Now we prove asserion (ii). Sine is a enered Gaussian proess and, due o (2), we have T + T [E 2 ]d = + E 2 = γ (H), (8) ollows by a simple appliaion o Proposiion 5. (ii) wihβ =2 2H and k =. Finally we urn o asserion (iii). A irs, we prove (9) or a bounded proess v. Sinev is (Y )-adaped and is enered, Ev =orall, i.e., he proess v is enered. Sine is Gaussian, or all and τ, he ondiional ovariane E[ +τ /Y +τ ] is deerminisi and hene Ev v +τ +τ = E [ E[v v +τ +τ /Y +τ ] ] = Ev v +τ E +τ. So, i sup v C< a.s., hen or all andτ Ev v +τ +τ = Ev v +τ E +τ C( τ 2H 2 ). Consequenly, by appliaion o Proposiion 5. (i) o v wih β = 2 2H, we ge ha (9) holds. Now, we onsider he ase o a possibly unbounded v. For any ixed posiive onsan K, weanwrie T v d = I K T (T )+IK 2 (T ), I K (T )= I( v K)v d ; I2 K (T )= v I( v >K) d. T T Sine he proess (I( v K)v, ) is bounded, by he irs sep above, we know ha I K zero as T goes o ininiy and so we obain ha or all K> (T ) ends a.s. o sup T + T v d sup I2 K (T ) a.s. T + 3 Furher deails an be obained upon reques o he auhor or orrespondene.

27 2 M.L. KLEPTSYNA, A. LE BRETON AND M. VIOT Hene, o prove ha (9) is sill valid, i suies o show ha K + sup I2 K T + (T ) = a.s. () Bu, using he Cauhy-Shwarz inequaliy, we see ha I2 K (T ) T v 2 T d T T 4 I( v >K)d 4 4 T T d. In he seond aor o he righ hand side, we have T I( v >K)d v d T v 2 d, T KT K T and so, we ge I2 K (T ) 4 [ v 2 K T d] 5 T 8 [ 4 T d] 4. () Sine is a enered Gaussian proess, we have E 4 =3(E 2 ) 2 and hene, due o (2), i ollows ha T + T [E 4 ]d = + E 4 =3γ (H). 2 Then, applying Proposiion 5. (ii) wihβ =2 2H and k = 2 o, i urns ha T 4 T + T d =3γ2 (H) a.s. Finally, hanks o he assumpion sup T + T T is valid and hene also (9) holds or v. v2 d<+ a.s., he inequaliy () shows ha () Lemma 5.3. Le he kernel Γ 2 (, s) be he (, 2)-enry o Γ (, s) deined by (2) wih A and C given by (38). Le z =( z, ) be he proess driven by he equaion Then he ollowing asserions hold: (i) or all τ + z = δ z s ds + 2 2H Γ (τ) =A { a γ Γ(2H +)sinπh δ H+ 2 Γ 2 (, s)dν s. (2) 2 H Γ 2 λ ( + τ,)=γ (τ), H + Γ(H + 2 ) (ii) here exiss a onsan C> suh ha or all and τ E z z +τ C( τ 2H 2 ), e δ s (τ + s) H 2 ds }, (3)

28 SEPARATION PRINCIPLE IN CONTROLLED FRACTIONAL GAUSSIAN SYSTEMS 2 (iii) he proess z possesses he ergodi ype propery [ ] T z 2 T + T d = A2 Γ(2H +) 2(δ 2 δ2) δ 2H δ 2H { δ 2H +A 2 Γ(2H +)sinπh (δ + δ ) 2 + δ 2 H δ 2 δ 2 H δ2 γ(δ + a) } a.s. [ γ 2 (δ + a) γ(δ + a) ] 2δ δ δ 2δ Proo. A irs we deal wih asserion (i). Here again, he omplee proo is raher long and very ehnial and so we jus give some hins. 4 The irs sep is o derive an appropriae represenaion o he kernel Γ 2 (, s). From [9], we know ha i X denoes he auxiliary proess deined on [,]by s X s = a X r dr + s K H (, r)dv r ; s, in whih V sands or he undamenal maringale assoiaed o he Bm V H hrough (5), hen Γ 2 (, s) =E( X s π s( X ))(Q s π s (Q)) ; s, Q is he proess deined by (9). Now, due o he represenaion (3) o Q, or s, hekernel Γ 2 (, s) isgivenby Γ 2 Aλ H (, s) = 2(2 2H) b H (s)jγ (, s), (4) Xζ he veors Γ Xζ (, s) inr 2 are he ovarianes Γ Xζ (, s) =E( X s π s ( X ))(ζ s π s (ζ)) ; s. I an be shown ha hese ovarianes saisy he equaion Γ Xζ (, s) = a + s s Γ Xζ (r, r)dr + 2 2H λ H s K H (, r)j b H (r)dr { a 2 A H(r) A2 λ H 4(2 2H) γ ζζ (r)b H(r) } Γ Xζ (, r)dr, γ ζζ (r) is he soluion o equaion (4). An explii expression an be obained or Γ Xζ (, s) and hen he asympoi behavior o Γ 2 ( + τ,) an be analyzed rom (4) by means o reinemens o he mehods used in [8]. Now, we prove asserion (ii). Due o equaion (2), we ge ha z an be represened as he kernel G(, s) isgivenby z = G(, s) =Γ 2 (, s) δ G(, s)dν s, (5) s e δ ( r) Γ 2 (r, s)dr. 4 Furher deails an be obained upon reques o he auhor or orrespondene.