A Mean Field Game Approach to Equilibrium Pricing with Market Clearing Condition
aa r X i v : . [ q -f i n . M F ] O c t A Mean Field Game Approach to Equilibrium Pricingwith Market Clearing Condition ∗ Masaaki Fujii † , Akihiko Takahashi ‡ First version: 6 March, 2020This version: 3 October, 2020
Abstract
In this work, we study an equilibrium-based continuous asset pricing problem whichseeks to form a price process endogenously by requiring it to balance the flow of sales-and-purchase orders in the exchange market, where a large number of agents 1 ≤ i ≤ N areinteracting through the market price. Adopting a mean field game (MFG) approach, we finda special form of forward-backward stochastic differential equations of McKean-Vlasov typewith common noise whose solution provides a good approximate of the market price. Weshow the convergence of the net order flow to zero in the large N -limit and get the order ofconvergence in N under some conditions. We also extend the model to a setup with multiplepopulations where the agents within each population share the same cost and coefficientfunctions but they can be different population by population. Keywords :
FBSDE of McKean-Vlasov type, common noise, general equilibrium
One of the most important problems in the financial economics is to understand how the assetprice processes are formed through the interaction among a large number of rational competitiveagents. In this paper, using a stylized model of security exchange, we try to explicitly form anapproximate market price process which balances the flow of sales-and-purchase orders from alarge number of rational financial institutions. If we directly force the price process to balancethe net order flow, the strategies of the agents become strongly coupled and the problem ishardly solvable. In fact, it is even unclear how to make the cost functions of the agents well-defined, since the market price results in a very complicated recursive functional of strategies ofall the agents that makes it difficult to guarantee the convexity of the cost functions. In orderto circumvent this problem, we make use of the recent developments of mean field games.Since its inception brought by the pioneering works of Lasry & Lions [20, 21, 22] and Huang,Malhame & Caines [18], mean field game has rapidly developed into one of the most activelystudied topics in the field of probability theory, applied mathematics, engineering, finance andeconomics. The greatest strength of the mean field game approach is to render notoriously ∗ All the contents expressed in this research are solely those of the author and do not represent any views oropinions of any institutions. The author is not responsible or liable in any manner for any losses and/or damagescaused by the use of any contents in this research. † Quantitative Finance Course, Graduate School of Economics, The University of Tokyo. ‡ Quantitative Finance Course, Graduate School of Economics, The University of Tokyo. ε -Nash equilibrium, we show that the solution of the mean-field limit problem ac-tually provides asymptotic market clearing in the large- N limit. Under additional integrabilityconditions, Glivenko-Cantelli convergence theorem in the Wasserstein distance even provides aspecific order of convergence in terms of the number of agents N . We also discuss the extensionof the model to the situation with multiple populations where the agents share the same cost andcoefficient functions within each population but they can be different population by population.This will provide an important tool to study the price formation in the presence of different typeof agents such as Buy-side and Sell-side institutions, for example.The organization of the paper is as follows: After explaining the notation in Section 2, wegive an intuitive derivation of the limit problem from the game of finite number of agents inSection 3, which motivates the readers to study the special type of FBSDEs of MKV-type. Thesolvability of the FBSDE is studied in Section 4. Using the derived regularity of the solution,we prove the asymptotic market clearing in Section 5. In Section 6, we discuss the extensionof the model to the setup with multiple populations. Finally, in Section 7, we give concludingremarks. We discuss further extensions of the model and future directions of research.2 Notation
We introduce (N+1) complete probability spaces:(Ω , F , P ) and (Ω i , F i , P i ) Ni =1 , endowed with filtrations F i := ( F it ) t ≥ , i ∈ { , · · · , N } . Here, F is the completion of thefiltration generated by d -dimensional Brownian motion W (hence right-continuous) and, foreach i ∈ { , · · · , N } , F i is the complete and right-continuous augmentation of the filtrationgenerated by d -dimensional Brownian motions W i as well as a W i -independent n -dimensionalsquare-integrable random variables ( ξ i ). ( ξ i ) Ni =1 are supposed to have the same law. We alsointroduce the product probability spacesΩ i = Ω × Ω i , F i , F i = ( F it ) t ≥ , P i , i ∈ { , · · · , N } where ( F i , P i ) is the completion of ( F ⊗ F i , P ⊗ P i ) and F i is the complete and right-continuous augmentation of ( F t ⊗ F it ) t ≥ . In the same way, we define the complete proba-bility space (Ω , F , P ) endowed with F = ( F t ) t ≥ satisfying the usual conditions as a product of(Ω i , F i , P i ; F i ) Ni =0 .Throughout the work, the symbol L and L ̟ denote given positive constants, the symbol C a general positive constant which may change line by line. When we want to emphasize that C depends only on some specific variables, say a and b , we use the symbol C ( a, b ). For a givenconstant T >
0, we use the following notation for frequently encountered spaces: • L ( G ; R d ) denotes the set of R d -valued G -measurable square integrable random variables. • S ( G ; R d ) is the set of R d -valued G -adapted continuous processes X satisfying || X || S := E (cid:2) sup t ∈ [0 ,T ] | X t | (cid:3) < ∞ . • H ( G ; R d ) is the set of R d -valued G -progressively measurable processes Z satisfying || Z || H := E h(cid:16)Z T | Z t | dt (cid:17)i < ∞ . • L ( X ) denotes the law of a random variable X . • P ( R d ) is the set of probability measures on ( R d , B ( R d )). • P p ( R d ) with p ≥ P ( R d ) with finite p -th moment; i.e., the set of µ ∈ P ( R d )satisfying M p ( µ ) := (cid:16)Z R d | x | p µ ( dx ) (cid:17) p < ∞ . We always assign P p ( R d ) with ( p ≥
1) the p -Wasserstein distance W p , which makes P p ( R d ) acomplete separable metric space. As an important property, for any µ, ν ∈ P p ( R d ), we have W p ( µ, ν ) = inf n E [ | X − Y | p ] p ; L ( X ) = µ, L ( Y ) = ν o , (2.1)where “inf” is taken over all random variables with laws equal to µ and ν , respectively. For3ore details, see Chapter 5 in [7]. We frequently omit the arguments such as ( G , R d ) in theabove definitions when there is no confusion from the context. In this section, in order to introduce the special form of forward-backward stochastic differentialequations of McKean-Vlasov type to be studied in this paper, we give a heuristic derivation ofthe mean-field limit problem from the corresponding equilibrium problem with finite number ofagents. As a motivating example, we consider the equilibrium-based pricing problem of n typesof securities, which are continuously traded in the security exchange in the presence of a largenumber of participating agents indexed by i ∈ { , · · · , N } . Every agent is supposed to havemany small clients who can only trade directly to the agent via over-the-counter markets (OTC)and have no access to the security exchange.We suppose that each agent i ∈ { , · · · , N } tries to solve the probleminf α i ∈ A i J i ( α i ) (3.1)with some cost functional J i ( α i ) := E hZ T f ( t, X it , α it , ̟ t , c t , c it ) dt + g ( X iT , ̟ T , c T , c iT ) i , subject to the dynamic constraint: dX it = (cid:16) α it + l ( t, ̟ t , c t , c it ) (cid:17) dt + σ ( t, ̟ t , c t , c it ) dW t + σ ( t, ̟ t , c t , c it ) dW it with X i = ξ i ∈ L ( F i ; R n ). Here, ( X it ) t ≥ is an R n -valued process denoting the time- t positionof the n securities for the agent i with the initial position ξ i . ( c t ) t ≥ ∈ H ( F ; R n ) denotesthe coupon payments from the securities or the market news commonly available to all theagents, while ( c it ) t ≥ ∈ H ( F i ; R n ) denotes some independent factors affecting only on the agent i . Moreover, ( c it ) t ≥ are also assumed to have the common law for all 1 ≤ i ≤ N . We furthersuppose c T and c iT are square integrable to handle the terminal cost g . Each agent controls( α it ) t ≥ denoting the trading speed though the security exchange. The remaining terms ( l, σ , σ )denote the order flow to the agent from his/her clients through over-the-counter (OTC) markets.( ̟ t ) t ≥ is the market price of the n securities. The space of admissible strategies A i of the agent i is the set of processes ( α it ) t ≥ adapted to the complete right-continuous augmentation of thefiltration (cid:0) σ { ̟ s : s ≤ t } ) ∨ F it (cid:1) t ≥ satisfying E Z T | α it | dt < ∞ . In contrast to the standard optimization problems with a given market price process, we wantto understand the fundamental mechanism of financial market which determines the marketprice by the equilibrium condition. The equilibrium price ( ̟ t ) t ≥ adapted to the filtration F isdetermined endogenously so that the optimal strategies of the agents ( b α it ) Ni =1 satisfy the market4learing condition for every t ∈ [0 , T ], P -a.s. N X i =1 b α it = 0 , (3.2)which denotes the balance point of demand and supply at the security exchange.Although we have already made simplistic assumptions such that the cost functions as wellas the coefficient functions of all the agents are common, the problem is still hardly solvable.Due to the clearing condition (3 . ̟ t ) t ≥ becomes a complicated functional of the agents’ trading strategies and hencethe problem for each agent is highly recursive with respect to ( α it ) t ≥ , ≤ i ≤ N . It is even unclearhow to guarantee the cost function well-defined by making it convex with respect to the controls.In order to obtain some insight, let us consider a much simpler situation. It is naturalto suppose that the impact to the market price ( ̟ t ) t ∈ [0 ,T ] from the individual agent becomesnegligibly small when N is sufficiently large. Moreover, ( ̟ t ) t ∈ [0 ,T ] is likely to be given by F -progressively measurable process since the effects from the idiosyncratic parts from many agentsare expected to be canceled out. If this is the case, the problem for each agent reduces to thestandard stochastic optimal control problem in a given random environment ( ̟ t , c t , c it ) t ∈ [0 ,T ] with F i -adapted trading strategy ( α it ) t ∈ [0 ,T ] . Let us first investigate this simple problem indetails. We introduce the cost functions: f : [0 , T ] × ( R n ) → R , g : ( R n ) → R , f : [0 , T ] × ( R n ) → R and g : ( R n ) → R , which are measurable functions such that f ( t, x, α, ̟, c , c ) := h ̟, α i + 12 h α, Λ α i + f ( t, x, ̟, c , c ) ,g ( x, ̟, c , c ) := − δ h ̟, x i + g ( x, c , c ) . Assumption 3.1. (MFG-a) (i) Λ is a positive definite n × n symmetric matrix with λI n × n ≤ Λ ≤ λI n × n in the sense of2nd-order form where λ and λ are some constants satisfying < λ ≤ λ .(ii) For any ( t, x, ̟, c , c ) , | f ( t, x, ̟, c , c ) | + | g ( x, c , c ) | ≤ L (1 + | x | + | ̟ | + | c | + | c | ) . (iii) f and g are continuously differentiable in x and satisfy, for any ( t, x, x ′ , ̟, c , c ) , | ∂ x f ( t, x ′ , ̟, c , c ) − ∂ x f ( t, x, ̟, c , c ) | + | ∂ x g ( x ′ , c , c ) − ∂ x g ( x, c , c ) | ≤ L | x ′ − x | , and | ∂ x f ( t, x, ̟, c , c ) | + | ∂ x g ( x, c , c ) | ≤ L (1 + | x | + | ̟ | + | c | + | c | ) .(iv)The functions f and g are convex in x in the sense that for any ( t, x, x ′ , ̟, c , c ) , f ( t, x ′ , ̟, c , c ) − f ( t, x, ̟, c , c ) − h x ′ − x, ∂ x f ( t, x, ̟, c , c ) i ≥ γ f | x ′ − x | ,g ( x ′ , c , c ) − g ( x, c , c ) − h x ′ − x, ∂ x g ( x, c , c ) i ≥ γ g | x ′ − x | , with some constants γ f , γ g ≥ .(v) l, σ , σ are the measurable functions defined on [0 , T ] × ( R n ) and are R n , R n × d and R n × d - alued, respectively. Moreover they satisfy the linear growth condition: | ( l, σ , σ )( t, ̟, c , c ) | ≤ L (1 + | ̟ | + | c | + | c | ) for any ( t, ̟, c , c ) .(vi) δ ∈ [0 , is a given constant. The first term h ̟, α i of f denotes the direct cost incurred by the sales and purchase of thesecurities and the second term h α, Λ α i is some fee to be paid to the exchange depending onthe trading speed, or may be interpreted as some internal cost. The first term of g denotes themark-to-market value at the closing time with some discount factor δ < f and g denote therunning as well as the terminal cost which are affected by the market price, coupon streams, orthe news. Remark 3.1.
If we think c as a coupon stream of the securities, one may consider for example, f ( t, x, ̟, c , c ) = −h c , x i + f ′ ( t, x, ̟, c ) as a running cost with an appropriate measurable function f ′ . For securities with a given ma-turity T with exogenously specified payoff c , such as bonds and futures, it is natural to consider g ( x, c ) = g ( x, c ) = −h c , x i as the terminal cost. For this problem, the (reduced) Hamiltonian is given by H ( t, x, y, α, ̟, c , c ) = h y, α + l ( t, ̟, c , c ) i + f ( t, x, α, ̟, c , c ) . Since ∂ α H ( t, x, y, α, ̟, c , c ) = y + ̟ + Λ α , the minimizer of the Hamiltonian is b α ( y, ̟ ) := − Λ( y + ̟ ) (3.3)where Λ := Λ − . The adjoint FBSDE associated with the stochastic maximal principle for eachagent 1 ≤ i ≤ N is thus given by, dX it = (cid:16)b α ( Y it , ̟ t ) + l ( t, ̟ t , c t , c it ) (cid:17) dt + σ ( t, ̟ t , c t , c it ) dW t + σ ( t, ̟ t , c t , c it ) dW it ,dY it = − ∂ x f ( t, X it , ̟ t , c t , c it ) dt + Z i, t dW t + Z it dW it , (3.4)with X i = ξ i and Y iT = ∂ x g ( X iT , ̟ T , c T , c iT ). Theorem 3.1.
Under Assumption (MFG-a) and a given ( ̟ t ) t ∈ [0 ,T ] ∈ H ( F ; R n ) , the problem (3 . for each agent is uniquely characterized by the FBSDE (3 . which is strongly solvable witha unique solution ( X i , Y i , Z i, , Z i ) ∈ S ( F i ; R n ) × S ( F i ; R n ) × H ( F i ; R n × d ) × H ( F i ; R n × d ) .Proof. Since the cost functions are jointly convex with ( x, α ) and strictly convex in α , theproblem is the special situation investigated in Section 1.4.4 in [8]. Note that, in our case, thediffusion terms σ , σ are independent of ( X i , α i ). The proof is the direct result of Theorem 1.60in the same reference. We shall see that the condition δ < b α it = − Λ( Y it + ̟ t ) , t ∈ [0 , T ] . Let us check the market clearing condition. In the current situation, (3 .
2) is equivalent to ̟ t = − N N X i =1 Y it which is of course inconsistent with the our simplifying assumption that requires ( ̟ t ) t ≥ to be an F -adapted process. However, in the current setup, for any t ∈ [0 , T ], ( Y it ) Ni =1 are exchangeablerandom variables due to the construction of the probability space, common coefficient functions,and the fact that ( ξ i ) Ni =1 as well as ( c it , t ∈ [0 , T ]) Ni =1 are assumed to be i.i.d. Thus De Finetti’stheory of exchangeable sequence of random variables tells,lim N →∞ N N X i =1 Y it = E h Y t | \ k ≥ σ { Y jt , j ≥ k } i a . s . See for example Theorem 2.1 in [8]. It also seems natural to expect that the tail σ -field is reducedto F t . Therefore we can expect that, in the large- N limit, the market price of the securitiesmay be given by ̟ t = − E [ Y t |F t ].The above observation motivates us to consider the following FBSDE: dX t = (cid:16)b α (cid:0) Y t , − E [ Y t |F t ] (cid:1) + l (cid:0) t, − E [ Y t |F t ] , c t , c t (cid:1)(cid:17) dt + σ (cid:0) t, − E [ Y t |F t ] , c t , c t (cid:1) dW t + σ (cid:0) t, − E [ Y t |F t ] , c t , c t (cid:1) dW t ,dY t = − ∂ x f (cid:0) t, X t , − E [ Y t |F t ] , c t , c t (cid:1) dt + Z t dW t + Z t dW t , with X = ξ with Y T = δ − δ E (cid:2) ∂ x g ( X T , c T , c T ) |F T (cid:3) + ∂ x g ( X T , c T , c T ). To simplify the notation,we have omitted the superscript 1 from Y , X , ξ and c . Let us remark on the terminalcondition. Y T = ∂ x g ( X T , − E [ Y T |F T ] , c T , c T ) is not yet fully specified. Taking the conditionalexpectation in the both sides gives E [ Y T |F T ] = δ E [ Y T |F T ] + E (cid:2) ∂ x g ( X T , c T , c T ) |F T (cid:3) , which implies E [ Y T |F T ] = − δ E (cid:2) ∂ x g ( X T , c T , c T ) |F T (cid:3) . Substituting this expression for E [ Y T |F T ]in ∂ x g , we get the above specification of the terminal condition.This is the FBSDE we are going to study in the following. It is of McKean-Vlasov typewith common noise, and similar to the FBSDEs relevant for the extended mean field games.In the following, we are going to prove the existence of a unique solution to the above FBSDEunder appropriate conditions and then show that − E [ Y t |F t ] is actually a good approximate ofthe market price by investigating how accurately it achieves the market clearing condition (3 . N increases. 7 Solvability of the mean-field FBSDE
We now investigate the solvability of the FBSDE derived in the last section dX t = (cid:16)b α (cid:0) Y t , − E [ Y t |F t ] (cid:1) + l (cid:0) t, − E [ Y t |F t ] , c t , c t (cid:1)(cid:17) dt + σ (cid:0) t, − E [ Y t |F t ] , c t , c t (cid:1) dW t + σ (cid:0) t, − E [ Y t |F t ] , c t , c t (cid:1) dW t ,dY t = − ∂ x f (cid:0) t, X t , − E [ Y t |F t ] , c t , c t (cid:1) dt + Z t dW t + Z t dW t , (4.1)with X = ξ with Y T = δ − δ E (cid:2) ∂ x g ( X T , c T , c T ) |F T (cid:3) + ∂ x g ( X T , c T , c T ). b α is defined as in (3 . c t ) t ≥ ∈ H ( F ; R n ) and ( c t ) t ≥ ∈ H ( F ; R n ) with square integrable c T , c T are given as inputs.Let us remind the notation to write ξ = ξ and c = c . T Assumption 4.1. (MFG-b)
For any ( t, x, c , c ) ∈ [0 , T ] × ( R n ) and any ̟, ̟ ′ ∈ R n , the coefficient functions l, σ , σ and f satisfy | ( l, σ , σ )( t, ̟, c , c ) − ( l, σ , σ )( t, ̟ ′ , c , c ) | + | ∂ x f ( t, x, ̟, c , c ) − ∂ x f ( t, x, ̟ ′ , c , c ) | ≤ L ̟ | ̟ − ̟ ′ | . Due to the Lipschitz continuity and the absence of ( Z , Z ) in the diffusion coefficients of theforward SDE, we have the following short-term existence result. Theorem 4.1.
Under Assumptions (MFG-a,b), there exists some constant τ > which dependsonly on ( L, L ̟ , λ, δ ) such that for any T ≤ τ , there exists a unique strong solution ( X, Y, Z , Z ) ∈ S ( F ; R n ) × S ( F ; R n ) × H ( F ; R n × d ) × H ( F ; R n × d ) to the FBSDE (4 . .Proof. Although there exist terms involving E [ Y t |F t ], one can adopt the standard technique forthe Lipschitz FBSDE. See, for example, the proof of Theorem 1.45 [8]. T In order to obtain existence result for general T , we are going to apply the technique developedby Peng & Wu [24]. In the case of the standard optimization problem, the joint convexity in thestate and control variables combined with strict convexity in the control variable are enough toobtain the unique existence. Interestingly however, we need a strict convexity also in the statevariable X in our problem. As we shall see, this is because the term − E [ Y t |F t ] which appearsdue to the clearing condition weakens the convexity. Assumption 4.2. (MFG-c1)(i) The functions σ and σ are independent of the argument ̟ .(ii) For any t ∈ [0 , T ] , any random variables x, x ′ , c , c ∈ L ( F ; R n ) and any sub- σ -field G ⊂ F ,the function l satisfies the monotone condition with some positive constant γ l > : E h h l ( t, E [ x |G ] , c , c ) − l ( t, E [ x ′ |G ] , c , c ) , x − x ′ i i ≥ γ l E (cid:2) E [ x − x ′ |G ] (cid:3) . iii) There exists a strictly positive constant γ satisfying < γ ≤ (cid:16) γ f − L ̟ γ l (cid:17) ∧ γ g . Moreover,for any x, x ′ , c , c ∈ L ( F ; R n ) and any sub- σ -field G ⊂ F , the function g satisfies γ g E [ | x − x ′ | ] + δ − δ E h h E (cid:2) ∂ x g ( x, c , c ) − ∂ x g ( x ′ , c , c ) |G (cid:3) , x − x ′ i i ≥ γ E [ | x − x ′ | ] . Remark 4.1. If l and ∂ x g have separable forms such as h ( x ) + h c ( c , c ) with some functions h and h c , then the conditions (ii) and (iii) are satisfied when the function h is monotone.Economically speaking, the condition (ii) implies that the demand from the individual OTCclients of each agent toward the security decreases when its market price rises. The next theorem is the first main existence result.
Theorem 4.2.
Under Assumptions (MFG-a,b,c1) , there exists a unique strong solution ( X, Y, Z , Z ) ∈ S ( F ; R n ) × S ( F ; R n ) × H ( F ; R n × d ) × H ( F ; R n × d ) to the FBSDE (4 . .Proof. In order to simplify the notation, let us define the functionals
B, F and G for any y, x, c , c ∈ L ( F ; R n ) by B ( t, y, c , c ) := (cid:16) − Λ( y − E [ y |F t ]) + l ( t, − E [ y |F t ] , c , c ) (cid:17) ,F ( t, x, y, c , c ) := − ∂ x f (cid:0) t, x, − E [ y |F t ] , c , c (cid:1) ,G ( x, c , c ) := δ − δ E (cid:2) ∂ x g ( x, c , c ) |F T (cid:3) + ∂ x g ( x, c , c ) . (4.2)With the convention ∆ y := y − y ′ , ∆ x := x − x ′ , one can easily confirms E (cid:2) h B ( t, y, c , c ) − B ( t, y ′ , c , c ) , ∆ y i (cid:3) ≤ − γ l E (cid:2) E [∆ y |F t ] (cid:3) , E (cid:2) h F ( t, x, y, c , c ) − F ( t, x ′ , y ′ , c , c ) , ∆ x i (cid:3) ≤ − (cid:16) γ f − L ̟ γ l (cid:17) E [ | ∆ x | ] + γ l E (cid:2) E [∆ y |F t ] (cid:3) , E (cid:2) h G ( x, c , c ) − G ( x ′ , c , c ) , ∆ x i (cid:3) ≥ γ E [ | ∆ x | ] , (4.3)where the first estimate follows from (MFG-c1)(ii) and Jensen’s inequality, the second from(MFG-a)(iv), (MFG-b) and Cauchy-Schwarz inequality. The third one is the direct consequenceof (MFG-c)(iii).We first make the following hypothesis: there exists some constant ̺ ∈ [0 ,
1) such that,for any ( I bt ) t ≥ , ( I ft ) t ≥ in H ( F ; R n ) and any η ∈ L ( F T ; R n ), there exists a unique solution( x ̺ , y ̺ , z ,̺ , z ̺ ) ∈ S ( F ; R n ) × S ( F ; R n ) × H ( F ; R n × d ) × H ( F ; R n × d ) to the FBSDE: dx ̺t = (cid:0) ̺B ( t, y ̺t , c t , c t ) + I bt (cid:1) dt + σ ( t, c t , c t ) dW t + σ ( t, c t , c t ) dW t ,dy ̺t = − (cid:0) (1 − ̺ ) γx ̺t − ̺F ( t, x ̺t , y ̺t , c t , c t ) + I ft (cid:1) dt + z ,̺t dW t + z ̺t dW t , (4.4)with x ̺ = ξ and y ̺T = ̺G ( x ̺T , c T , c T ) + (1 − ̺ ) x ̺T + η . Note that when ̺ = 0 we have a decoupledset of SDE and BSDE and hence the hypothesis trivially holds. Our goal is to extend the ̺ up to 1 by following Peng-Wu’s continuation method [24]. Now, for an arbitrary set of inputs9 x, y, z , z ) ∈ S ( F ; R n ) × H ( F ; R n × d ) × H ( F ; R n × d ) and constant ζ ∈ (0 , dX t = (cid:2) ̺B ( t, Y t , c t , c t ) + ζB ( t, y t , c t , c t ) + I bt (cid:3) dt + σ ( t, c t , c t ) dW t + σ ( t, c t , c t ) dW t ,dY t = − (cid:2) (1 − ̺ ) γX t − ̺F ( t, X t , Y t , c t , c t ) + ζ ( − γx t − F ( t, x t , y t , c t , c t )) + I ft (cid:3) dt + Z t dW t + Z t dW t , (4.5)with X = ξ and Y T = ̺G ( X T , c T , c T ) + (1 − ̺ ) X T + ζ ( G ( x T , c T , c T ) − x T ) + η . The existenceof the solution ( X, Y, Z , Z ) ∈ S × S × H × H is guaranteed by the previous hypothesis. Weare going to prove the map ( x, y, z , z ) ( X, Y, Z , Z ) defined above becomes strict contractionwhen ζ > x, y, z , z ) and ( x ′ , y ′ , z ′ , z ′ ), let us denote the corresponding solutionsto (4 .
5) by (
X, Y, Z , Z ) and ( X ′ , Y ′ , Z ′ , Z ′ ), respectively. We put ∆ X t := X t − X ′ t , ∆ Y t := Y t − Y ′ t and similarly for the others. Applying Itˆo’s formula to h ∆ X t , ∆ Y t i and using the estimates(4 . E (cid:2) h ∆ X T , ∆ Y T i (cid:3) ≤ − γ E Z T | ∆ X t | dt + ζC E Z T h | ∆ Y t | ( | ∆ y t | + E [∆ y t |F t ]) + | ∆ X t | ( | ∆ x t | + E [ | ∆ y t |F t ]) i dt , E (cid:2) h ∆ X T , ∆ Y T i (cid:3) ≥ ( ̺γ + (1 − ̺ )) E [ | ∆ X T | ] − ζC E (cid:2) | ∆ X T | ( | ∆ x T | + E [ | ∆ x T ||F T ]) (cid:3) , with some ̺ -independent constant C . Let us set γ c := min(1 , γ ) >
0. Then one easily confirms0 < γ c ≤ ̺γ + (1 − ̺ ) for any ̺ ∈ [0 , γ c E h | ∆ X T | + Z T | ∆ X t | dt i ≤ ζC E (cid:2) | ∆ X T | ( | ∆ x T | + E [ | ∆ x T ||F T ]) (cid:3) + ζC E Z T h | ∆ Y t | ( | ∆ y t | + E [∆ y t |F t ]) + | ∆ X t | ( | ∆ x t | + E [ | ∆ y t |F t ]) i dt . Using Young’s inequality and and a new constant C , we get E [ | ∆ X T | ] + E Z T | ∆ X t | dtd ≤ ζC E Z T (cid:0) | ∆ Y t | + ( | ∆ x t | + | ∆ y t | ) (cid:1) dt + ζC E [ | ∆ x T | ] . (4.6)Treating X, X ′ as inputs, the standard estimates for the Lipschitz BSDEs (see, for example,Theorem 4.2.3 in [25]) gives E h sup t ∈ [0 ,T ] | ∆ Y t | + Z T ( | ∆ Z t | + | ∆ Z t | ) dt i ≤ C E h | ∆ X T | + Z T | ∆ X t | dt i + ζC E h | ∆ x T | + Z T ( | ∆ x t | + | ∆ y t | ) dt i . Combining with (4 .
6) and choosing ζ > E h sup t ∈ [0 ,T ] | ∆ Y t | + Z T ( | ∆ Z t | + | ∆ Z t | ) dt i ≤ ζC E h | ∆ x T | + Z T ( | ∆ x t | + | ∆ y t | ) dt i . (4.7)10y the similar procedures, we also have E h sup t ∈ [0 ,T ] | ∆ X t | i ≤ ζC E h | ∆ x T | + Z T ( | ∆ x t | + | ∆ y t | ) dt i . (4.8)From (4 .
7) and (4 . E h sup t ∈ [0 ,T ] | ∆ X t | + sup t ∈ [0 ,T ] | ∆ Y t | + Z T ( | ∆ Z t | + | ∆ Z t | ) dt i ≤ ζC E h sup t ∈ [0 ,T ] | ∆ x t | + sup t ∈ [0 ,T ] | ∆ y t | + Z T ( | ∆ z t | + | ∆ z t | ) dt i . Thus there exists ζ >
0, being independent of the size of ̺ , that makes the map ( x, y, z , z ) ( X, Y, Z , Z ) strict contraction. Therefore the initial hypothesis holds true for ( ̺ + ζ ), whichestablishes the existence. The uniqueness follows from the next proposition. Proposition 4.1.
Given two set of inputs ( ξ, c , c ) , ( ξ ′ , c ′ , c ′ ) , coefficients ( δ, Λ) , ( δ ′ , Λ ′ ) and thecoefficient functions ( l, σ , σ, f , g ) , ( l ′ , σ ′ , σ ′ , f ′ , g ′ ) satisfying Assumptions (MFG-a,b,c1) , let usdenote the corresponding solutions to (4 . by ( X, Y, Z , Z ) and ( X ′ , Y ′ , Z ′ , Z ′ ) , respectively.We also define the functionals ( B, F, G ) and ( B ′ , F ′ , G ′ ) by (4 . with corresponding coefficients,respectively. Then, we have the following stability result: E h sup t ∈ [0 ,T ] | ∆ X t | + sup t ∈ [0 ,T ] | ∆ Y t | + Z T ( | ∆ Z t | + | ∆ Z t | ) dt i ≤ C E h | ∆ ξ | + | G | + Z T (cid:16) | F ( t ) | + | B ( t ) | + | σ ( t ) | + | σ ( t ) | (cid:17) dt i , where C is a constant depending only on T as well as the Lipschitz constants of the system, and B ( t ) := B ( t, Y ′ t , c t , c t ) − B ′ ( t, Y ′ t , c ′ t , c ′ t ) ,F ( t ) := F ( t, X ′ t , Y ′ t , c t , c t ) − F ′ ( t, X ′ t , Y ′ t , c ′ t , c ′ t ) , ( σ , σ )( t ) = ( σ ( t, c t , c t ) − σ ′ ( t, c ′ t , c ′ t ) , σ ( t, c t , c t ) − σ ′ ( t, c ′ t , c ′ t )) ,G := G ( X ′ T , c T , c T ) − G ′ ( X ′ T , c ′ T , c ′ T ) , and ∆ ξ := ξ − ξ ′ , ∆ X t := X t − X ′ t and similarly for the other variables.Proof. Let us put ∆ B ( t ) := B ( t, Y t , c t , c t ) − B ( t, Y ′ t , c t , c t ), ∆ F ( t ) := F ( t, X t , Y t , c t , c t ) − F ( t, X ′ t , Y ′ t , c t , c t )and ∆ G := G ( X T , c T , c T ) − G ( X ′ T , c T , c T ). We get by Itˆo’s formula that E (cid:2) h ∆ X T , ∆ G + G i (cid:3) = E h h ∆ ξ, ∆ Y i + Z T (cid:16) h F ( t ) , ∆ X t i + h B ( t ) , ∆ Y t i + h σ ( t ) , ∆ Z t i + h σ ( t ) , ∆ Z t i + (cid:0) h ∆ F ( t ) , ∆ X t i + h ∆ B ( t ) , ∆ Y t i (cid:1)(cid:17) dt i . . γ E h | ∆ X T | + Z T | ∆ X t | dt i ≤ E h h ∆ ξ, ∆ Y i − h ∆ X T , G i + Z T (cid:16) h F ( t ) , ∆ X t i + h B ( t ) , ∆ Y t i + h σ ( t ) , ∆ Z t i + h σ ( t ) , ∆ Z t i (cid:17) dt i . (4.9)On the other hand, the standard estimates for Lipschitz SDEs and BSDEs give E h sup t ∈ [0 ,T ] | ∆ Y t | + Z T ( | ∆ Z t | + | ∆ Z t | ) dt i ≤ C E h | G | + Z T | F ( t ) | dt i + C E h | ∆ X T | + Z T | ∆ X t | dt i , (4.10) E h sup t ∈ [0 ,T ] | ∆ X t | i ≤ C E h | ∆ ξ | + Z T (cid:2) | B ( t ) | + | σ ( t ) | + | σ ( t ) | (cid:3) dt i + C E Z T | ∆ Y t | dt . Combining the above inequalities (4 .
9) and (4 .
10) gives E h sup t ∈ [0 ,T ] | ∆ X t | + sup t ∈ [0 ,T ] | ∆ Y t | + Z T ( | ∆ Z t | + | ∆ Z t | ) dt i ≤ C E h | ∆ ξ | + | G | + Z T (cid:2) | F ( t ) | + | B ( t ) | + | σ ( t ) | + | σ ( t ) | (cid:3) dt i + C E h h ∆ ξ, ∆ Y i − h ∆ X T , G i + Z T (cid:2) h F ( t ) , ∆ X t i + h B ( t ) , ∆ Y t i + h σ ( t ) , ∆ Z t i + h σ ( t ) , ∆ Z t i (cid:3) dt i . Now simple application of Young’s inequality establishes the claim.
Corollary 4.1.
Under Assumptions (MFG-a,b,c1) , the solution ( X, Y, Z , Z ) to the FBSDE (4 . satisfies the following estimate: E h sup t ∈ [0 ,T ] | X t | + sup t ∈ [0 ,T ] | Y t | + Z T ( | Z t | + | Z t | ) dt i ≤ C E h | ξ | + | ∂ x g (0 , c T , c T ) | + Z T (cid:16) | ∂ x f ( t, , , c t , c t ) | + | l ( t, , c t , c t ) | + | ( σ , σ )( t, c t , c t ) | (cid:17) dt i , where C is a constant depending only on T, δ and Lipschitz constants of the system.Proof.
By quick inspection of the proof for Proposition 4.1, one can confirm that as long asthere exists a solution ( X ′ , Y ′ , Z ′ , Z ′ ) ∈ S × S × H × H , their coefficients need not sat-isfy Assumption (MFG-a,b,c1). In particular, by putting ξ ′ and ( l ′ , σ ′ , σ ′ , f ′ , g ′ ) all zero, wehave a trivial solution ( X ′ , Y ′ , Z ′ , Z ′ ) = (0 , , , Securities of maturity T with exogenously specified payoff If we consider the exchange markets of bonds and futures, or other financial derivatives withmaturity T , those securities cease to exist at T after paying exogenously specified amount of12ash c T . In this case, it is natural to consider with δ = 0 and g ( x, c ) = g ( x, c ) := −h c , x i , (4.11)since there is no reason to put penalty on the outstanding volume at T . In this case, the terminalfunction g in (4 .
11) does not have the strict convexity. Fortunately, even in this case, we canprove the unique existence as well as the stability result of the same form.
Assumption 4.3. (MFG-c2)(i) The functions σ and σ are independent of the argument ̟ .(ii) For any t ∈ [0 , T ] , any random variables x, x ′ , c , c ∈ L ( F ; R n ) and any sub- σ -field G ⊂ F ,the function l satisfies the monotone condition with some positive constant γ l > : E h h l ( t, E [ x |G ] , c , c ) − l ( t, E [ x ′ |G ] , c , c ) , x − x ′ i i ≥ γ l E (cid:2) E [ x − x ′ |G ] (cid:3) . (iii) γ := γ f − L ̟ γ l is strictly positive and the terminal function g is given by (4 . with δ = 0 . Theorem 4.3.
Under Assumptions (MFG-a,b,c2) , there exists a unique strong solution ( X, Y, Z , Z ) ∈ S ( F ; R n ) × S ( F ; R n ) × H ( F ; R n × d ) × H ( F ; R n × d ) to the FBSDE (4 . . Moreover, the sameform of stability and L estimates given in Proposition 4.1 and Corollary 4.1 hold.Proof. Note that, in this case, the terminal condition for the BSDE is independent of X T . Thus,as in Theorem 2.3 [24], we put y ̺T = Y T = − c T in (4 .
4) and (4 . h ∆ X T , ∆ Y T i = 0, one can follow the same arguments to get the desired result. The proofof the stability result can also be done in almost exactly the same way. We are now ready to investigate if our FBSDE (4 .
1) actually provides a good approximate ofthe market price and if so, how accurate it is. By Theorem 3.1, if we use ( − E (cid:2) Y t |F t (cid:3) ) t ∈ [0 ,T ] as the input ( ̟ t ) t ∈ [0 ,T ] , where ( Y t ) t ∈ [0 ,T ] is the unique solution to the FBSDE (4 .
1) with theconvention ξ = ξ and c = c , the optimal strategy of the individual agent is given by b α i mf ( t ) := b α ( Y it , − E [ Y t |F t ]) = − Λ( Y it − E [ Y t |F t ]) (5.1)where ( Y it ) t ∈ [0 ,T ] is the solution to (3 .
4) with ( ̟ t = − E (cid:2) Y t |F t (cid:3) ) t ∈ [0 ,T ] . Theorem 5.1.
If the conditions for Theorem 4.1, Theorem 4.2 or Theorem 4.3 are satisfiedthen we have lim N →∞ E Z T (cid:12)(cid:12)(cid:12) N N X i =1 b α i mf ( t ) (cid:12)(cid:12)(cid:12) dt = 0 . Moreover if there exists some constant Γ such that sup t ∈ [0 ,T ] E (cid:2) | Y t | q (cid:3) q ≤ Γ < ∞ for some q > ,then there exists some constant C independent of N such that E Z T (cid:12)(cid:12)(cid:12) N N X i =1 b α i mf ( t ) (cid:12)(cid:12)(cid:12) dt ≤ C Γ ǫ N , (5.2)13 here ǫ N := N − / max( n, (cid:0) N ) { n =4 } (cid:1) .Proof. Let us consider the following set of FBSDEs with 1 ≤ i ≤ N on the filtered probabilityspace (Ω , F , P ; F ) constructed in Section 2. dX it = (cid:16) − Λ( Y it − E [ Y it |F t ]) + l ( t, − E [ Y it |F t ] , c t , c it ) (cid:17) dt + σ ( t, − E [ Y it |F t ] , c t , c it ) dW t + σ ( t, − E [ Y it |F t ] , c t , c it ) dW it ,dY it = − ∂ x f ( t, X it , − E [ Y it |F t ] , c t , c it ) dt + Z i, t dW t + Z it dW it , with X i = ξ i and Y iT = δ/ (1 − δ ) E [ ∂ x g ( X iT , c T , c iT ) |F T ] + ∂ x g ( X iT , c T , c iT ). Thanks to theexistence of unique strong solution, Yamada-Watanabe Theorem for FBSDEs (see, Theorem1.33 [8]), there exists some measurable function Φ such that for every 1 ≤ i ≤ N ,( X it , Y it ) t ∈ [0 ,T ] = Φ (cid:16) ( c t ) t ∈ [0 ,T ] , ( W t ) t ∈ [0 ,T ] , ξ i , ( c it ) t ∈ [0 ,T ] , ( W it ) t ∈ [0 ,T ] (cid:17) . Hence, conditionally on F , the set of proceses ( X it , Y it ) t ∈ [0 ,T ] with 1 ≤ i ≤ N are independentlyand identically distributed. In particular, we have P -a.s. E [ Y it |F t ] = E [ Y t |F t ] , ∀ t ∈ [0 , T ] , E [ ∂ x g ( X iT , c T , c iT ) |F T ] = E [ ∂ x g ( X T , c T , c T ) |F T ] . (5.3)Note that, under the convention ξ = ξ and c = c , we actually have ( X , Y ) = ( X, Y ). From(5 . X it , Y it , Z i, t , Z it ) t ∈ [0 ,T ] = ( X it , Y it , Z i, t , Z it ) t ∈ [0 ,T ] in S ( F i ) × S ( F i ) × H ( F i ) × H ( F i ). Therefore,1 N N X i =1 b α i mf ( t ) = − Λ (cid:16) N N X i =1 Y it − E [ Y t |F t ] (cid:17) . (5.4)We can easily check that E h W (cid:16) N N X i =1 δ Y it , L ( Y t |F t ) (cid:17) (cid:12)(cid:12)(cid:12) F t i ≤ N N X i =1 E (cid:2) | Y it | |F t (cid:3) + 2 E (cid:2) | Y t | |F t (cid:3) = 4 E (cid:2) | Y t | |F t (cid:3) . Since ( Y it ) ≤ i ≤ N are F t -conditionally independently and identically distributed and also Y ∈ S , the same arguments leading to (2 .
14) in [8] imply that the pointwise convergence holds:lim N →∞ E h W (cid:16) N N X i =1 δ Y it , L ( Y t |F t ) (cid:17) i = 0 . (5.5)We are now going to show that the set of functions, ( f N ) N ∈ N defined by[0 , T ] ∋ t f N ( t ) := E (cid:2) W (cid:0) µ t , µ t (cid:1) (cid:3) ∈ R with µ t := N P Ni =1 δ Y it and µ t := L ( Y t |F t ) are precompact in the set C ([0 , T ]; R ) endowed with14he topology of uniform convergence. In fact, uniformly in N ,sup t ∈ [0 ,T ] | f N ( t ) | ≤ t ∈ [0 ,T ] E (cid:2) | Y t | (cid:3) ≤ C < ∞ (5.6)where C is given by the estimate in Corollary 4.1. Moreover, for any 0 ≤ t, s ≤ T , Cauchy-Schwarz, (5 .
6) and the triangular inequalities give | f N ( t ) − f N ( s ) | ≤ E h(cid:16) W ( µ t , µ t ) + W ( µ s , µ ) (cid:17) i E h(cid:16) W ( µ t , µ t ) − W ( µ s , µ ) (cid:17) i ≤ C E h(cid:16) W ( µ t , µ t ) − W ( µ s , µ s ) (cid:17) i ≤ C E h W ( µ t , µ s ) + W ( µ t , µ s ) i . ≤ C E h N N X i =1 | Y it − Y is | + | Y t − Y s | i ≤ C E (cid:2) | Y t − Y s | (cid:3) , uniformly in N , where we have used the fact that ( Y i ) i ≥ are conditionally i.i.d at the lastinequality. Since ( Y t ) t ∈ [0 ,T ] is a continuous process, the above estimate tells that ( f N ) N ∈ N is equicontinuous, which is also uniformly equicontinuous since we are working on the finiteinterval. Now, Arzela-Ascoli theorem implies the desired precompactness.Combining with the pointwise convergence (5 . N →∞ sup t ∈ [0 ,T ] E h W (cid:16) N N X i =1 δ Y it , L ( Y t |F t ) (cid:17) i = 0 . (5.7)From the definition of Wasserstein distance (2 . (cid:12)(cid:12)(cid:12) N N X i =1 Y it − E [ Y t |F t ] (cid:12)(cid:12)(cid:12) ≤ W (cid:16) N N X i =1 δ Y it , L ( Y t |F t ) (cid:17) , and hence, from (5 . E Z T (cid:12)(cid:12)(cid:12) N N X i =1 b α i mf ( t ) (cid:12)(cid:12)(cid:12) dt ≤ C sup t ∈ [0 ,T ] E h W (cid:16) N N X i =1 δ Y it , L ( Y t |F t ) (cid:17) i . (5.8)The first conclusion now follows from (5 . .
8) and the (Fourth Step) in the proof of Theorem 2.12 in [8].Theorem 5.1 justifies our intuitive understanding and a special type of FBSDEs (4 .
1) derivedin Section 3 as a reasonable model to approximate the market clearing price. When there existshigher integrability, Glivenko-Cantelli convergence theorem in the Wasserstein distance evenprovides a specific order ǫ N of convergence in terms of the number of agents N (5 . Remark 5.1.
Consider the situation treated in Theorem 4.3, for example, a market model ofa Futures contract. If the contract pays unit amount of the underlying asset per contract whosevalue is exogenously given by c T , our mean-field limit model (4 . gives Y T = − c T . This means hat the modeled Futures price satisfies ̟ T = − E [ Y T |F T ] = c T , which guarantees the convergenceof the modeled price to the value of the underlying asset at the maturity T . This is a cruciallyimportant feature that any market model of this type of securities must satisfy. The main limitation of the last model is that there exists only one type of agents who share thecommon cost functions as well as the coefficient functions for their state dynamics. Interest-ingly, it is rather straightforward to extend the model to the situation with multiple populations,where the agents in each population share the same cost and coefficient functions but they canbe different population by population. From the perspective of the practical applications, thisis a big advantage since we can analyze, for example, the interactions between the Sell-side andBuy-side institutions for financial applications, or consumers and producers for economic appli-cations. For general issues of mean field games as well as mean field type control problems in thepresence of multiple populations without common noise, see Fujii [13]. Although there exists acommon noise in the current model, the conditional law only enters as a form of expectation.Therefore, as long as the system of FBSDEs is Lipschitz continuous, there exists a unique strongsolution at least for small T . For general T , although it is rather difficult to find appropriateset of assumptions, it is still possible for some simple cases. In this section, our main task is tofind an appropriate limit model that extends (4 .
1) for multiple populations and the sufficientconditions that make appropriate monotone conditions hold, which guarantees the existence ofunique solution.In the following, we shall treat m populations indexed by p ∈ { , · · · , m } . For each p , N p ≥ p, i ) the ithagent in the population p . First, let us enlarge the probability space constructed in Sec-tion 2. In addition to (Ω , F , P ; F ), we introduce (Ω p,i , F p,i , P p,i ; P p,i ) with 1 ≤ i ≤ N p and 1 ≤ p ≤ m , each of which is generated by ( ξ p,i , W p,i ) with d -dimensional Brownian motion W p,i and a W p,i -independent R n -valued square integrable random variable ξ p,i . For each p ,( ξ p,i ) N p i =1 are assumed to have the common law. We define (Ω p,i , F p,i , P p,i ; F p,i ) as the product of(Ω , F , P ; F ) and (Ω p,i , F p,i , P p,i ; P p,i ). Finally (Ω , F , P ; F ) is defined as a product of all thespaces (Ω , F , P ; F ) and (Ω p,i , F p,i , P p,i ; F p,i ), 1 ≤ i ≤ N p , ≤ p ≤ m , and (Ω i , F i , P i ; F i ) as aproduct of (Ω , F , P ; F ) and (Ω p,i , F p,i , P p,i ; F p,i ) with 1 ≤ p ≤ m . Every probability space isassumed to be complete and every filtration is assumed to be complete and right-continuouslyaugmented to satisfy the usual conditions.As we have done in Section 3, we first assume that the market price of n securities is givenexogenously by ̟ t ∈ H ( F ; R n ). Under this setup, we consider the control problem for each( p, i ) agent defined by inf α p,i ∈ A p,i J p,i ( α p,i ) , (6.1)with J p,i ( α p,i ) := E hZ T f p ( t, X p,it , α p,it , ̟ t , c t , c p,it ) dt + g p ( X p,iT , ̟ T , c T , c p,iT ) i , dX p,it = (cid:16) α p,it + l p ( t, ̟ t , c t , c p,it ) (cid:17) dt + σ p, ( t, ̟ t , c t , c p,it ) dW t + σ p ( t, ̟ t , c t , c p,it ) dW p,it with X p,i = ξ p,i . As before we assume ( c t ) t ≥ ∈ H ( F ; R n ) and ( c p,it ) t ≥ ∈ H ( F p,i ; R n ). Inaddition, within each population p , the random sources ( c p,it ) t ≥ are assumed to have a commonlaw 1 ≤ i ≤ N p . Admissible strategies A p,i is the space H ( F p,i ; R n ). The measurable functions f p : [0 , T ] × ( R n ) → R , g p : ( R n ) → R , f p : [0 , T ] × ( R n ) → R and g p : ( R n ) → R are given by f p ( t, x, α, ̟, c , c ) := h ̟, α i + 12 h α, Λ p α i + f p ( t, x, ̟, c , c ) ,g p ( x, ̟, c , c ) := − δ h ̟, x i + g p ( x, c , c ) . Assumption 6.1. (MFG-A)
We assume the following conditions uniformly in p ∈ { , · · · , m } .(i) Λ p is a positive definite n × n symmetric matrix with λI n × n ≤ Λ p ≤ λI n × n in the sense of2nd-order form where λ and λ are some constants satisfying < λ ≤ λ .(ii) For any ( t, x, ̟, c , c ) , | f p ( t, x, ̟, c , c ) | + | g p ( x, c , c ) | ≤ L (1 + | x | + | ̟ | + | c | + | c | ) . (iii) f p and g p are continuously differentiable in x and satisfy, for any ( t, x, x ′ , ̟, c , c ) , | ∂ x f p ( t, x ′ , ̟, c , c ) − ∂ x f p ( t, x, ̟, c , c ) | + | ∂ x g p ( x ′ , c , c ) − ∂ x g p ( x, c , c ) | ≤ L | x ′ − x | , and | ∂ x f p ( t, x, ̟, c , c ) | + | ∂ x g p ( x, c , c ) | ≤ L (1 + | x | + | ̟ | + | c | + | c | ) .(iv)The functions f p and g p are convex in x in the sense that for any ( t, x, x ′ , ̟, c , c ) , f p ( t, x ′ , ̟, c , c ) − f p ( t, x, ̟, c , c ) − h x ′ − x, ∂ x f p ( t, x, ̟, c , c ) i ≥ γ f | x ′ − x | ,g p ( x ′ , c , c ) − g p ( x, c , c ) − h x ′ − x, ∂ x g p ( x, c , c ) i ≥ γ g | x ′ − x | , with some constants γ f , γ g ≥ .(v) l p , σ p, , σ p are the measurable functions defined on [0 , T ] × ( R n ) and are R n , R n × d and R n × d -valued, respectively. Moreover they satisfy the linear growth condition: | ( l p , σ p, , σ p )( t, ̟, c , c ) | ≤ L (1 + | ̟ | + | c | + | c | ) for any ( t, ̟, c , c ) .(vi) δ ∈ [0 , is a given constant. Under Assumption (MFG-A), Theorem 3.1 guarantees that the control problem (6 .
1) foreach agent ( p, i ) is uniquely characterized by dX p,it = (cid:16)b α p ( Y p,it , ̟ t ) + l p ( t, ̟ t , c t , c p,it ) (cid:17) dt + σ p, ( t, ̟ t , c t , c p,it ) dW t + σ p ( t, ̟ t , c t , c p,it ) dW p,it ,dY p,it = − ∂ x f p ( t, X p,it , ̟ t , c t , c p,it ) dt + Z p,i, t dW t + Z p,it dW p,it , (6.2)with X p,i = ξ p,i and Y p,iT = − δ̟ T + ∂ x g p ( X p,iT , c T , c p,iT ). We have defined b α p ( y, ̟ ) := − Λ p ( y + ̟ )17nd Λ p := (Λ p ) − as before. There exists a unique strong solution ( X p,it , Y p,it , Z p,i, t , Z p,it ) t ∈ [0 ,T ] ∈ S ( F p,i ; R n ) × S ( F p,i ; R n ) × H ( F p,i ; R n × d ) × H ( F p,i ; R n × d ), and the optimal trading strategyfor the agent ( p, i ) is given by b α p,it = b α p ( Y p,it , ̟ t ) , ∀ t ∈ [0 , T ] . Let us check the market clearing condition under this setup. In order to balance the demandand supply of securities at the exchange, we need to have P mp =1 P N p i =1 b α ( Y p,it , ̟ t ) = 0. Thisrequires the market price to satisfy ̟ t = − (cid:16) m X p =1 n p Λ p (cid:17) − m X p =1 n p Λ p (cid:16) N p N p X i =1 Y p,it (cid:17) , where N = P mp =1 N p and n p := N p /N . At the moment, this is inconsistent to the initialassumption that requires ( ̟ t ) t ≥ to be F -adapted. However, since for each 1 ≤ p ≤ m ,( Y p,it ) N p i =1 are F -conditionally independently and identically distributed, we may follow the samearguments used in Section 3. If we take N → ∞ while keeping the relative size of populations n p constant, we can expect to obtain ̟ t = − ˆΞ m X p =1 ˆΛ p E [ Y p, t |F t ] (6.3)in the large population limit whereˆΛ p := n p Λ p , ˆΞ := (cid:16) m X p =1 ˆΛ p (cid:17) − . Remark 6.1.
When Λ p = Λ for every population p , one can easily check that (6 . becomes ̟ t = − m X p =1 n p E [ Y p, t |F t ] . Since Y of the adjoint equation represents the marginal cost i.e., the first order derivative ofthe value function with respect to the state variable x , the above expression of ̟ implies thatthe market price may be given by the population-weighted average of the marginal benefit (-cost)across the entire populations. By the observation we have just made, we are motivated to study the following limit problemwith 1 ≤ p ≤ m : dX pt = (cid:16)b α p (cid:0) Y pt , ̟ ( E [ Y t |F t ]) (cid:1) + l p (cid:0) t, ̟ ( E [ Y t |F t ]) , c t , c pt (cid:1)(cid:17) dt + σ p, (cid:0) t, ̟ ( E [ Y t |F t ]) , c t , c pt (cid:1) dW t + σ p (cid:0) t, ̟ ( E [ Y t |F t ]) , c t , c pt (cid:1) dW p, t ,dY pt = − ∂ x f p (cid:0) t, X pt , ̟ ( E [ Y t |F t ]) , c t , c pt (cid:1) dt + Z p, t dW t + Z pt dW p, t , (6.4)18ith X p = ξ p and Y pT = δ − δ ˆΞ m X p =1 ˆΛ p E (cid:2) ∂ x g p ( X pT , c T , c pT ) |F T (cid:3) + ∂ x g p ( X pT , c T , c pT ) . We put as before ξ p := ξ p, and c p := c p, to lighten the notation. Here, ̟ ( E [ Y t |F t ]) := − ˆΞ m X p =1 ˆΛ p E [ Y pt |F t ] , b α p ( y, ̟ ) := − Λ p ( y + ̟ )and hence (6 .
4) is actually an m -coupled system of FBSDEs of McKean-Vlasov type. One canderive the terminal condition from Y pT = − δ̟ ( E [ Y T |F T ]) + ∂ x g p ( X pT , c T , c pT ) , (6.5)by summing over 1 ≤ p ≤ m after taking conditional expectation given F T . In the following,we use the notation( X t , Y t , Z t , Z t ) t ∈ [0 ,T ] = (cid:16) ( X pt ) mp =1 , ( Y pt ) mp =1 , ( Z p, t ) mp =1 , ( Z pt ) mp =1 (cid:17) t ∈ [0 ,T ] . (6.6) T For small T , Lipschitz continuity suffices to guarantee the existence of a unique solution. Assumption 6.2. (MFG-B)
Uniformly in p ∈ { , · · · , m } , for any ( t, x, c , c ) ∈ [0 , T ] × ( R n ) and any ̟, ̟ ′ ∈ R n , thecoefficient functions l p , σ p, , σ p and f p satisfy | ( l p , σ p, , σ p )( t, ̟, c , c ) − ( l p , σ p, , σ p )( t, ̟ ′ , c , c ) | + | ∂ x f p ( t, x, ̟, c , c ) − ∂ x f p ( t, x, ̟ ′ , c , c ) | ≤ L ̟ | ̟ − ̟ ′ | . The following theorem follows exactly in the same way as Theorem 4.1.
Theorem 6.1.
Under Assumptions (MFG-A,B), there exists some constant τ > which de-pends only on ( L, L ̟ , δ, n p , Λ p ) such that for any T ≤ τ , there exists a unique strong solution ( X, Y, Z , Z ) ∈ S (cid:0) F ; ( R n ) m (cid:1) × S (cid:0) F ; ( R n ) m (cid:1) × H (cid:0) F ; ( R n × d ) m (cid:1) × H (cid:0) F ; ( R n × d ) m (cid:1) to theFBSDE (6 . . Remark 6.2.
Note that the above system of FBSDEs becomes a linear-quadratic form by choos-ing ( l p , σ p, , σ p , f p , g p ) appropriately. In this case, the problem reduces to solving ordinary dif-ferential equations of Riccati type. Therefore, the existence of a solution for a given T can betested, at lest numerically, by checking the absence of a “blow up” in its solution. T We now move on to the existence result of a unique solution for general T . It is very difficult tofind general existence criteria for fully-coupled multi-dimensional FBSDEs. A the moment, inorder to apply well-known Peng-Wu’s method, let us put the following simplifying assumptions.19 ssumption 6.3. (MFG-C1) (i) For every ≤ p ≤ m , the functions σ p, and σ p are independent of the argument ̟ .(ii) Λ p = Λ and n p = 1 /m for every p .(iii) For any t ∈ [0 , T ] , any random variables x p , x p ′ , c , c p ∈ L ( F ; R n ) and any sub- σ -field G ⊂ F , the functions ( l p ) mp =1 satisfy with some positive constant γ l > , m X p =1 E h(cid:10) l p (cid:0) t, E [ x |G ] , c , c (cid:1) − l p (cid:0) t, E [ x ′ |G ] , c , c p (cid:1) , x p − x p ′ (cid:11)i ≥ mγ l E (cid:2) E [ x − x ′ |G ] (cid:3) , where x := m P mp =1 x p and similarly for x ′ .(iv) There exists a strictly positive constant γ satisfying < γ ≤ (cid:16) γ f − L ̟ γ l (cid:17) ∧ γ g . Moreover,the functions ( g p ) mp =1 satisfy for any x p , x p ′ , c , c p ∈ L ( F ; R n ) and any sub- σ -field G ⊂ F , δ − δ m − E h(cid:10) m X p =1 E [ ∂ x g p ( x p , c , c p ) − ∂ x g p ( x p ′ , c , c p ) |G ] , m X p =1 ( x p − x p ′ ) (cid:11)i + γ g m X p =1 E [ | x p − x p ′ | ] ≥ γ m X p =1 E [ | x p − x p ′ | ] . Remark 6.3.
The conditions (iii) and (iv) in the above assumption are rather restrictive. Thecondition (iii) is satisfied, for example, if l p has a separable form l p = h ( x )+ h p ( c t , c pt ) with somefunction h , which is common to every population and strictly monotone. (iv) is also satisfied byrequiring similar structure. Or, since ∂ x g p is Lipschitz continuous in x , the absolute value ofthe first term is bounded by δ − δ max(( L p ) mp =1 ) P mp =1 E | x p − x p ′ | , where the L p is the Lipschitzconstant for ∂ x g p . Thus the condition (iv) is satisfied if δ max(( L p ) mp =1 ) is sufficiently small. The next result is the counterpart of Theorem 4.2.
Theorem 6.2.
Under Assumptions (MFG-A,B,C1) , there exists a unique strong solution ( X, Y, Z , Z ) ∈ S (cid:0) F ; ( R n ) m (cid:1) × S (cid:0) F ; ( R n ) m (cid:1) × H (cid:0) F ; ( R n × d ) m (cid:1) × H (cid:0) F ; ( R n × d ) m (cid:1) to the FBSDE (6 . .Moreover, the same form of stability and L estimates given in Proposition 4.1 and Corollary 4.1hold.Proof. Under Assumption (MFG-C1), (6 .
4) can be written as dX pt = n − Λ (cid:16) Y pt − m m X p =1 E [ Y pt |F t ] (cid:17) + l p (cid:16) t, − m m X p =1 E [ Y pt |F t ] , c t , c pt (cid:17)o dt + σ p, ( t, c t , c pt ) dW t + σ p ( t, c t , c pt ) dW p, t ,dY pt = − ∂ x f p (cid:16) t, X pt , − m m X p =1 E [ Y pt |F t ] , c t , c pt (cid:17) dt + Z p, t dW t + Z pt dW p, t , with X p = ξ p and Y pT = δ − δ m m X p =1 E (cid:2) ∂ x g p ( X pT , c T , c pT ) |F T (cid:3) + ∂ x g p ( X pT , c T , c pT ) . p , let us define the functionals B p , F p and G p for any y p , x p , c , c p ∈ L ( F ; R n ) with y := ( y p ) mp =1 , x := ( x p ) mp =1 and c := ( c p ) mp =1 by B p ( t, y, c , c p ) := − Λ (cid:16) y p − m m X p =1 E [ y p |F t ] (cid:17) + l p (cid:16) t, − m m X p =1 E [ y p |F t ] , c , c p (cid:17) F p ( t, x p , y, c , c p ) := − ∂ x f (cid:16) t, x p , − m m X p =1 E [ y p |F t ] , c , c p (cid:17) ,G p ( x, c , c ) := δ − δ m m X p =1 E [ ∂ x g p ( x p , c , c p ) |F T ] + ∂ x g p ( x p , c , c p ) , and set B ( t, y, c , c ) := ( B p ( t, y, c , c p )) mp =1 , F ( t, x, y, c , c ) := ( F p ( t, x p , y, c , c p )) mp =1 and G ( x, c , c ) :=( G p ( x, c , c )) mp =1 . With ∆ y := y − y ′ and ∆ x := x − x ′ , we have from (MFG-C1)(iii), E h h B ( t, y, c , c ) − B ( t, y ′ , c , c ) , ∆ y i i := m X p =1 E h h B p ( t, y, c , c ) − B p ( t, y ′ , c , c ) , ∆ y p i i ≤ − m X p =1 E [ h ∆ y p , Λ∆ y p i ] + 1 m E h(cid:10) m X p =1 E [∆ y p |F t ] , Λ m X p =1 ∆ y p (cid:11)i − mγ l E h(cid:16) m m X p =1 E [∆ y p |F t ] (cid:17) i ≤ − mγ l E h(cid:16) m m X p =1 E [∆ y p |F t ] (cid:17) i . (6.7)There exists a orthogonal matrix P such that P ⊤ Λ P becomes diagonal. Then working on thenew basis ˆ y p = P ⊤ ∆ y p , 1 ≤ p ≤ m , the last inequality of (6 .
7) can be checked componentby component 1 ≤ i ≤ n by the fact ( P mp =1 ˆ y pi ) ≤ m P mp =1 | ˆ y pi | . Second, from (MFG-A)(iv),(MFG-B) and Cauchy-Schwarz inequality, E h h F ( t, x, y, c , c ) − F ( t, x ′ , y ′ , c , c ) , ∆ x i i ≤ − (cid:16) γ f − L ̟ γ l (cid:17) E [ | ∆ x | ] + mγ l E h(cid:16) m m X p =1 E [∆ y pt |F t ] (cid:17) i . (6.8)Finally, from (MFG-A, C1)(iv), we immediately get E h h G ( x, c , c ) − G ( x ′ , c , c ) , ∆ x i i ≥ γ E [ | ∆ x | ] . Now we have established the monotone conditions corresponding to (4 .
3) for the current model.We can now repeat the same procedures in the proof of Theorem 4.2 and Proposition 4.1.Let us give the results for the securities of maturity T with exogenously specified payoff. Assumption 6.4. (MFG-C2) (i) For every ≤ p ≤ m , the functions σ p, and σ p are independent of the argument ̟ .(ii) Λ p = Λ and n p = 1 /m for every p .(iii) For any t ∈ [0 , T ] , any random variables x p , x p ′ , c , c p ∈ L ( F ; R n ) and any sub- σ -field ⊂ F , the functions ( l p ) mp =1 satisfy with some positive constant γ l > , m X p =1 E h(cid:10) l p (cid:0) t, E [ x |G ] , c , c (cid:1) − l p (cid:0) t, E [ x ′ |G ] , c , c p (cid:1) , x p − x p ′ (cid:11)i ≥ mγ l E (cid:2) E [ x − x ′ |G ] (cid:3) , where x := m P mp =1 x p and similarly for x ′ .(iv) γ := γ f − L ̟ γ l is strictly positive. Moreover, δ = 0 and the terminal function g p is given by g p ( x, c ) = g p ( x, c ) := −h c , x i (6.9) for every ≤ p ≤ m . Theorem 6.3.
Under Assumptions (MFG-A,B,C2) , there exists a unique strong solution ( X, Y, Z , Z ) ∈ S (cid:0) F ; ( R n ) m (cid:1) × S (cid:0) F ; ( R n ) m (cid:1) × H (cid:0) F ; ( R n × d ) m (cid:1) × H (cid:0) F ; ( R n × d ) m (cid:1) to the FBSDE (6 . .Moreover, the same form of the stability and L estimates given in Proposition 4.1 and Corol-lary 4.1 holds.Proof. Using the inequalities (6 .
7) and (6 .
8) with P mp =1 h ∆ X pT , ∆ Y pT i = 0, we can follow thesame arguments in the proof of Theorem 4.3. At the last part of this section, we investigate the asymptotic market clearing in the presence ofmultiple populations. As in Section 5, we define ( ̟ t ) t ∈ [0 ,T ] using the solution to the system ofthe mean-field FBSDEs: ̟ t = ̟ ( E [ Y t |F t ]) := − ˆΞ m X p =1 ˆΛ p E (cid:2) Y pt |F t (cid:3) where ( Y pt ) mp =1 is the solution of (6 . ̟ t ) t ∈ [0 ,T ] as amarket clearing price, we solve the individual agent problem (6 .
1) with this ̟ as an input. Thecorresponding individual problem (6 .
1) for the agent ( p, i ) is given by the unique strong solution( X p,i , Y p,i , Z p,i, , Z p,i ) of (6 . p, i ) is then given by b α p,i mf ( t ) := − Λ p (cid:16) Y p,it − ˆΞ m X q =1 ˆΛ q E (cid:2) Y qt |F t (cid:3)(cid:17) , ∀ t ∈ [0 , T ] . Theorem 6.4.
If the conditions for Theorem 6.1, Theorem 6.2 or Theorem 6.3 are satisfiedthen we have lim N →∞ E Z T (cid:12)(cid:12)(cid:12) N m X p =1 N p X i =1 b α p,i mf ( t ) (cid:12)(cid:12)(cid:12) dt = 0 , where N := P mp =1 N p and the limit is taken while keeping ( n p := N p /N ) ≤ p ≤ m constant. More-over if there exists some constant Γ such that sup t ∈ [0 ,T ] E (cid:2) | Y t | q (cid:3) q ≤ Γ < ∞ for some q > , then here exists some constant C independent of N such that E Z T (cid:12)(cid:12)(cid:12) N m X p =1 N p X i =1 b α p,i mf ( t ) (cid:12)(cid:12)(cid:12) dt ≤ C Γ ǫ N , where ǫ N := N − / max( n, (cid:0) N ) { n =4 } (cid:1) .Proof. By the definition of b α p,i mf , we have1 N m X p =1 N p X i =1 b α p,i mf ( t ) = − N m X p =1 N p X i =1 Λ p (cid:16) Y p,it − ˆΞ m X q =1 ˆΛ q E [ Y qt |F t ] (cid:17) = − m X p =1 ˆΛ p (cid:16) N p N p X i =1 Y p,it − E [ Y pt |F t ] (cid:17) . (6.10)On the other hand, we have for each 1 ≤ p ≤ m , 1 ≤ i ≤ N p , dX p,it = (cid:16)b α p (cid:0) Y p,it , ̟ ( E [ Y t |F t ]) (cid:1) + l p (cid:0) t, ̟ ( E [ Y t |F t ]) , c t , c p,it (cid:1)(cid:17) dt + σ p, (cid:0) t, ̟ ( E [ Y t |F t ]) , c t , c p,it (cid:1) dW t + σ p (cid:0) t, ̟ ( E [ Y t |F t ]) , c t , c p,it (cid:1) dW p,it ,dY p,it = − ∂ x f p (cid:0) t, X p,it , ̟ ( E [ Y t |F t ]) , c t , c p,it (cid:1) dt + Z p,i, t dW t + Z p,it dW p,it , with X p,i = ξ p,i , Y p,iT = − δ̟ ( E [ Y T |F T ]) + ∂ x g p ( X p,iT , c T , c p,iT ) . By the unique strong solvability, Yamada-Watanabe theorem implies that there exists somefunction Φ p for each 1 ≤ p ≤ m such that for every 1 ≤ i ≤ N p ,( Y p,it ) t ∈ [0 ,T ] = Φ p (cid:16) c , ( W t ) t ∈ [0 ,T ] , ( E [ Y qt |F t ] t ∈ [0 ,T ] ) ≤ q ≤ m , ξ p,i , ( c p,it ) t ∈ [0 ,T ] , ( W p,it ) t ∈ [0 ,T ] (cid:17) . Hence ( Y p,it ) t ∈ [0 ,T ] , ≤ i ≤ N p are independently and identically distributed conditionally on F . Inparticular, we have E [ Y p,it |F t ] = E [ Y p, t |F t ].We now compare ( X p, t , Y p, t , Z p, , t , Z p, t ) t ∈ [0 ,T ] with ( X pt , Y pt , Z p, t , Z pt ) t ∈ [0 ,T ] by treating ̟ ( E [ Y t |F t ])as external inputs. Note that the terminal condition of the latter satisfies the relation (6 . Y p, t ) t ∈ [0 ,T ] = ( Y pt ) t ∈ [0 ,T ] in S ( F p, ; R n ). As a result we have obtained E [ Y pt |F t ] = E [ Y p, t |F t ]. Using the expression (6 . N m X p =1 N p X i =1 b α p,i mf ( t ) = − m X p =1 ˆΛ p (cid:16) N p N p X i =1 Y p,it − E [ Y p, t |F t ] (cid:17) . We can now repeat the last part of the proof for Theorem 5.1.23
Concluding Remarks and Further Extensions
In this work, we have studied endogenous formation of market clearing price using a stylizedmodel of a security exchange. We have derived a special type of FBSDE of McKean-Vlasovtype with common noise whose solution provides a good approximate of the equilibrium price.In addition to the existence of strong unique solution to the FBSDE, we have proved that themodeled price asymptotically clear the market in the large N -limit. We also gave the order ofconvergence ǫ N when the solution of the FBSDE possesses higher order of integrability. In thefollowing, let us list up of a further extension of our technique and some interesting topics forfuture projects: • Dependence on the conditional law of the state : For applications to energy and com-modity markets, or economic models with producers and consumers, one may want to studythe cost functions ( f , g ) depending on the empirical distribution of the sate X of the agentssuch as f (cid:16) t, X it , N P Nj =1 δ X jt , ̟ t , c t , c it (cid:17) . Under the setup with conditional independence, thecost function for the limit problem is naturally given by f (cid:16) t, X t , L ( X t |F t ) , ̟ t , c t , c t (cid:17) . Even inthis case, the resultant FBSDE (4 .
1) is solvable, at least for small T , if ( ∂ x f , ∂ x g ) are Lipschitzcontinuous in the measure argument with respect to W -distance. Under the stronger assump-tion guaranteeing the monotone conditions (4 . T . As long as the source of common noise is solely from the filtration F generated by W , we can avoid subtleties regarding the admissibility (so-called H -hypothesis).See Remark 2.10 in [8] as a useful summary for this issue. • Explicit solution : If we chose f , g as quadratic functions and l, σ , σ as affine functions,we obtain a linear-quadratic mean field game with common noise. In this case, an explicit so-lution may be available where the coefficients functions are given as the solutions to differentialequations of Riccati type. • Property of market price process : It seems interesting to study the properties of themarket clearing price theoretically and numerically. For example, if n = d the equivalentmartingale measure (EMM) can be uniquely determined. Based on the payoff distribution c and the cost functions of the agents ( f , g ), one may study how the market price process underthe EMM behaves, for example, the relation between the skew of its implied volatility and therisk-averseness of the agents. References [1] Achdou, Y., J.Buera, F., Lasry, J., Lions, P. and Moll, B., 2014,
Partial differential equation modelsin macroeconomics , Philosophical Transaction of The Royal Society, A 372:20130397.[2] Alasseur, C., Ben Taher, I., Matoussi, A., 2020,
An extended mean field games for storage in smartgrids , Journal of Optimization Theory and Applications, 184: 644-670.[3] Bensoussan, A., Frehse, J. and Yam, P., 2013,
Mean field games and mean field type control theory ,SpringerBriefs in Mathematics, NY.[4] Carmona. R. and Delarue, F., 2013,
Mean field forward-backward stochastic differential equations ,Electron. Commun. Probab., Vol. 18, No. 68, pp. 1-15.
5] Carmona, R. and Delarue, F., 2013,
Probabilistic analysis of mean-field games , SIAM J. Control.Optim., Vol. 51, No. 4, pp. 2705-2734.[6] Carmona, R. and Delarue, F., 2015,
Forward-backward stochastic differential equations and controlledMcKean-Vlasov dynamics , The Annals of Probability, Vol. 43, No. 5, pp. 2647-2700.[7] Carmona, R. and Delarue, F., 2018,
Probabilistic Theory of Mean Field Games with ApplicationsI , Springer International Publishing, Switzerland.[8] Carmona, R. and Delarue, F., 2018,
Probabilistic Theory of Mean Field Games with ApplicationsII , Springer International Publishing, Switzerland.[9] Djehiche, B., Barreiro-Gomez, J. and Tembine, H., 2018,
Electricity price dynamics in the smartgrid: a mean-field-type game perspective , 23rd International Symposium on Mathematical Theoryof Networks and Systems Hong Kong University of Science and Technology, Hong Kong, July 16-20,2018.[10] Fu, G., Graewe, P., Horst, U. and Popier, A., 2019,
A mean field game of optimal portfolio liquida-tion , available from https://arxiv.org/pdf/1804.04911.pdf.[11] Fu, G., Horst, U., 2018,
Mean-Field Leader-Follower Games with terminal state constraint , avialablefrom https://arxiv.org/pdf/1809.04401.pdf.[12] Fu, G., 2019,
Extended mean field games with singular controls , available athttps://arxiv.org/pdf/1909.04154.pdf.[13] Fujii, M., 2019,
Probabilistic approach to mean field games and mean field type control problems withmultiple populations , working paper, available from https://arxiv.org/pdf/1911.11501.pdf.[14] Gomes, D.A., Nurbekyan, L. and Pimentel, E.A., 2015,
Economic models and mean-field gamestheory , Publicaoes Matematicas, IMPA, Rio, Brazil.[15] Gomes, D.A., Pimental, E.A. and Voskanyan, V., 2016,
Regularity Theory for Mean-field gamesystems , SpringerBriefs in Mathematicsm.[16] Gomes, D.A. and Saude, J., 2020,
A mean-field game approach to price formation , Dyn Games Appl(2020). https://doi.org/10.1007/s13235-020-00348-x.[17] Gueant, O., Lasry, J., Lions, P., 2010,
Mean field games and Oil production , Economica. TheEconomics of Sustainable Development.[18] Hunag, M., Malhame and R., Caines, P.E., 2006,
Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , Commun. Inf. Syst., Vol.6, No. 3, pp. 221-252.[19] Kolokoltsov, V.N. and Malafeyev, O.A., 2019,
Many agent games in socio-economic systems: corrup-tion, inspection, coalition building, network growth, security , Springer Series in Operations Researchand Financial Engineering.[20] Lasry, J. M. and Lions, P.L., 2006,
Jeux a champ moyen I. Le cas stationnaire , C. R. Sci. Math.Acad. Paris, 343 pp. 619-625.[21] Lasry, J. M. and Lions, P.L., 2006,
Jeux a champ moyen II. Horizon fini et controle optimal , C. R.Sci. Math. Acad. Paris, 343, pp. 679-684.[22] Lasry, J.M. and Lions, P.L., 2007,
Mean field games , Jpn. J. Math., Vol. 2, pp. 229-260.
23] Lehalle, C.A. and Mouzouni, C., 2019,
A mean field game of portfolio trading and its consequenceson perceived correlations , available at https://arxiv.org/pdf/1902.09606.pdf.[24] Peng, S. and Wu, Z., 1999,
Fully coupled forward-backward stochastic differential equations andapplications to optimal control . SIAM J. Control Optim. , pp. 825-843.[25] Zhang, J., 2017, Backward Stochastic Differential Equations , Springer, NY., Springer, NY.