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Partial Information Differential Games for Mean-Field SDEs 1 2 HuaXIAO ,ShuaiqiZHANG , 1.SchoolofMathematicsandStatistics,ShandongUniversity,Weihai264209,China E-mail:xiao [email protected] 2.SchoolofEconomicsandCommerce,GuangdongUniversityofTechnology,Guangzhou510520,China E-mail:[email protected] Abstract: Thispaperisconcernedwithnon-zerosumdifferentialgamesofmean-fieldstochasticdifferentialequationswithpartial 6 information and convex control domain. First, applying the classical convex variations, we obtain stochastic maximum principle 1 forNashequilibriumpoints. Subsequently, under additionalassumptions, verificationtheoremforNashequilibriumpointsisalso 0 derived. Finally,asanapplication, alinearquadraticexampleisdiscussed. TheuniqueNashequilibriumpointisrepresentedina 2 feedbackformofnotonlytheoptimalfilteringbutalsoexpectedvalueofthesystemstate,throughout thesolutionsoftheRiccati equations. n a Key Words: Partial information, Mean-field games, Backward stochastic differential equations, Maximum principle, Verification J theorem 8 ] 1 Introduction information and convex control domain. The distinguishing C feature is the information available to the two players is the O Inthispaper,westudypartialinformationstochasticdiffer- sub-filtrationoffullinformation. Theproblemwestudymay h. entialgameproblemsinwhichsystemstatesaregovernedby cover many control and game problems of mean-field SDEs stochasticdifferentialequations(SDEs)ofmean-fieldtype,in t withcompleteinformationasspecialcases. Thepresentwork a thesensethatthecoefficientsoftheSDEsdependnotonlyon willalsoenrichtherelevanttheoryofstochasticfiltering. m thesystemstates, butalsoontheirexpectedvalues. Also,the The rest of this paper is organized as follows. In Section [ SDEsofmean-fieldtypeareoftenusedtodescribetheaggre- 2, we specify the problem considered. Section 3 is devoted gatebehavioroflotsofmutuallyinteractingparticlesatmeso- 1 toderivingthestochasticmaximumprincipleandverification v scopiclevelandplayanimportantroleinphysics,finance,eco- theorem for Nash equilibrium points. Finally, in Section 4, 2 nomics,etc. Formoreinformation,wereferthereader,forin- we solve an LQ example to explain our application. By in- 9 stance,to[6,12]aswellasthereferencestherein. Recently,a troducingthesystemsofsomeRiccatiequationsandforward- 9 newkindofbackwardSDEs(BSDEs) ofmean-fieldtypehas 1 backwardstochasticfilteringequationsofmean-filedtype,we beenstudiedbyBuckdahnetal. [3, 4]whichiscalledmean- 0 givethefeedbackrepresentationfortheuniqueNash equilib- 1. fieldBSDEs. ForclassicalcontrolproblemsofSDEswithout riumpoint. meanfield,wereferthereadersto[13,16],etc. 0 Mean-fieldgamesandmean-fieldcontrolproblemshavere- 2 FormulationofProblem 6 1 ceived considerable attention in the probability and optimal Let|x|denotetheEuclidiannormofx ∈ Rn andhx,yibe : controlliteraturein recentyears. Li[11] studiedthe stochas- v the innerproductofx,y ∈ Rn. The transposeandEuclidian Xi ttircolmdaoxmimaiunmapnrdinaclispolegfootrtmheeavne-rfiifielcdatSioDnEtshweoitrhemcounvnedxercoadn-- normofamatrixM =(mij)11≤≤ij≤≤nd =(m1,··· ,md)∈Rn×d r ditional conditions. Buckdahn, Djehiche and Li [2] used the are expressed as M∗ and |M| = trace(MM∗), respec- a classical spike perturbation and derived a Peng-type general tively. Similarly,hM ,M i=trace(M M∗)withM ,M ∈ 1 2 p1 2 1 2 stochasticmaximumprinciple. Yong[15]investigatedlinear- Rn×d. Let T > 0 be a fixed constant and C be a posi- quadratic(LQ)optimalcontrolproblemsformean-fieldSDEs tive constant which can be different from line to line. Let and a feedback representation was obtained for the optimal (Ω,F,(F ) ,P)beacompletefilteredprobabilityspace t 0≤t≤T control. Lasry and Lions [10] presented three examples of on which F denotesa naturalfiltration generatedby a stan- t mean-fieldapproachtomodellinginEconomicsandFinance, dardBrownianmotion(w ,w )withvaluesinRd1+d2. 1 2 derivednonlinearmean-fieldSDEsandestablishedtheirlinks Weonlyconsiderthecaseoftwoplayersanddefinethead- with various fields of Analysis. More recent developments missiblecontrolsetU forPlayeri(i=1,2)by i and their applications of mean-fiels games of SDEs can be found in Bensoussan, Sung and Yam [1], Carmona, Delarue U = v (·)|v (·):[0,T]×Ω−→U , isa Gi-adapted andLachapelle[5],Guant[8],etc.,andthereferencestherein. i i i i t Differentfromtheaboveworks,weconsidertwoplayersnon- n T processsatisfyingE v (t)2dt<∞ , (1) zero sum differential games of mean-field SDEs with partial i Z0 o ThisworkispartiallysupportedbyNationalNaturalScienceFoundation whereUi is a nonemptyconvexsubsetofRri, and Gti ⊆ Ft (NNSF)ofChinaunderGrant11371228,11471192,61573217and11501129. denotes the information available to Player i. Every element ofU iscalledanopen-loopadmissiblecontrolforPlayerion 3 NashEquilibriumPoint i [0,T](i = 1,2). AndU ×U iscalledthesetofopen-loop 1 2 3.1 NecessaryConditions admissible controls for the players. Unless otherwise stated, Player1controlsv andPlayer2controlsv . In this subsection, we establish a necessary conditions for 1 2 In the following, we consider the controlledstate equation NashequilibriumpointsofProblem(MF).Letussupposenow ofmean-fieldtype that u (·),u (·) isanequilibriumpointwiththecorrespond- 1 2 ingoptimalstatex(·). Thenwedefinetheperturbedcontrolas dxv1,v2(t)=f(t,xv1,v2(t),Exv1,v2(t),v1(t),v2(t))dt follo(cid:0)ws: (cid:1) +σ (t,xv1,v2(t),Exv1,v2(t),v (t),v (t))dw (t)  1 1 2 1 u (t)=u (t)+ε (v (t)−u (t)), (5)  +σ2(t,xv1,v2(t),Exv1,v2(t),v1(t),v2(t))dw2(t), εi i i i i xv1,v2(0)=x0, t≥0, where εi > 0 is sufficiently small and vi(t) is an arbi- andthecostfunctional (2) ttrhaartyUadmisiscsoinbvleexc,otnhternolfoorf 0Pla≤yerεi ≤(i 1=,0 1≤,2)t. ≤NTot,iciet i i yields u (t) ∈ U . We denote by x (·) resp. x (·) the εi i ε1 ε2 Ji(v1(·),v2(·)) state xuε1,u2 resp. xu1,uε2 associated w(cid:0)ith uε1(·),u(cid:1)2(·) T resp. (u (·),u (·)) . For simplicity, we set g(t) = =E li(t,xv1,v2(t),Exv1,v2(t),v1(t),v2(t))dt g(t,x(t),E1x((cid:0)t),uε12(t),u2(t)(cid:1)), g =f,σ1,σ2,l1(cid:0),l2. (cid:1) "Z0 (cid:0) Weintroducethefoll(cid:1)owingvariationalequations: +ϕ (xv1,v2(T),Exv1,v2(T)) , (3) i # dxi(t)= fx(t)xi(t)+fx˜(t)Exi(t) wherethemappings  (cid:2)+fvi(t)(vi(t)−ui(t)) dt σf1(t(,t,xx,,x˜x˜,,vv11,,vv22))::ΩΩ××[0[0,,TT]]××RRnn++nn××UU11××UU22→→RRnn,×d1,  ++(cid:2)σσ+21xxσ((tt1))vxxi(iit(()tt())v++i(tσσ)21−xx˜˜((ttu))iEE(txx)(cid:3)ii)(((cid:3)ttd))w1(t) (6) σ (t,x,x˜,v ,v ):Ω×[0,T]×Rn+n×U ×U →Rn×d2, lϕi2(i(tx,x,x,˜x˜),:vΩ11,×v22R):n+Ωn×→[0R,T,]×Rn+n×U11×U22→R,  xi(0)=(cid:2)0,+σi2=vi(1t),(2v.i(t)−ui(t))(cid:3)dw2(t), Fori=1,2,weset satisfythefollowingassumptions: (A1) thecoefficientsf,σ andσ areF -adaptedandbounded 1 2 t x¯ (t)=ε−1 x (t)−x(t) −xi(t), byC(1+|x|+|x˜|+|v1|+|v2|). Theyarealsocontinu- εi i εi ouslydifferentiablewithrespectto(x,x˜,v1,v2)andtheir ψε1(t)= xε1(cid:0)(t),Exε1(t),u(cid:1)ε1(t),u2(t) , partialderivativesareLipschitzcontinuousanduniformly ψλ (t)= (cid:0)x(t)+λε x1(t)+x¯ (t) ,E(cid:1)x(t) bounded. ε1 1 ε1 (A2) l1 and l2 are Ft-adapted and continuously differen- (cid:16) (cid:0) +λε E x1((cid:1)t)+x¯ (t) , tiable with respect to (x,x˜,v ,v ). ϕ and ϕ are F - 1 ε1 1 2 1 2 T measurable and continuously differentiable with respect φλ (t)= x(t)+λε x1(t)+x¯(cid:0)(t) ,Ex(t) (cid:1)(cid:17) ε1 1 ε1 to(x,x˜). Moreover,theirpartialderivativesareLipschitz continuousandboundedbyC(1+|x|+|x˜|+|v1|+|v2|). (cid:16) (cid:0)+λε1E x1(t)+(cid:1)x¯ε1(t) ,u1(t) Ouraimistofind(u1,u2)∈U1×U2suchthat +λε(cid:0)1 v1(t)−u1(t)(cid:1) ,u2(t) . J (u (·),u (·))≤J (v (·),u (·)), (cid:0) (cid:1) (cid:17) 1 1 2 1 1 2 (4) ThenbyasimilarmethodasshowninLi[11]andHuiandXiao (J2(u1(·),u2(·))≤J2(u1(·),v2(·)), [9]with a minormodification,we havethefollowingconver- genceresult. for all (v ,v ) ∈ U × U . We call (u ,u ) an open-loop 1 2 1 2 1 2 Nashequilibriumpointofthegameproblem(ifitexists). Lemma3.1. UnderAssumption(A1),wehave SinceG1 andG2 arethe sub-informationofF , itimplies t t t thisisthepartialinformationgameproblem. Onthecontrary, lim E sup |x (t)−x(t)|2 =0, when Gt1 = Gt2 = Ft,t ∈ [0,T], it reduces to be a com- ε1→0 0≤t≤T ε1 (7) pleteinformationcase. Sotheproblem(1)-(4)denotesthepar- lim E sup |x (t)−x(t)|2 =0. tialinformationnonzero-sumdifferentialgameproblemofthe ε2→0 0≤t≤T ε2 mean-field-typeSDEs. Forsimplicity,wedenoteditbyProb- lem(MF). Proof. By Assumption (A1) and the Burkholder-Davis- Gundyinequality,wederive Thenwecangetthefirstconvergenceresultof(8)fromGron- wall’sinequality. (cid:3) E sup |x (t)−x(t)|2 ε1 Since (u (·),u (·)) is the Nash equilibrium point, then it 0≤t≤T 1 2 followsthat T ≤3TE |f t,ψ (t) −f(t)|2dt ε1 Z0T (cid:0) (cid:1) ε−11[J1(uε1(·),u2(·))−J1(u1(·),u2(·))]≥0 (9) +12E |σ t,ψ (t) −σ (t)|2dt 1 ε1 1 and Z0 T (cid:0) (cid:1) +12E |σ t,ψ (t) −σ (t)|2dt ε−1[J (u (·),u (·))−J (u (·),u (·))]≥0. (10) 2 ε1 2 2 2 1 ε2 2 1 2 Z0 T (cid:0) (cid:1) ≤C E |x (t)−x(t)|2dt Lemma 3.3. Let Assumptions (A1) and (A2) hold. Then the T ε1 followingvariationalinequalityholdsfori=1,2: Z0 T +ε21CTE |v1(t)−u1(t)|2dt, T Z0 E lix(t)xi(t)+lix˜(t)Exi(t)+livi(t)(vi(t)−ui(t)) dt where CT > 0 is a constant only depending on T > 0 and Z0 h i the Lipschitz coefficients of f, σ1 and σ2. From Gronwall’s +E[ϕix(x(T),Ex(T))xi(T)+ϕix˜(x(T),Ex(T))Exi(T)] inequalitywegetthedesiredresult. (cid:3) ≥0. (11) Lemma3.2. UnderAssumption(A1),ityields Proof. Wefirstlyprove(11)holdsfori=1andtheanother εl1i→m0E0≤sut≤pT|x¯ε1(t)|2 =0, casecanbesimilarlyderived.From(9),ityields (8) εl2i→m0E0≤sut≤pT|x¯ε2(t)|2 =0. 0≤ε−11[J1(uε1(·),u2(·))−J1(u1(·),u2(·))] T Proof. Without loss of generality, we prove the first result =ε−1E l (t,ψ (t))−l (t) dt 1 1 ε1 1 of (8) and the latter one can be similarly derived. For g = Z0 f,σ1,σ2,weset +ε−11E ϕ(cid:2)1(xε1(T),Exε1(T))(cid:3)−ϕ1(x(T),Ex(T)) 1 =I1+I2.(cid:2) (cid:3) a1(t)= g (t,φλ (t))dλ, g x ε1 Z0 FromAssumptions(A1),(A2)andLemma3.2,wederive 1 a2(t)= g (t,φλ (t))dλ, g Z01 x˜ ε1 I1 =E T 1l1x(t,φλε1(t))dλ· x1(t)+x¯ε1(t) Duae3g(tto++)=AZZ00sZ11s0u(cid:0)(cid:0)mgg(cid:0)vx˜pg1(tx(tito,(,φtnφ,λεφλε1(1A(λε(1tt1)())t)),)−)−a−g1gx˜vg(1axt(n)(t(cid:1)dt))d(cid:1)(cid:1)λadd2·λλE··axrx(cid:0)e1v1(1(tbt())to)th−uun1i(fto)r(cid:1)m.ly ++Z−Z0Z0→01(cid:20)1lZ1lE1v0x1˜((tt,T,φφλελεl111((xtt())t)))ddxλλ1(··t(cid:0)E)v(cid:0)+1x((cid:0)1tl1)(x˜t−)(t+u)E1x¯(xεt11)((cid:1)(tt(cid:21)))d(cid:1)t(cid:1) g g Z0 h boundedand lim E sup |a3g(t)|2 =0.Thenwehave +l1v1(t) v1(t)−u1(t) dt, (12) ε1→0 (cid:18)0≤t≤T (cid:19) (cid:0) (cid:1)i dx¯ (t)= [a1(t)x¯ (t)+a2(t)Ex¯ (t)+a3(t)]dt ε1 f ε1 f ε1 f  +[a1σ1(t)x¯ε1(t)+a2σ1(t)Ex¯ε1(t)+a3σ1(t)]dw1(t) I =E 1ϕ ψλ (T) dλ· x¯ (T)+x1(T)  +[a1σ2(t)x¯ε1(t)+a2σ2(t)Ex¯ε1(t)+a3σ2(t)]dw2(t), 2 (cid:20)Z0 1x(cid:16) ε1 (cid:17) (cid:16) ε1 (cid:17) 1 weAhpax¯pvεley1(in0g)=Itoˆ’0s.formula to |x¯ε1(t)|2 and Assumption (A1), +Z0 ϕ−1→x˜(cid:16)ψEελ1ϕ(T1x)((cid:17)xd(λT·),EE(cid:16)xx¯(εT1()T)x)1+(Tx)1(T)(cid:17)(cid:21) T h E sup |x¯ε1(t)|2 ≤CE |x¯ε1(t)|2dt +ϕ1x˜(x(T),Ex(T))Ex1(T) . (13) (cid:20)0≤t≤T (cid:21) Z0 i Combining(12)with(13),theinequality(11)followsfori = +CE sup |a3(t)|2+|a3(t)|2+|a3 (t)|2 . f σ σ2 1. (cid:3) (cid:20)0≤t≤T (cid:21) (cid:0) (cid:1) Next,wedefinetheHamiltonianfunctionH :[0,T]×Rn× 3.2 SufficientConditions i Rn×U1×U2×Rn×Rn×d1 ×Rn×d2 →Rasfollows: Inwhatfollows,weproceedtoestablishthesufficientcon- ditionsofNashequilibriumpoints(alsocalledverificationthe- H (t,a,a˜,v ,v ,q ,k ,k ) i 1 2 i i1 i2 orem). ,hq ,f(a,a˜,v ,v )i+hk ,σ (a,a˜,v ,v )i i 1 2 i1 1 1 2 Theorem3.2(VerificationTheorem). Let(A1)and(A2)hold. +hki2,σ2(a,a˜,v1,v2)i+li(a,a˜,v1,v2), Let (u1(·),u2(·)) ∈ U1 × U2 with the corresponding solutions x and (q ,k ,k ) to the equations (2) and (14). Sup- i i1 i2 and denote Hi(t) = Hi t,x(t),Ex(t),u1(t),u2(t),qi(t), poseHi t,a,a˜,v1,v2,qi(t),ki1(t),ki2(t) andϕi areconvex ki1(t),ki2(t) ,i = 1,2.Let(cid:0)usconsiderthefollowingadjoint withresp(cid:0)ectto(a,a˜,vi). Moreover, (cid:1) BSDEofmean-fieldtype (cid:1) E H t,x(t),Ex(t),u (t),u (t),q (t),k (t), 1 1 2 1 11 −dqi(t)= fx∗(t)qi(t)+E fx˜∗(t)qi(t) +σ1∗x(t)ki1(t) k(cid:2)12(t(cid:0)) Gt1 = inf E H1 t,x(t),Ex(t),v1,u2(t),  +(cid:2) E σ1∗x˜(t)ki1(t(cid:0)) +σ2∗x(t(cid:1))ki2(t) (cid:1)(cid:12) (cid:3) v1∈U1 (cid:2)q ((cid:0)t),k (t),k (t) G1 , (19)  +−Eki(cid:0)(cid:0)1σ(t2∗)x˜d(wt)1k(it2)(t−)(cid:1)(cid:1)k+i2(lti∗x)d(tw)2+(tE),li∗x˜(t)(cid:3)dt E H t,(cid:12)x(t),Ex(t),u (t),u1 (t),q11(t),k12(t)(cid:12)(cid:12), t (cid:3) 2 1 2 2 21  qi(T)=ϕ∗ix(cid:0)x(T),Ex(T)(cid:1)+Eϕ∗ix˜(cid:0)x(T),Ex(T)(cid:1)(,14) (cid:2)k22((cid:0)t) Gt2 =v2i∈nfU2E H2 t,x(t),Ex(t),u1(t),v2, which coupled with (2) constitutes a forward-backwardSDE (cid:1)(cid:12) (cid:3) (cid:2)q ((cid:0)t),k (t),k (t) G2 . (20) (cid:12) 2 21 22 t (FBSDE)ofmean-fieldtype. Inthesequel,westatenecessaryconditionsofNashequilib- Then (u1(·),u2(·)) is a Nash equilibrium point(cid:12)(cid:12)of P(cid:3)roblem riumpoints,i.e. stochasticmaximumprincipleasfollows: (MF). Theorem 3.1 (Maximum Principle). Suppose (A1) and (A2) Proof. Letvi(·)∈Ui,i=1,2.Wedenotebyxv1 andxthe hold. Let (u1(·),u2(·)) be a Nash equilibrium point of solutionsto(2)associatedwiththeadmissiblecontrols(v1,u2) Problem (MF) with the corresponding solutions x(·) and and(u1,u2),respectively.Weset q (·),k (·),k (·) of (2)and(14). Thenitfollowsthat i i1 i2 H (t)=H t,x(t),Ex(t),u (t),u (t),q (t),k (t), 1 1 1 2 1 11 (cid:0) E[H(cid:1)1v1(t)|Gt1](v1−u1(t))≥0 (15) k12(t) , H(cid:0)1v1(t)=H1 t,xv1(t),Exv1(t),v1(t),u2(t), q1(t),k1(cid:1)1(t),k12(t) , g(t)(cid:0)=g t,x(t),Ex(t),u1(t),u2(t) , and E[H (t)|G2](v −u (t))≥0, (16) gv1(t)=g t,xv1,Ex(cid:1)v1(t),v1(t)(cid:0),u2(t) , g =f,σ1,σ2,l1(cid:1). 2v2 t 2 2 Byvirtueo(cid:0)ftheconvexityofϕ ,wehav(cid:1)eforanyv (·)∈U 1 1 1 dtdP−a.e.,foranyv ∈U andv ∈U . 1 1 2 2 J (v (·),u (·))−J (u (·),u (·))≥I +I (21) 1 1 2 1 1 2 1 2 Proof. We firstly prove (15). Applying Itoˆ’s formula to hx1(t),q (t)i,foranyv (·)∈U weobtain with 1 1 1 I =E ϕ (x(T),Ex(T))(xv1(T)−x(T)) 1 1x E ϕ1x x(T),Ex(T) x1(T)+ϕ1x˜ x(T),Ex(T) Ex1(T) (cid:2)+ϕ1x˜(x(T),Ex(T))E(xv1(T)−x(T)) , T (cid:2) (cid:0)= E −(cid:1)l1x(t)x1(t)−(cid:0)l1x˜(t)Ex1(t)(cid:1) (cid:3) I =E T lv1(t)−l (t) dt (cid:3) Z0 h 2 Z0 (cid:16)1 1 (cid:17) −l (t) v (t)−u (t) dt T 1v1 1 1 =E Hv1(t)−H (t)−hq (t),fv1(t)−f(t)i +E (cid:0)T H1v1(t) v1((cid:1)t)i−u1(t) dt. (17) −hkZ0(t)h,σv11(t)−σ 1(t)i 1 Z0 11 1 1 (cid:0) (cid:1) −hk (t),σv1(t)−σ (t)i dt. (22) Thistogetherwiththevariationalinequality(11)derives 12 2 2 i ApplyingItoˆ’sformulatohq (t),xv1(t)−x(t)i,wehave 1 T E E H (t) G1 v (t)−u (t) dt T Z0 1v1 t 1 1 I1 =E hq1(t),fv1(t)−f(t)i (cid:2) =E (cid:12)(cid:12) T H(cid:3)(cid:0)1v1(t) v1(t)−(cid:1)u1(t) dt≥0, (18) +Zh0k11h(t),σ1v1(t)−σ1(t)i+hk12(t),σ2v1(t)−σ2(t)i Z0 −hH∗ (t),xv1(t)−x(t)i (cid:0) (cid:1) 1x for all v1(·) ∈ U1, which implies (15) holds. By the similar −hEH∗ (t),xv1(t)−x(t)i dt. (23) methodabove,(16)isalsotrue. (cid:3) 1x˜ i Substituting(22)and(23)into(21)andapplyingtheconvexity ThentheuniqueNashequilibriumpointisdenotedby ofH ,weget 1 u (t)=−m−1(t)b (t) τ (t)xˆ(t)+δ (t)Ex(t) , J (v (·),u (·))−J (u (·),u (·)) 1 1 1 1 1 (28) 1 1T 2 1 1 2 (u2(t)=−m−21(t)b2(t)(cid:0)τ2(t)xˆ(t)+δ2(t)Ex(t)(cid:1), ≥E H1v1(t)−H1(t)−hH1∗x(t),xv1(t)−x(t)i whereEx,(τ ,τ ),(δ ,δ )an(cid:0)dxâredeterminedby(3(cid:1)8),(45), Z0 (cid:16) (46)and(47)1,re2spect1ivel2y. −hEH∗ (t),xv1(t)−x(t)i dt 1x˜ Proof. Weshallcompletetheproofbytwoparts. T (cid:17) ≥E H (t) v (t)−u (t) dt Part1. WefirstneedtoprovetheuniqueNashequilibrium 1v1 1 1 pointcanberepresentedby Z0 T (cid:0) (cid:1) =E E H1v1(t) Gt1 v1(t)−u1(t) dt. (24) u1(t)=−m−11(t)b1(t)qˆ1(t), (29) ConditioZn0(19)(cid:2)impliesE(cid:12)(cid:12) H(cid:3)(cid:0) (t) G1 v ((cid:1)t)−u (t) ≥ 0, (u2(t)=−m−21(t)b2(t)qˆ2(t), 1v1 t 1 1 dtdP−a.e.on[0,T],whichderives wherex, q ,k ,k and q ,k ,k aretheuniquesolu- (cid:2) (cid:12) (cid:3)(cid:0) (cid:1) 1 11 12 2 21 22 (cid:12) tionofthefollowingcoupledFBSDEofmean-fieldtype J1(v1(·),u2(·))−J1(u1(·),u2(·))≥0. (cid:0) (cid:1) (cid:0) (cid:1) dx(t)= a(t)x(t)+a¯(t)Ex(t)−m−1(t)b2(t)qˆ (t) Similarly, we can also derive J (u (·),v (·)) − 1 1 1 2 1 2 J2(u1(·),u2(·))≥0.Theproofiscompleted. (cid:3)  −m−21(cid:2)(t)b22(t)qˆ2(t) dt+c1(t)dw1(t)+c2(t)dw2(t), 4 LQExample x(0)=x , 0 (cid:3) (30) Inthissection,weworkoutanLQexampletoillustratethe  −dq (t)= a(t)q (t)+a¯(t)Eq (t)+g (t)x(t) 1 1 1 1 theoreticalresult. Withoutlossofgenerality,weonlyconsider the following case: n = d = d = 1,b (t)b (t) 6= 0,t ∈  +g¯1(t)E(cid:2)x(t) dt−k11(t)dw1(t)−k12(t)dw2(t), 1 2 1 2 [0,T]. Throughoutthis section, we assume additional condi-  q1(T)=h1x(T(cid:3))+h¯1Ex(T), tion. (31) (A3) m−1(t)b2(t)=m−1(t)b2(t),t∈[0,T]. and 1 1 2 2 Example4.1. Considerthesystemoflinearmean-fieldSDE −dq2(t)= a(t)q2(t)+a¯(t)Eq2(t)+g2(t)x(t) dxv1,v2(t)= a(t)xv1,v2(t)+a¯(t)Exv1,v2(t)  +g¯2(t)E(cid:2)x(t) dt−k21(t)dw1(t)−k22(t)dw2(t),  q (T)=h x(T)+h¯ Ex(T).  (cid:2)+b1(t)v1(t)+b2(t)v2(t) dt (25) 2 2 (cid:3) 2 (32)  +c1(t)dw1(t)+c2(t)dw2(cid:3)(t), Here we denote by pˆ(t) the mathematical expectation of xv1,v2(0)=x , p(t) with respect to Fw1, i.e., pˆ(t) , E p(t) Fw1 ,p = 0 t t withthequadraticcostfunctional qin1g,qt2w,ok1s1te,pks2.1,x.TherestofPart1isdivide(cid:2)dint(cid:12)(cid:12)othef(cid:3)ollow- Step (i) (u ,u ) of the form (29) is the Nash equilibrium J v (·),v (·) = 1E T g (t) xv1,v2(t) 2 pointindeed.1 2 i 1 2 i (cid:0) (cid:1) 2 "Z0 (cid:16) (cid:0) (cid:1) WefirstwritedowntheHamiltonianfunction +g¯i(t) Exv1,v2(t) 2+mi(t) vi(t) 2 dt Hi t,x,x˜,v1,v2,qi,ki1,ki2 +(cid:0)h xv1,v2(T(cid:1)) 2+h¯ E(cid:0)xv1,v(cid:1)2((cid:17)T) 2 (26) ,qi(cid:0)a(t)x+a¯(t)x˜+b1(t)v1(cid:1)+b2(t)v2 +ki1c1 i i 1 # +(cid:0)k (t)c + g x2+g¯x˜2+m v2 (cid:1). (33) (cid:0) (cid:1) (cid:0) (cid:1) i2 2 2 i i i i and the information available to two players G1 = G1 = (cid:0) (cid:1) t t Applying Theorem 3.1, we derive the candidate Nash equi- Fw1 =σ{w (s),0≤s≤t}. t 1 librium point of the form (29) and the coupled FBSDE of Here, all coefficientswith respect to t in (25) and (26) are mean-field type (30)-(32). We can check that ϕ (x,x˜) = deterministic and uniformly bounded. In addition, g and g¯ i i i 1(h x2+h¯ x˜2)andH t,x,x˜,v ,v ,q ,k ,k in(33)sat- are non-negative, h and h¯ are non-negative constants, and 2 i i i 1 2 i i1 i2 i i isfy the conditionsin Theorem3.2. Therefore, u (·),u (·) mi is positive. The set of admissible controls for Player i is (cid:0) (cid:1)1 2 oftheform(29)istheNashequilibriumpointindeed. definedby (cid:0) (cid:1) Based on the arguments in Step (i), we conclude that the U ={v (·)|v (·)isanR-valuedFw1-adaptedprocess existence and uniqueness of the Nash equilibrium points are i i i t equivalenttothe existenceanduniquenessofthe solutionsto T satisfyingE v2(t)dt<∞},i=1,2. (27) (30)-(32). i Z0 Step(ii)Thesolutionsof(30)-(32)areexistentandunique. Takingmathematicalexpectationonbothsidesof(30)-(32), Similarly, based on the known Ex,Eq ,Eq ,qˆ 1 2 1 wehavethefollowingforward-backwardordinaryequations and qˆ , there also exists the unique solution 2 x,(q ,k ,k ),(q ,k ,k ) to(30)-(32). 1 11 12 2 21 22 dEx(t)= (a(t)+a¯(t))Ex(t) Part 2. We need to verify the feedback form of the Nash (cid:0) (cid:1)  −m−11(t(cid:2))b21(t)Eq1(t)−m−21(t)b22(t)Eq2(t) dt, (34) equilibriumpointin(29)isrepresentedby(28). Ex(0)=x , Basedontheterminalconditionsin(42)and(43),weset 0 (cid:3)  −dEq (t)= (a(t)+a¯(t))Eq (t) qî(t)=τi(t)xˆ(t)+δi(t)Ex(t), i=1,2, (44) 1 1  (cid:2)+(g1(t)+g¯1(t))Ex(t) dt, (35) subjecttoτi(T)=hiandδi(T)=¯hi. ApplyingItoˆ’sformula toqˆ (t) resp.qˆ (t) in(44)andcomparingthecoefficientsof  Eq1(T)=(h1+h¯1)Ex(T), (cid:3) xˆ(t)1andEx(t)b2etweenitand(42) resp. (43) ,respectively, (cid:0) (cid:1) and  weget −dEq (t)= (a(t)+a¯(t))Eq (t) (cid:0) (cid:1) 2 2 τ˙ +2aτ −m−1b2τ2−m−1b2τ τ +g =0,  (cid:2)+(g2(t)+g¯2(t))Ex(t) dt, (36) 1 1 1 1 1 2 2 1 2 1 (45)  Eq2(T)=(h2+h¯2)Ex(T). (cid:3) (τ˙2+2aτ2−m−21b22τ22−m−11b21τ1τ2+g2 =0, ApplyingthemethodasshowninChangandXiao[7]andAs- δ˙1+ 2a+a¯−m−11b21α1−m−21b22α2−m1−1b21τ1 δ1 s(Eumx(ptt)io,Enq(A(t3)),,Ewqe(ct)a)ntpor(o3v4e)-t(h3e6r)ewexitihststhae ruenliaqtiuoenssoalsutfiooln-  (cid:0) −m−21b22τ1δ2+a¯τ1+a¯α1+g¯1(cid:1)=0, lows: 1 2 δ˙2+ 2a+a¯−m−11b21α1−m−21b22α2−m−21b22τ2 δ2 Eqi(t)=αi(t)Ex(t), i=1,2, (37) (cid:0) −m−11b21τ2δ1+a¯τ2+a¯α2+g¯2(cid:1)=0, with τ (T) = h and δ (T) = h¯ . Applying the method(4a6s) Ex(t)=x0eR0t[a(s)+a¯(s)−m−11(s)b21(s)(α1(s)+α2(s))]ds, (38) showniinChangaindXiaoi [7]andAissumption(A3),thereexist theuniquesolutionsto(45)and(46). where(α ,α )istheuniquesolutionofthefollowingRiccati 1 2 Substituting(44) into(41), we canderivetheexplicitsolu- equations tionasfollow: α˙ +2(a+a¯)α −m−1b2α2−m−1b2α α t 1 1 1 1 1 2 2 1 2 xˆ(t)=x Φt + Φtc (s)dw (s) 0 0 s 1 1 +g1+g¯1 =0, (39) Z0 t + Φt a¯−m−1b2δ −m−1b2δ (s)Ex(s)ds (47) s 1 1 1 2 2 2 α˙2+2(a+a¯)α2−m−11b21α1α2−m−21b22α22 with ΦZt0= e(cid:0)xp{ t(a − m−1b2τ − m(cid:1)−1b2τ )(r)dr}. The +g +g¯ =0, (40) s s 1 1 1 2 2 2 2 2 proofiscompleted. (cid:3) R subjecttoα (T)=h +h¯ . 5 ConclusionRemarks i i i Substituting (37) into (31) and (32), taking conditional Inthispaper,westudynon-zerosummean-fieldgamewith mathematical expectation on both sides of (30)-(32) with re- specttoFw1 andapplyingLemma5.4inXiong[14],wehave partialinformationandderivethestochasticmaximumprinci- t ple and verification theorem for the Nash equilibrium points. dxˆ(t)= a(t)xˆ(t)+a¯(t)Ex(t)−m−1(t)b2(t)qˆ (t) Comparedwiththeexistingliterature,thecontributionsofthis 1 1 1 paperare:  −m−21(cid:2)(t)b22(t)qˆ2(t) dt+c1(t)dw1(t), • Partialinformationismoregeneralcasethancompletein- xˆ(0)=x0, (cid:3) formation.Theresultspartlygeneralizetherelatedmean- (41) −dqˆ (t)= a(t)qˆ (t)+g (t)xˆ(t) fieldcontrolorgameproblemswithcompleteinformation 1 1 1 (seee.g. [1,2,11,15]);  + a¯(t)α1(t(cid:2))+g¯1(t) Ex(t) dt−kˆ11(t)dw1(t), (42) • Mean-field-type forward-backward stochastic filtering qˆ (T)=h xˆ(T)+h¯ Ex(T), equations are found, which enriches the theory of clas- 1 (cid:0) 1 1(cid:1) (cid:3) sicalfiltering;  −dqˆ2(t)= a(t)qˆ2(t)+g2(t)xˆ(t) • TheuniqueNashequilibriumpointintheLQexampleis  + a¯(t)α2(t(cid:2))+g¯2(t) Ex(t) dt−kˆ21(t)dw1(t), (43) representedinthefeedbackformofnotonlytheoptimal filtering but also the expected value of the system state, qˆ (T)=h xˆ(T)+h¯ Ex(T), 2 (cid:0) 2 2(cid:1) (cid:3) throughthesolutionsofsomeRiccatiequations. which constitute a kind of fully coupled forward-backward Inaddition,sincetherearemanypartialinformationmean- stochastic filtering equationsof mean-field type and exist the field game problems in finance and economics, we hope the uniquesolution(xˆ,qˆ ,kˆ ,qˆ ,kˆ ). resultshaveapplicationsintheserelatedareas. 1 11 2 21 References [1] A.Bensoussan, K.C.J.Sung,andS.C.P.Yam,LinearCquadratic time-inconsistent mean field games, Dyn. 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