An Extension of Major-Minor Mean Field Game Theory

An Extension of Ma jor-Minor Mean Field Game Theory Agustín Muñoz González 1 1 Departamen to de Matemáticas, F acultad de Ciencias Exactas y Naturales, Univ ersidad de Buenos Aires, Buenos Aires, Argen tina Marc h 18, 2026 Abstract This w ork extends the theory presen ted in Me an Field Games with a Dominating Player b y Bensoussan, Chau and Y am [ 2 ] on mean field games with a dominating pla yer, to the case in which the utility and cost functions dep end not only on the la w of the states, but on the join t state–control la w. W e incorp orate the unified notation Π t = L ( X 1 t , U 1 t | F 0 t ) ∈ P 2 ( R n 1 × A ) , whic h describ es the conditional distribution of the state–control pair of the represen tative agent given the common noise of the dominating pla y er. In addition, w e generalize the role of the dominating pla y er to include the direct impact of its controls u 0 on the dynamics and functionals of the system. The optimization problems are reform ulated in terms of Π t , the necessary optimalit y conditions are established via sto chastic maximum principles, and a coupled SHJB–FP system of equations is obtained that synthesizes the equilibrium conditions. This framework pro vides a significant extension of the existing literature on MF G with a dominating play er. Keywor ds— mean field games, dominating pla y er, Lions deriv ativ e, join t state–con trol measure, SHJB–FP equations, sto chastic maxim um principle 1 Con ten ts 1 In tro duction 3 2 W orking spaces and assumptions 5 3 Preliminaries 7 3.1 Deriv atives on the space of measures: Lions, linear, and W asserstein gradien t 7 3.2 F okker–Planc k equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Problems 9 4.1 Problem 1: Con trol of the Representativ e Agent . . . . . . . . . . . . . . . . 9 4.2 Problem 2: Equilibrium Condition . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3 Problem 3: Con trol of the Dominating Play er . . . . . . . . . . . . . . . . . 9 5 Main results 9 6 Extension of the results 12 7 Conclusion 19 2 1 In tro duction In this section w e extend the framew ork of me an field games with a dominating player (ma jor–minor mean field games), developed in [ 2 ], to the case in whic h the utilit y and cost functions dep end not only on the la w of the states, but on the joint state–control la w. This extension allows us to capture phenomena where the interaction among representativ e agen ts is not describ ed solely through their state tra jectories, but also through the distributions induced by their control decisions. In addition, w e generalize the role of the dominating pla y er: in the classical model, its influence on the system app ears only through its state x 0 ; here we also consider the direct impact of its controls u 0 . In this wa y , the extended framew ork encompasses b oth dep endencies via the dominating state and dep endencies via its actions, a crucial asp ect in applications where the decisions of the ma jor play er determine the dynamics and rewards of the rest of the system. W e introduce the notation Π t = L  X 1 t , U 1 t   F 0 t  ∈ P 2 ( R n 1 × A ) , whic h describ es the conditional distribution of the pair ( x 1 , u 1 ) of the representativ e agent giv en the common noise of the dominating pla y er. As sp ecial cases, the law of the states and the law of the controls are obtained as marginals: µ t = (pr X ) # Π t , q t = (pr U ) # Π t . The adv an tage of w orking with Π t is that it unifies b oth dep endencies in a single measure on the pro duct space, which simplifies b oth the notation and the application of calculus on the space of probability measures (Lions deriv atives). In the MF G literature it is customary for the mean field interaction to o ccur through the la w of the states, µ , and not through the law of the controls. Some authors, such as R. Carmona and D. Lac ker, analyze this case by introducing the la w of the con trols into the cost function optimized by the agen ts [ 1 ]. Ho wev er, when a new pla yer with dominating c haracteristics is in tro duced, the results found in the asso ciated literature refer to the classical MF G setting. F or example, in [ 2 ] a game is analyzed with one dominating agen t and many small agents, where the functionals dep end only on the law of the states: J 0 ( u 0 ) = E  Z T 0 f 0 ( x 0 ( t ) , µ ( t ) , u 0 ( t )) dt + h 0 ( x 0 ( T ) , µ ( T ))  , J 1 ( u 1 , x 0 , µ ) = E  Z T 0 f 1 ( x 1 ( t ) , x 0 ( t ) , µ ( t ) , u 1 ( t )) dt + h 1 ( x 1 ( T ) , x 0 ( T ) , µ ( T ))  . In this work, we adapt the theory of [ 2 ] to the case in which the mean field interactions o ccur through the joint state–con trol law Π t . The ob jectives of this work are: 1. T o reformulate the optimization problems of the represen tativ e agent and the domi- nating play er in terms of Π (Problems 1–3). 3 2. T o recall the necessary technical notions: diff usion op erators and F okker–Planc k equa- tions, as well as the Lions deriv ative for functions dep ending on join t measures. 3. T o extend the results on necessary optimality conditions to the state–control case, stating and proving the corresp onding lemmas (analogues of Lemmas 24–26 in [ 2 ]). The pro ofs follow the same strategy as in the classical case: v ariations of the con trols are considered, the v ariations of the ob jective functions are expressed in terms of the measure Π and the induced dynamics, and via duality adjoin t pro cesses are introduced that allow the minimizers to b e characterized in terms of sto c hastic Hamilton–Jacobi–Bellman (SHJB) problems coupled with F okk er–Planck equations. 4 2 W orking spaces and assumptions W e work on a filtered probability space (Ω , F , P ; F 0 = ( F 0 t ) t ∈ [0 ,T ] , F 1 = ( F 1 t ) t ∈ [0 ,T ] ) , with tw o indep endent adapted Brownian motions W 0 and W 1 (of dimensions d 0 and d 1 ), and initial conditions ξ 0 ∈ R n 0 , ξ 1 ∈ R n 1 indep enden t of ( W 0 , W 1 ) . Let x 0 ( t ) ∈ R n 0 and x 1 ( t ) ∈ R n 1 denote the state pro cesses for the dominating play er and a representativ e agent, resp ectiv ely , whose dynamics are given b y the follo wing sto c hastic differen tial equations: dx 0 = g 0 ( x 0 ( t ) , Π t , u 0 ( x 0 ( t ) , t )) dt + σ 0 ( x 0 ( t )) dW 0 ( t ) , x 0 (0) = ξ 0 , dx 1 = g 1 ( x 1 ( t ) , x 0 ( t ) , u 0 ( t ) , Π t , u 1 ( x 1 ( t ))) dt + σ 1 ( x 1 ( t )) dW 1 ( t ) , x 1 (0) = ξ 1 . The functional co efficients are defined as follo ws: g 0 : R n 0 × P 2 ( R n 1 × A ) × R m 0 → R n 0 , g 1 : R n 1 × R n 0 × R m 0 × P 2 ( R n 1 × A ) × R m 1 → R n 1 , σ 0 : R n 0 → R n 0 × d 0 , σ 1 : R n 1 → R n 1 × d 1 . The dominating play er and the representativ e agents also ha v e the follo wing ob jective functionals, resp ectively: J 0 ( u 0 ) = E  Z T 0 f 0 ( x 0 ( t ) , Π t , u 0 ( t )) dt + h 0 ( x 0 ( T ) , Π T )  , (1) J 1 ( u 1 , x 0 , u 0 , Π) = E  Z T 0 f 1 ( x 1 ( t ) , x 0 ( t ) , u 0 ( t ) , Π t , u 1 ( t )) dt + h 1 ( x 1 ( T ) , x 0 ( T ) , u 0 ( T ) , Π T )  . (2) The functions are defined as follows: f 0 : R n 0 × P 2 ( R n 1 × A ) × R m 0 → R , f 1 : R n 1 × R n 0 × R m 0 × P 2 ( R n 1 × A ) × R m 1 → R , h 0 : R n 0 × P 2 ( R n 1 × A ) → R , h 1 : R n 1 × R n 0 × R m 0 × P 2 ( R n 1 × A ) → R . Here, u 0 ∈ R m 0 and u 1 ∈ R m 1 represen t the resp ectiv e con trols of the dominating play er and the representativ e agents. T o guarantee the existence of optimal controls and the equilibrium of the game, we imp ose the following assumptions on the functions defining the state pro cesses and pay offs. Unlik e the original form ulation of Bensoussan, Chau and Y am [ 2 ], where the functions depend on the la w of the states µ t ∈ P ( R n 1 ) , in our extension the functions dep end on the joint state–con trol la w Π t ∈ P ( R n 1 × A ) . The assumptions are adapted accordingly . 5 • (A.1) Lipschitz con tinuit y: The functions g 0 , σ 0 , g 1 and σ 1 are globally Lipsc hitz contin uous in all their argumen ts. In particular, there exists a constant K > 0 such that: | g 0 ( x 0 , Π , u 0 ) − g 0 ( x ′ 0 , Π ′ , u ′ 0 ) | ≤ K ( | x 0 − x ′ 0 | + W 2 (Π , Π ′ ) + | u 0 − u ′ 0 | ) ; | σ 0 ( x 0 ) − σ 0 ( x ′ 0 ) | ≤ K | x 0 − x ′ 0 | ; | g 1 ( x 1 , x 0 , u 0 , Π , u 1 ) − g 1 ( x ′ 1 , x ′ 0 , u ′ 0 , Π ′ , u ′ 1 ) | ≤ K ( | x 1 − x ′ 1 | + | x 0 − x ′ 0 | + | u 0 − u ′ 0 | ) + K ( W 2 (Π , Π ′ ) + | u 1 − u ′ 1 | ) ; | σ 1 ( x 1 ) − σ 1 ( x ′ 1 ) | ≤ K | x 1 − x ′ 1 | . • (A.2) Linear growth: The functions g 0 , σ 0 , g 1 and σ 1 gro w at most linearly in all their argumen ts. In particular, there exists a constant K > 0 such that: | g 0 ( x 0 , Π , u 0 ) | ≤ K (1 + | x 0 | + M 2 (Π) + | u 0 | ); | σ 0 ( x 0 ) | ≤ K (1 + | x 0 | ); | g 1 ( x 1 , x 0 , u 0 , Π , u 1 ) | ≤ K (1 + | x 0 | + | x 1 | + | u 0 | + M 2 (Π) + | u 1 | ); | σ 1 ( x 1 ) | ≤ K (1 + | x 1 | ) . • (A.3) Quadratic growth condition on the cost function: There exists a constant K > 0 such that: | f 1 ( x 1 , x 0 , u 0 , Π , u 1 ) − f 1 ( x ′ 1 , x ′ 0 , u ′ 0 , Π ′ , u ′ 1 ) | ≤ K (1 + | x 1 | + | x ′ 1 | + | x 0 | + | x ′ 0 | + | u 0 | + | u ′ 0 | + M 2 (Π) + M 2 (Π ′ ) + | u 1 | + | u ′ 1 | ) · ( | x 1 − x ′ 1 | + | x 0 − x ′ 0 | + | u 0 − u ′ 0 | + W 2 (Π , Π ′ ) + | u 1 − u ′ 1 | ) ; | h 1 ( x 1 , x 0 , u 0 , Π) − h 1 ( x ′ 1 , x ′ 0 , u ′ 0 , Π ′ ) | ≤ K (1 + | x 1 | + | x ′ 1 | + | x 0 | + | x ′ 0 | + | u 0 | + | u ′ 0 | + M 2 (Π) + M 2 (Π ′ )) · ( | x 1 − x ′ 1 | + | x 0 − x ′ 0 | + | u 0 − u ′ 0 | + W 2 (Π , Π ′ )) . • (A.4) Differentiabilit y: The functions g 0 , f 0 , h 0 , g 1 , f 1 and h 1 are con tinuously differentiable in x 0 ∈ R n 0 , x 1 ∈ R n 1 , u 0 ∈ R m 0 , u 1 ∈ R m 1 with b ounded deriv atives. W e denote, for example, the deriv ative of g 0 with resp ect to x 0 b y g 0 ,x 0 . They are also Gâteaux differen tiable in Π = mdλ , for example, for m ∈ L 2 ( R n 1 ) : d dθ     θ =0 g 0 ( x i , ( m + θ ˜ m ) dλ, u i ) = Z R n ∂ g 0 ∂ m ( x i , mdλ, u i )( ξ ) ˜ m ( ξ ) dξ . 6 • (A.5): The functions σ 0 (resp. σ 1 ) are twice contin uously differen tiable in x 0 (resp. x 1 ) with b ounded first and second order deriv ativ es. 3 Preliminaries 3.1 Deriv ativ es on the space of measures: Lions, linear, and W asser- stein gradien t Let P 2 ( R d ) b e the space of probability measures with finite second moment. There are three equiv alent notions (under regularity) of the deriv ativ e of functionals f : P 2 ( R d ) → R that w e will use as reference. W e follow the notes of [ 4 , 5 ]. Lions deriv ative (via lift ). One identifies µ ∈ P 2 with a random v ariable ϑ ∈ L 2 (Ω; R d ) suc h that P ϑ = µ , and lifts f to ˜ f : L 2 → R b y ˜ f ( ϑ ) = f ( P ϑ ) . W e say that f is differentiable at µ 0 if ˜ f is F réchet differen tiable at some ϑ 0 with law µ 0 , and then there exists h 0 : R d → R d suc h that D ˜ f ( ϑ 0 ) = h 0 ( ϑ 0 ) . The Lions gr adient of f at µ 0 is the Borel version ( ∂ µ f )( µ 0 , · ) := h 0 ( · ) . Linear deriv ativ e. The linear deriv ative δ f δ µ ( µ, · ) is (defined up to an additive constan t) the function that represents the v ariation along the geo desic segmen t µ h = hµ + (1 − h ) µ ′ , via f ( µ ) − f ( µ ′ ) = Z 1 0 Z R d δ f δ µ ( µ h , x ) ( µ − µ ′ )( dx ) dh. Under regularity , the t w o notions are related b y ∂ µ f ( µ, x ) = ∂ x  δ f δ µ ( µ, x )  . Recall that W 2 denotes the W asserstein metric of order 2 on P 2 ( R d ) , defined by W 2 ( µ, ν ) 2 = inf π ∈ Γ( µ,ν ) Z R d × R d | x − y | 2 dπ ( x, y ) , where Γ( µ, ν ) is the set of couplings of µ and ν . W asserstein gradient (in trinsic). Another notion (from optimal transp ort) defines the differen tiabilit y of f at µ by the existence of a vector ξ ∈ T µ P 2 suc h that f ( µ n ) − f ( µ ) − R ξ ( x ) ( y − x ) π n ( dx, dy ) W 2 ( µ n , µ ) → 0 for µ n → µ in W 2 and π n optimal transport plans. This ξ is denoted ∇ W f ( µ ) and is equiv alent to ∂ µ f ( µ, · ) . Remark. W e use the Lions derivative sinc e it is esp e cial ly c onvenient b e c ause: 7 1. It is formulate d in the Hilb ert sp ac e L 2 , which facilitates chain rules, BSDEs, and the Dynamic Pr o gr amming Principle; 2. It dir e ctly admits an Itô formula over flows of laws; 3. It is explicitly r elate d to the other notions (line ar and ∇ W ), so no gener ality is lost. Notation for join t state–control la ws. In our extension we work on the pro duct space with Π t ∈ P 2 ( R n 1 × A ) and disintegration Π t ( dx, du ) = m t ( dx ) Λ x ( du ) . W e define the joint Lions derivative D Π f (Π)( x, u ) as the Lions gradient of f on the pro duct space. T o align with the rest of the work, w e use the shorthand  D Π f (Π) , δ x  := Z A D Π f (Π)( x, u ) Λ x ( du ) , whic h coincides with the pro jection onto the state when f dep ends on Π only through m t . If moreo v er f (Π) = F ( m t , q t ) , the pro jections of D Π f recov er the marginal deriv ativ es ∂ m F and the dep endence on q t . This conv en tion simplifies the writing of the optimality conditions and the SHJB–FP systems. 3.2 F okker–Planc k equation In general, when working with probability measures and comparing them, it is more con- v enien t to do so when they p ossess density functions on R n 1 . W e define the second-order op erator A 1 and its adjoint A ∗ 1 b y A 1 Φ( x, t ) = − tr( a 1 ( x ) D 2 Φ( x, t )) , A ∗ 1 Φ( x, t ) = − n 1 X i,j =1 ∂ 2 ∂ x i ∂ x j  a ij 1 ( x )Φ( x, t )  , (3) where a 1 ( x ) = 1 2 σ 1 ( x ) σ 1 ( x ) ∗ is a p ositive definite matrix. Let x 1 ( t ) b e the solution of the SDE of the represen tative agent under a control u 1 , and supp ose that the conditional densit y p u 1 ( · , t ) of x u 1 1 ( t ) exists. Then p u 1 satisfies the F okker–Planc k equation with dep endence on the joint law Π , ( ∂ t p u 1 ( x, t ) = − A ∗ 1 p u 1 ( x, t ) − div( g 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , u 1 ( x, t )) p u 1 ( x, t )) , p u 1 ( x, 0) = ω ( x ) , (4) where ω is the initial density of ξ 1 . W e further assume that p u 1 ( · , t ) ∈ L 2 ( R n 1 ) and that p u 1 ( · , t ) dλ ∈ P 2 ( R n 1 ) . F or any densit y m w e write m dλ = m when the context admits no am biguit y . W e can then rewrite the cost functional of the representativ e agent as J 1 ( u 1 , x 0 , u 0 , Π) = E  Z T 0 Z R n 1 f 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , u 1 ( x, t )) p u 1 ( x, t ) dx dt + Z R n 1 h 1 ( x, x 0 ( T ) , u 0 ( T ) , Π T ) p u 1 ( x, T ) dx  . 8 4 Problems In this section we reformulate the optimization problems of the representativ e agent and the dominating play er within the extended framework, in which the cost functions dep end on the joint law of states and controls. 4.1 Problem 1: Con trol of the Represen tativ e Agen t Giv en the pro cess x 0 , the control u 0 , and an exogenous flo w of join t measures Π = (Π t ) t ∈ [0 ,T ] , find a control u 1 ∈ A 1 that minimizes the cost functional J 1 ( u 1 ; x 0 , u 0 , Π) := E  Z T 0 f 1  x 1 ( t ) , x 0 ( t ) , u 0 ( t ) , Π t , u 1 ( t )  dt + h 1  x 1 ( T ) , x 0 ( T ) , u 0 ( T ) , Π T   . (5) 4.2 Problem 2: Equilibrium Condition Let x u 1 1 b e the dynamics induced by an optimal control u 1 of Problem 1. W e denote by M (Π) the joint law induced b y the pair ( x u 1 1 , u 1 ) , conditioned on the filtration F 0 t . The equilibrium problem consists in finding a flo w of measures Π suc h that the fixed-p oint property is satisfied: M (Π)( t ) = Π t , ∀ t ∈ [0 , T ] . 4.3 Problem 3: Con trol of the Dominating Pla y er Giv en the solution Π obtained in Problem 2, find a control u 0 ∈ A 0 that minimizes the cost functional J 0 ( u 0 ; Π) := E  Z T 0 f 0  x 0 ( t ) , Π t , u 0 ( t )  dt + h 0  x 0 ( T ) , Π T   . (6) 5 Main results Lemma 5.1 (Necessary condition for Problem 1) . Given x 0 , u 0 and an exo genous flow of joint me asur es Π as in Pr oblem 1, the optimal c ontr ol ˆ u 1 ∈ A 1 is optimal if and only if it satisfies the fol lowing SHJB e quation: ( − ∂ t Ψ = ( H 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , D Ψ( x, t )) − A 1 Ψ( x, t )) dt − K Ψ ( x, t ) dW 0 ( t ) , Ψ( x, T ) = h 1 ( x, x 0 ( T ) , u 0 ( T ) , Π T ) , wher e H 1 ( x, x 0 , u 0 , Π , q ) = inf u 1 n f 1 ( x, x 0 , u 0 , Π , u 1 ) + q g 1 ( x, x 0 , u 0 , Π , u 1 ) o . The infimum is attaine d uniquely at ˆ u 1 , that is, H 1 ( x, x 0 , u 0 , Π , q ) = f 1 ( x, x 0 , u 0 , Π , ˆ u 1 ) + q g 1 ( x, x 0 , u 0 , Π , ˆ u 1 ) . 9 Remark (Martingale pro cess K Ψ ) . The pr o c ess K Ψ ( x, t ) app e aring in the SHJB e quation is the c o efficient of the martingale p art in the Do ob–Meyer de c omp osition of Ψ with r esp e ct to the filtr ation F 0 gener ate d by the Br ownian motion of the dominating player W 0 . Mor e pr e cisely, the value function Ψ( x, t ) satisfies a BSDE of the form − d Ψ( x, t ) = F ( x, t, D Ψ) dt − K Ψ ( x, t ) dW 0 ( t ) , wher e K Ψ is the squar e-inte gr able inte gr and that ensur es the adapte dness of the solution. The existenc e of K Ψ is guar ante e d by the martingale r epr esentation the or em in the Br ownian filtr ation F 0 . Pr o of. W e apply the sto chastic maximum principle. F or any p erturbation ˜ u 1 ∈ A 1 , 0 = d dθ     θ =0 J 1 ( ˆ u 1 + θ ˜ u 1 , x 0 , u 0 , Π) . (7) Rewriting the exp ectation in terms of the conditional density p ˆ u 1 ( x, t ) of the state x 1 and differen tiating, w e obtain 0 = E " Z T 0 Z R n 1 ˜ p ( x, t ) f 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) dx dt + Z T 0 Z R n 1 p ˆ u 1 ( x, t ) f 1 ,u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) ˜ u 1 ( x, t ) dx dt + Z R n 1 ˜ p ( x, T ) h 1 ( x, x 0 ( T ) , u 0 ( T ) , Π T ) dx # , where ˜ p = d dθ   θ =0 p ˆ u 1 + θ ˜ u 1 . Differen tiating with resp ect to θ in the F okk er–Planc k equation (4), ˜ p satisfies        ∂ ˜ p ∂ t = − A ∗ 1 ˜ p ( x, t ) − div ( ˜ u 1 ( x, t ) g 1 ,u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) p ˆ u 1 ( x, t )) − div ( g 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) ˜ p ( x, t )) , ˜ p ( x, 0) = 0 . W e introduce the adjoint pro cess Ψ as the solution of the BSDE        − ∂ t Ψ =  f 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , u 1 ) + D Ψ( x, t ) g 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , u 1 ) − A 1 Ψ( x, t )  dt − K Ψ ( x, t ) dW 0 ( t ) , Ψ( x, T ) = h 1 ( x, x 0 ( T ) , u 0 ( T ) , Π T ) . 10 Consider the inner pro duct d Z R n 1 ˜ p ( x, t )Ψ( x, t ) dx = Z R n 1 {− A ∗ 1 ˜ p ( x, t ) − div [ ˜ u 1 ( x, t ) g 1 ,u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) p ˆ u 1 ( x, t )] − div [ g 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) ˜ p ( x, t )] } Ψ( x, t ) dxdt − Z R n 1 ˜ p ( x, t ) { f 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , u 1 ( x, t )) + D Ψ( x, t ) g 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , u 1 ( x, t )) − A 1 Ψ( x, t ) } dxdt + Z R n 1 ˜ p ( x, t ) K Ψ ( x, t ) dxdW 0 ( t ) = Z R n 1 ( ˜ u 1 ( x, t ) g 1 ,u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) p ˆ u 1 ( x, t )) D Ψ( x, t ) dxdt − Z R n 1 ˜ p ( x, t ) f 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , u 1 ( x, t )) dxdt + Z R n 1 ˜ p ( x, t ) K Ψ ( x, t ) dxdW 0 ( t ) . In tegrating o v er [0 , T ] and taking exp ectations on b oth sides we get E  Z R n 1 ˜ p ( x, T ) h 1 ( x, x 0 ( T ) , u 0 ( T ) , Π T ) dx  = E  Z T 0 Z R n 1 ( ˜ u 1 ( x, t ) g 1 ,u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) p ˆ u 1 ( x, t )) D Ψ( x, t ) dxdt − Z T 0 Z R n 1 ˜ p ( x, t ) f 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , u 1 ( x, t )) dxdt  , where the term inv olving K Ψ v anishes up on taking exp ectations since it is a martingale term with zero exp ectation. Com bining this with equation (7), we obtain 0 = E  Z T 0 Z R n 1 ˜ u 1 ( x, t ) [ g 1 ,u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) D Ψ( x, t ) + f 1 ,u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t ))] p ˆ u 1 ( x, t ) dxdt ] . Recall that p ˆ u 1 ( · , t ) is a conditional probability densit y function and hence nonnegative, and ˜ u 1 is an arbitrary Marko vian control. Therefore ˆ u 1 is optimal only if g 1 ,u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) D Ψ( x, t ) + f 1 ,u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t )) = 0 , a.e. ( x, t ) . This is equiv alen t to L u 1 ( x, x 0 ( t ) , u 0 ( t ) , Π t , ˆ u 1 ( x, t ) , D Ψ( x, t )) = 0 , a.e. ( x, t ) , whic h provides a necessary condition for the minimization problem. Since the minimizer is assumed to b e attained at ˆ u 1 , which dep ends on x, x 0 , u 0 , Π and D Ψ , w e obtain the SHJB equation. 11 W e replace the exogenous flo w Π by the mean field measure Π x 0 ,u 0 , identifying m x 0 ,u 0 := p ˆ u 1 dλ with the conditional density of the optimal state of the representativ e agent given F 0 t . Com bining equations (4) and (7) we obtain the follo wing corollary . Corollary 5.2 (Necessary condition for Problems 1 and 2) . The c ontr ol for the r epr esentative agent is optimal and the e quilibrium c ondition is satisfie d if and only if the c ouple d SHJB–FP system holds:              − ∂ t Ψ =  H 1 ( x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )) − A 1 Ψ( x, t )  dt − K Ψ ( x, t ) dW 0 ( t ) , Ψ( x, T ) = h 1 ( x, x 0 ( T ) , u 0 ( T ) , Π x 0 ,u 0 T ) , ∂ t m x 0 ,u 0 = − A ∗ 1 m x 0 ,u 0 ( x, t ) − div  G 1 ( x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )) m x 0 ,u 0 ( x, t )  , m x 0 ,u 0 ( x, 0) = ω ( x ) , wher e: G 1 ( x, x 0 , u 0 , Π , q ) = g 1  x, x 0 , u 0 , Π , ˆ u 1 ( x, x 0 , u 0 , Π , q )  , Π x 0 ,u 0 denotes the state–c ontr ol law and m x 0 ,u 0 ( x, t ) denotes the c onditional density of x 1 ( t ) , given x 0 , u 0 . In p articular, the p assage fr om Pr oblem 1 to Pr oblem 2 is interpr ete d as: Π x 0 ,u 0 t = L  x 1 ( t ) , ˆ u 1 ( x 1 ( t ) , x 0 ( t ) , u 0 ( t ) , Π , D Ψ)   F 0 t  , that is, the joint me asur e induc e d by the optimal state and c ontr ol of the r epr esentative agent c onditione d on the information of the dominating player. 6 Extension of the results Prop osition 6.1 (Necessary condition for Problem 3 in the extended case) . L et Π x 0 ,u 0 t = L  x 1 ( t ) , ˆ u 1 ( t )   F 0 t  ∈ P 2 ( R n 1 × A ) b e the joint law induc e d by the r epr esentative agent (Pr ob- lems 1–2). The c ontr ol of the dominating player ˆ u 0 ∈ A 0 is optimal if and only if f 0  x 0 , Π x 0 ,u 0 t , ˆ u 0 ( t )  + p ( t ) · g 0  x 0 , Π x 0 ,u 0 t , ˆ u 0 ( t )  = inf u 0 ∈A 0 n f 0 ( x 0 , Π x 0 ,u 0 t , u 0 )+ p ( t ) · g 0 ( x 0 , Π x 0 ,u 0 t , u 0 ) o W e set H 0  x 0 , Π x 0 ,u 0 t , p ( t )  := inf u 0 ∈A 0 n f 0 ( x 0 , Π x 0 ,u 0 t , u 0 ) + p ( t ) · g 0 ( x 0 , Π x 0 ,u 0 t , u 0 ) o . The ad- joint pr o c ess p and the adjoint fields q ( · , t ) , r ( · , t ) satisfy, for t ∈ [0 , T ] , the system 12 − dp ( t ) =  g 0 ,x 0  x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )  p ( t ) + f 0 ,x 0  x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )  + Z R n 1 G 1 ,x 0  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  · D q ( x, t ) m x 0 ,u 0 ( x, t ) dx + Z R n 1 r ( x, t ) H 1 ,x 0  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  dx  dt − d 0 X ℓ =1 K ℓ p ( t ) dW ℓ 0 ( t ) + d 0 X ℓ =1 σ ℓ ∗ 0 ,x 0  x 0 ( t )  K ℓ p ( t ) dt, p ( T ) = h 0 ,x 0  x 0 ( T ) , Π x 0 ,u 0 T  + Z R n 1 r ( x, T ) h 1 ,x 0  x, x 0 ( T ) , u 0 ( T ) , Π x 0 ,u 0 T  dx, (A dj-p) − ∂ t q ( x, t ) = h − A 1 q ( x, t ) + p ( t ) D D Π g 0  x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )  ; δ x E + D q ( x, t ) · G 1  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  + Z R n 1 × A D q ( ξ , t ) · D D Π G 1  ξ , x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( ξ , t )  ; δ x E Π x 0 ,u 0 t ( dξ , du ) + Z R n 1 × A r ( ξ , t ) D D Π H 1  ξ , x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( ξ , t )  ; δ x E Π x 0 ,u 0 t ( dξ , du ) + D D Π f 0  x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )  ; δ x Ei dt − K q ( x, t ) dW 0 ( t ) , q ( x, T ) = D D Π h 0  x 0 ( T ) , Π x 0 ,u 0 T  ; δ x E + Z R n 1 × A r ( ξ , T ) D D Π h 1  ξ , x 0 ( T ) , u 0 ( T ) , Π x 0 ,u 0 T  ; δ x E Π x 0 ,u 0 T ( dξ , du ) , (A dj-q) ∂ t r ( x, t ) = − A ∗ 1 r ( x, t ) − div x  r ( x, t ) H 1 ,q  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  + G 1 ,q  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  D q ( x, t ) m x 0 ,u 0 ( x, t )  , r ( x, 0) = 0 . (A dj-r) Remark. In the adjoint e quations, the derivative G 1 ,x 0 denotes the derivative of the drift evaluate d at the optimal c ontr ol: G 1 ,x 0 := ∂ ∂ x 0 g 1 ( · , ˆ u 1 ( · )) , wher e ˆ u 1 is determine d fr om the SHJB e quation of the r epr esentative agent. Analo gously, G 1 ,q denotes G 1 ,q := ∂ ∂ q g 1 ( · , ˆ u 1 ( · )) . Sketch of pr o of. The pro of follows the sc heme of Theorem 4.1 of Bensoussan, Chau and Y am [ 2 ], adapted to the dep endence on the join t law Π t ∈ P 2 ( R n 1 × A ) . 13 F or clarity we first presen t the case in which the cost functions dep end only on x 0 (not on u 0 ); the extended case is discussed in the Remark at the end of this section. (i) Gâte aux variation. W e p erturb u 0 7→ ˆ u 0 + θ ˜ u 0 and differentiate in θ : 0 = d dθ     θ =0 J 0 ( ˆ u 0 + θ ˜ u 0 ) = E  Z T 0  f 0 ,x 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )) ˜ x 0 ( t ) + Z R n 1 × A D D Π f 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )); δ ξ E d ˜ Π t ( ξ , u ) + f 0 ,u 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )) ˜ u 0 ( t )] dt + h 0 ,x 0 ( x 0 ( T ) , Π x 0 ,u 0 T ) ˜ x 0 ( T ) + Z R n 1 × A D D Π h 0 ( x 0 ( T ) , Π x 0 ,u 0 T ); δ ξ E d ˜ Π T ( ξ , u )  , (8) where ˜ x 0 = d dθ   θ =0 x 0 ( ˆ u 0 + θ ˜ u 0 ) ; ˜ m x 0 ,u 0 = d dθ   θ =0 m x 0 ,u 0 ( ˆ u 0 + θ ˜ u 0 ) ; ˜ Ψ = d dθ   θ =0 Ψ( ˆ u 0 + θ ˜ u 0 ) . (ii) Line arize d e quations. The v ariations satisfy d ˜ x 0 =  g 0 ,x 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )) ˜ x 0 ( t ) + Z R n 1 × A D D Π g 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )); δ ξ E d ˜ Π t ( ξ , u ) + g 0 ,u 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )) ˜ u 0 ( t )] dt + d 0 X l =1 σ l 0 ,x 0 ˜ x 0 ( t ) dW l 0 ( t ); ˜ x 0 (0) = 0; ∂ ˜ m x 0 ,u 0 ∂ t = − A ∗ 1 ˜ m x 0 ,u 0 ( x, t ) − div { [ G 1 ,x 0 ( x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )) ˜ x 0 ( t ) + Z R n 1 × A D D Π G 1 ( x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )); δ ξ E d ˜ Π t ( ξ , u ) + G 1 ,q ( x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )) D ˜ Ψ( x, t ) i m x 0 ,u 0 ( x, t ) + G 1 ( x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )) ˜ m x 0 ,u 0 ( x, t ) } , ˜ m x 0 ,u 0 ( x, 0) = 0; − ∂ t ˜ Ψ = [ H 1 ,x 0 ( x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )) ˜ x 0 ( t ) + Z R n 1 × A D D Π H 1 ( x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )); δ ξ E d ˜ Π t ( ξ , u ) + H 1 ,q ( x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )) D ˜ Ψ( x, t ) − A 1 ˜ Ψ( x, t ) i dt − K ˜ Ψ ( x, t ) dW 0 ( t ) , ˜ Ψ( x, T ) = h 1 ,x 0 ( x, x 0 ( T ) , u 0 ( T ) , Π x 0 ,u 0 T ) ˜ x 0 ( T ) + Z R n 1 × A D D Π h 1 ( x, x 0 ( T ) , u 0 ( T ) , Π x 0 ,u 0 T ); δ ξ E d ˜ Π T ( ξ , u ) . (iii) A djoint c oupling. W e introduce the adjoin t pro cesses p ( t ) , q ( x, t ) and r ( x, t ) as in the Prop osition and consider the dual pro ducts d  p ∗ ˜ x 0  , d Z q ( x, t ) ˜ m x 0 ,u 0 ( x, t ) dx, d Z r ( x, t ) ˜ Ψ( x, t ) dx. In tegrating by parts and using that Π x 0 ,u 0 t app ears only through D Π f 0 , D Π g 0 , D Π H 1 , D Π G 1 , all terms in ˜ x 0 , ˜ m x 0 ,u 0 , ˜ Ψ cancel b y the c hoice of ( p, q , r ) solving (A dj-p)–(A dj-q)–(Adj-r). 14 Summing the three differentials and taking exp ectations we obtain d { p ∗ ˜ x 0 + Z q ( x, t ) ˜ m x 0 ,u 0 ( x, t ) dx − Z r ( x, t ) ˜ Ψ( x, t ) dx } = [ − ˜ x ∗ 0 f 0 ,x 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )) + p ∗ ( t ) g 0 ,u 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )) ˜ u 0 ( t ) − Z R n 1 × A D D Π f 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )); δ x E d ˜ Π t ( x, u )  dt + {· · · } dW 0 ( t ) . In tegrating and taking exp ectations on eac h side, and considering (8), w e observe that the terms with r cancel each other, leaving 0 = E Z T 0 { f 0 ,u 0 ( x 0 , Π x 0 ,u 0 t , ˆ u 0 ) + p ∗ g 0 ,u 0 ( x 0 , Π x 0 ,u 0 t , ˆ u 0 ) } ˜ u 0 dt. Since ˜ u 0 is arbitrary , the con trol is optimal for the dominating play er only if f 0 ,u 0 ( x 0 , Π x 0 ,u 0 t , ˆ u 0 ) + p ∗ g 0 ,u 0 ( x 0 , Π x 0 ,u 0 t , ˆ u 0 ) = 0 , a.e. t. Since w e are assuming that a unique minimizer ˆ u 0 exists, we conclude that ˆ u 0 satisfies the infim um condition f 0 ( x 0 , Π x 0 ,u 0 t , ˆ u 0 ) + p · g 0 ( x 0 , Π x 0 ,u 0 t , ˆ u 0 ) = inf u 0 { f 0 ( x 0 , Π x 0 ,u 0 t , u 0 ) + p · g 0 ( x 0 , Π x 0 ,u 0 t , u 0 ) } . The rigorous v erification that the Lions deriv ativ e D Π and the Itô formula ov er flo ws of measures extend to the pro duct space P 2 ( R n 1 × A ) under h yp otheses (A.1)–(A.5) is detailed in [ 3 ]. Remark. In this extende d version, the adjoint pr o c esses r etain their fundamental r ole, but their interpr etation is br o adene d: • p ( t ) : c ontinues to b e the c ostate of the dominating player, c apturing the sensitivity of the functional with r esp e ct to variations in its state and c ontr ol x 0 , u 0 . • q ( x, t ) : now enc o des the sensitivity with r esp e ct to the extende d me asur e Π x 0 ,u 0 t . In pr actic e, this me ans that q c aptur es b oth the variations with r esp e ct to the state distri- bution and those induc e d indir e ctly by the law of the c ontr ols, given by the mar ginals of Π . In this sense, q absorbs the Lions derivatives with r esp e ct to b oth mar ginals and acts as the true multiplier for the c onsistency c onstr aint on the p opulation dynamics. • r ( x, t ) : as in the classic al c ase, it r emains the multiplier asso ciate d with the b ackwar d e quation of the agents (the SHJB). In the extende d version, its r ole is to ensur e that the variations in Ψ (which now dep end on Π and not only on m t ) ar e c orr e ctly tr ansmitte d to the system of ne c essary c onditions. Thus, the pr o c esses ( p, q , r ) form a dual system that r efle cts the trip artite structur e of the c onstr aints: dynamics of the dominating player, evolution of the p opulation, and c onsistency of the individual value. The key differ enc e with r esp e ct to the classic al c ase is that the sensitivity c aptur e d by q unifies in a single obje ct the dep endencies on the state and on the c ontr ol thr ough the me asur e Π x 0 ,u 0 t . 15 Remark. When the functions f 1 , g 1 also dep end on the c ontr ol of the dominating player u 0 , the optimality c ondition for the c ontr ol b e c omes f 0 ,u 0 ( x 0 , Π x 0 ,u 0 t , u 0 ) + p · g 0 ,u 0 ( x 0 , Π x 0 ,u 0 t , u 0 ) + Z R n 1 r ( x, t ) H 1 ,u 0  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  dx + Z R n 1 G 1 ,u 0  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  D q ( x, t ) m x 0 ,u 0 ( x, t ) dx = 0 , a.e. t. This me ans that the first-or der c ondition for u 0 is not obtaine d fr om the classic al Hamiltonian H 0 but fr om an effe ctive Hamiltonian that inc orp or ates the r e action of the r epr esentative agents: H 0 ( x 0 ( t ) , Π x 0 ,u 0 t ; p ( t ) , q ( · , t ) , r ( · , t )) := inf u 0 ∈A 0 { f 0 ( x 0 , Π x 0 ,u 0 t , u 0 ) + p · g 0 ( x 0 , Π x 0 ,u 0 t , u 0 ) + Z R n 1 r ( x, t ) H 1  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  dx + Z R n 1 D q ( x, t ) G 1  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  m x 0 ,u 0 ( x, t ) dx  . The c ontr ol of the dominating player ˆ u 0 ∈ A 0 is then optimal if and only if 0 = ∂ u 0 H 0 ( x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t ); p ( t ) , q ( · , t ) , r ( · , t )) , a.e. t ∈ [0 , T ] . Her e: • p is the adjoint pr o c ess asso ciate d with the state of the dominating player x 0 , • q and r ar e the adjoint pr o c esses c apturing the sensitivity of the r epr esentative agents’ system, • the terms with H 1 ,u 0 and G 1 ,u 0 describ e the mar ginal imp act of u 0 on the e quilibrium of the r epr esentative agent (c ost and dynamics r esp e ctively). Remark. In our extende d formulation the c ost functional dep ends on the joint me asur e Π x 0 ,u 0 t of the p air ( x, u ) . It might se em natur al that, when c omputing the Gâte aux derivative in the dir e ction of a variation of the c ontr ol, a variational pr o c ess ˜ Π should app e ar describing the evolution of the p erturb e d me asur e. However, this do es not o c cur, for two fundamental r e asons: 1. The Lions derivative with r esp e ct to Π alr e ady inc orp or ates automatic al ly the sensi- tivities in b oth c o or dinates: the state m t and the induc e d distribution of c ontr ols Λ x 0 . Thus, what would c onc eptual ly c orr esp ond to a term in ˜ Π de c omp oses into variations of ˜ m and ˜ Λ , which ar e pr e cisely those that app e ar in the c omputations. 2. F r om the dynamic al p oint of view, the c ontr ol has no diffusion of its own: it is deter- mine d as a fe e db ack fr om the tr aje ctory of x 0 and the c ommon information. Conse- quently, in the evolution of Π factor e d as Π t ( dx, du ) = m t ( dx ) Λ x ( du ) , 16 m t is the only me asur e satisfying a genuine F okker–Planck e quation. The p art asso ci- ate d with the c ontr ol is obtaine d dir e ctly fr om the e quilibrium p olicy without r e quiring an indep endent dynamics. In summary, the variation of the functional under p erturb ations of the c ontr ol is expr esse d solely in terms of ˜ m and ˜ Λ , which ar e absorb e d up on intr o ducing the adjoint pr o c ess q , and the only F okker–Planck e quation that app e ars is that of m t . This explains why the c omputations r etain the same structur e as in the classic al c ase. Let G 0 ( x 0 , Π , p ) := g 0 ( x 0 , Π , ˆ u 0 ( x 0 , Π , p )) . W e conclude with the main result of this work. Theorem 6.2 (Extended necessary condition for Problems 1, 2 and 3) . The system forme d by the r epr esentative agent and the dominating player admits as ne c essary optimality c ondition the fol lowing c ouple d system of sto chastic p artial differ ential e quations:                                  dx 0 = G 0  x 0 ( t ) , Π x 0 ,u 0 t , p ( t )  dt + σ 0 ( x 0 ( t )) dW 0 ( t ) , x 0 (0) = ξ 0 ; ∂ m x 0 ,u 0 ∂ t = − A ∗ 1 m x 0 ,u 0 ( t ) − div  G 1  x, x 0 , u 0 , Π x 0 ,u 0 t , D Ψ( x, t )  m x 0 ,u 0 ( t )  , m x 0 ,u 0 ( x, 0) = ω ( x ); − ∂ t Ψ =  H 1  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  − A 1 Ψ( x, t )  dt − K Ψ ( x, t ) dW 0 ( t ) , Ψ( x, T ) = h 1  x, x 0 ( T ) , u 0 ( T ) , Π x 0 ,u 0 T  . 17                                                                                                                − dp ( t ) =  G 0 ,x 0  x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )  p ( t ) + f 0 ,x 0  x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )  + Z R n 1 G 1 ,x 0  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  · D q ( x, t ) m x 0 ,u 0 ( x, t ) dx + Z R n 1 r ( x, t ) H 1 ,x 0  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  dx  dt − d 0 X ℓ =1 K ℓ p ( t ) dW ℓ 0 ( t ) + d 0 X ℓ =1 σ ℓ ∗ 0 ,x 0  x 0 ( t )  K ℓ p ( t ) dt, p ( T ) = h 0 ,x 0  x 0 ( T ) , Π x 0 ,u 0 T  + Z R n 1 r ( x, T ) h 1 ,x 0  x, x 0 ( T ) , u 0 ( T ) , Π x 0 ,u 0 T  dx ; − ∂ t q = h − A 1 q ( x, t ) + p ( t ) D D Π G 0  x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )  ; δ x E + D q ( x, t ) · G 1  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  + Z R n 1 × A D q ( ξ , t ) · D D Π G 1  ξ , x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( ξ , t )  ; δ x E Π x 0 ,u 0 t ( dξ , du ) + Z R n 1 × A r ( ξ , t ) D D Π H 1  ξ , x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( ξ , t )  ; δ x E Π x 0 ,u 0 t ( dξ , du ) + D D Π f 0  x 0 ( t ) , Π x 0 ,u 0 t , ˆ u 0 ( t )  ; δ x Ei dt − K q ( x, t ) dW 0 ( t ) , q ( x, T ) = D D Π h 0  x 0 ( T ) , Π x 0 ,u 0 T  ; δ x E + Z R n 1 × A r ( ξ , T ) D D Π h 1  ξ , x 0 ( T ) , u 0 ( T ) , Π x 0 ,u 0 T  ; δ x E Π x 0 ,u 0 T ( dξ , du ); ∂ t r = − A ∗ 1 r ( x, t ) − div x  r ( x, t ) H 1 ,q  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  + G 1 ,q  x, x 0 ( t ) , u 0 ( t ) , Π x 0 ,u 0 t , D Ψ( x, t )  D q ( x, t ) m x 0 ,u 0 ( x, t )  , r ( x, 0) = 0 . Remark. In the adjoint e quations, the derivative G 0 ,x 0 denotes the derivative of the drift evaluate d at the optimal c ontr ol: G 0 ,x 0 := ∂ ∂ x 0 g 0 ( x 0 , Π , ˆ u 0 ( x 0 , Π , p )) , wher e ˆ u 0 is the solution of ∂ u 0 H 0 = 0 . Remark. The system of e quations obtaine d c onstitutes the extende d version of the classic al adjoint SHJB–FP scheme. Inde e d, it c ombines: • The dynamics of the dominating state x 0 , governe d by a c ontr ol u 0 that inter acts with the joint law of states and c ontr ols Π x 0 ,u 0 = ( m x 0 ,u 0 , Λ x 0 ,u 0 ) . • The F okker–Planck e quation for the density of the r epr esentative agent, which describ es the aggr e gate evolution of the p opulation in terms of the c onditione d me asur e m x 0 ,u 0 . 18 • The sto chastic Hamilton–Jac obi–Bel lman e quation for the r epr esentative agent, whose dep endenc e on Π explicitly r efle cts the pr esenc e of the law of the states and of the c ontr ols in the c ost function. • The adjoint pr o c esses p ( t ) , q ( x, t ) , r ( x, t ) asso ciate d with the pr oblem of the dominating player, which enc o de the derivatives of f 0 , h 0 with r esp e ct to x 0 , the state me asur e, and the law of the c ontr ols. This c ouple d system simultane ously synthesizes the optimality c onditions for the r epr e- sentative agent (Pr oblem 1), the fixe d-p oint e quilibrium c ondition (Pr oblem 2), and the opti- mality c ondition for the dominating player (Pr oblem 3). In analytic terms, it is a F orwar d– Backwar d Sto chastic PDE system that gener alizes pr evious appr o aches to Me an Field Games, explicitly inc orp or ating the dep endenc e on the extende d law Π . 7 Conclusion In this w ork we extended the theory of mean field games with a dominating play er of Ben- soussan, Chau and Y am [ 2 ] to the case where the cost functionals and dynamics dep end on the join t state–con trol law Π t = L ( X 1 t , U 1 t | F 0 t ) ∈ P 2 ( R n 1 × A ) , rather than solely on the marginal distribution of states µ t . The main contributions are threefold. Reform ulation in terms of Π t . The optimization problems of the represen tativ e agent (Problems 1 and 2) and of the dominating play er (Problem 3) w ere reform ulated in terms of the join t measure Π t . This unifies the dep endencies on state and control in to a single measure on the pro duct space R n 1 × A , simplifying b oth the notation and the application of differen tial calculus on probability spaces. Extension of the Lions deriv ati v e. The Lions deriv ativ e extends naturally to the space P 2 ( R n 1 × A ) when A ⊂ R m is compact, leveraging the pro duct space structure. This extension is essen tial for formulating the optimalit y conditions, as it allows one to compute deriv atives of functionals with resp ect to the measure Π and decompose them into contributions from the state and control marginals. Coupled SHJB–FP–adjoin t system. Prop osition 6.1 establishes the necessary optimal- it y conditions for the dominating play er through a system of three adjoin t pro cesses ( p, q , r ) , and Theorem 6.2 synthesizes the equilibrium conditions of the full system in to a coupled F orward–Bac kw ard sto chastic–partial differential equation system. This result generalizes Theorem 4.1 of [ 2 ] to the new measure space. The dep endence on Π (rather than on µ alone) mo difies the adjoint equation for q ( x, t ) , which now captures simultaneously the sensitivities with resp ect to b oth state and con trol through the join t measure. 19 F uture directions. The extension presented here op ens several lines of in vestigation. First, the question of existence and uniqueness of solutions for the extended SHJB–FP– adjoin t system requires deep er analysis, particularly regarding the regularit y of the Lions deriv ative on the pro duct space. Second, the application of this framework to concrete mo dels—suc h as decentralized markets where a liquidit y provider acts as a dominating play er facing a con tinuum of traders [ 3 ]—constitutes a natural motiv ation and a test of the general- it y of the theory . Finally , the extension to games where m ultiple dominating play ers interact with eac h other and with the field of minor agents is an op en problem of b oth theoretical and applied interest. References [1] R. Carmona and D. Lack er. A probabilistic weak form ulation of mean field games and applications. The Annals of Applie d Pr ob ability , 25(3):1189–1231, 2015. [2] A. Bensoussan M.H.M. Chau and S.C.P . Y am. Mean field games with a dominating pla y er. 2014. [3] A. Muñoz Gonzalez. Mean field games in complex mark ets: Extensions and applications. 2025. [4] X. Guo and C. Liang. Ito’s formula for flo w of measures on semimartingales, 2022. [5] S. Korici. Lions deriv ative and its applications, 2024. Lecture notes. 20