Equilibrium Price Formation with a Major Player and its Mean Field Limit
aa r X i v : . [ q -f i n . M F ] F e b Equilibrium Price Formation with a Major Playerand its Mean Field Limit ∗ Masaaki Fujii † , Akihiko Takahashi ‡ First version: 22 February, 2021
Abstract
In this article, we consider the problem of equilibrium price formation in an incom-plete securities market consisting of one major financial firm and a large number of minorfirms. They carry out continuous trading via the securities exchange to minimize their costwhile facing idiosyncratic and common noises as well as stochastic order flows from theirindividual clients. The equilibrium price process that balances demand and supply of thesecurities, including the functional form of the price impact for the major firm, is derivedendogenously both in the market of finite population size and in the corresponding meanfield limit.
Keywords : equilibrium price formation, market clearing, mean field game, major agent,controlled-FBSDEs
In the traditional setups for financial derivatives and portfolio theories, a security price processis given exogenously as a part of model inputs. On the other hand, in the field of financialeconomics, the problem of equilibrium price formation has been one of the central issues,which seeks an appropriate price process that balances demand and supply of securities amonga large number of agents endogenously based on their preferences and rational actions. Therealready exist many established results for complete markets. See, for example, Karatzas &Shreve [38] and Kramkov [39] and references therein. On the other hand, there still remainmany issues for incomplete markets. A good summary of the problem can be found in Part 2and 3 in the monograph written by Jarrow [37]. See, for example, [15, 16, 17, 36] for importantcontributions. Let us also mention the recent works [52, 57, 59] which treat the coupled systemof quadratic backward-stochastic differential equations.The progress in the mean field game (MFG) theory in the last decade has opened a newpromising approach to study the long-standing problem of multi-agent games. Since the pub-lication of seminal works by Lasry & Lions [43, 44, 45] and Huang, Malhame & Caines [35], ∗ 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.
We use the same notation adopted in the work [29]. We introduce (N+1) complete probabilityspaces: (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, for3ach 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 ). We also introduce 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. For a given constant T >
0, we usethe following notation for frequently encountered spaces: • S n + denotes the space of n × n strictly positive definite matrices. • S n denotes the space of n × n positive semidefinite matrices. • 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. It is defined by, for any µ, ν ∈ P p ( R d ), W p ( µ, ν ) := inf π ∈ Π p ( µ,ν ) h(cid:16)Z R d × R d | x − y | p π ( dx, dy ) (cid:17) p i (2.1)where Π p ( µ, ν ) denotes the set of probability measures in P p ( R d × R d ) with marginals µ and ν . For more details, see Chapter 5 in [9]. • For any N variables ( x i ) Ni =1 , we write its empirical mean as m (( x i )) := 1 N N X i =1 x i .
4e frequently omit the arguments such as ( G , R d ) in the above definitions when there is noconfusion from the context. In the preceding works, we have been interested in the equilibrium price formation in a financialmarket among a large number of security firms. Every firm (agent) is supposed to have manyindividual clients who cannot directly access to the exchange. Therefore, every agent supposedto face the stochastic order flows from his individual clients in addition to the idiosyncratic aswell as common market shocks. Under such an environment, they carry out optimal trading viathe common exchange to minimize their cost functions. Importantly, since there exist very largenumber of agents, every agent considers that his market share is negligibly small and hencethat there is no direct market impact from his trading. In other words, they behave as pricetakers . The problem of equilibrium price formation is to search an appropriate price processof securities which equalize the demand and supply based on the agents’ cost functions andthe state dynamics. In the presence of common shocks, the price process inevitably becomesstochastic. Such a problem has been investigated in our two preceding papers [28, 29], wherethe former treats the mean-field limit and the latter proves the strong convergence to themean-field limit from the corresponding equilibrium of finite population.The new twist in the current paper is the presence of one major agent, a huge financialfirm, who knows that her trading volume has a significant market share. For a given order flowfrom the major agent, a properly functioning market is expected to produce an equilibriumprice process so that it matches the net demand and supply among all the agents. Throughthis function of the market, the equilibrium price process of the securities becomes dependenton the trading strategy of the major agent. Therefore, her optimization problem ends up inminimizing the cost with her own feedback effects into account. We then finally obtain themarket equilibrium price process by solving the major agent’s optimal strategy. In the follow-ing, we first solve this problem in the market with finite population size. The minor agents areallowed to be heterogeneous so that the coefficients functions for their state processes as wellas the cost functions can be different each other. The large population limit of minor agentswill be studied in later sections.Let us now describe the setup more concretely. There are N minor agents indexed by i = 1 , · · · , N . The major agent is always labeled by the index 0. The number of securitiestraded in the market is assume to be n ∈ N . Each minor agent i ∈ { , · · · , N } tries to solvethe cost minimization problem among the admissible strategies A i := H ( F ; R n )inf α i ∈ A i J i ( α i ) (3.1)with some functions f i and g i , which denotes the running as well as terminal costs, respectively: J i ( α i ) := E hZ T f i ( t, X it , α it , ̟ t , Λ t , c t , c it ) dt + g i ( X iT , ̟ T , c T , c iT ) i . i thagent, is given by dX it = (cid:0) α it + l i ( t, c t , c it ) (cid:1) dt + σ i ( t, c t , c it ) dW t + σ i ( t, c t , c it ) dW it , t ∈ [0 , T ]with X i = ξ i . Here, ξ i ∈ L ( F i ; R n ) denotes the size of the initial position, which is assumedto have the common law for every 1 ≤ i ≤ N . ( ̟ t ) t ∈ [0 ,T ] ∈ H ( F ; R n ) denotes the market priceprocess of the n securities. In the end, we want to determine ( ̟ t ) t ∈ [0 ,T ] endogenously so that itequalize the amount of demand and supply. ( c t ) t ≥ ∈ H ( F ; R n ) denotes the coupon paymentsfrom the securities or the market news affecting all the agents, while ( c it ) t ≥ ∈ H ( F i ; R n )denotes some idiosyncratic shocks affecting only the i th agent. Moreover, ( c it ) t ≥ are alsoassumed to have the common law for all 1 ≤ i ≤ N . (Λ t ) t ∈ [0 ,T ] is an F -adapted process relatedto the trading fee to be paid to the exchange. The terms involving ( l i , σ i , σ i ) denote the orderflow to the i th agent from his individual clients through the over-the-counter (OTC) market.Each minor agent controls ( α it ) t ∈ [0 ,T ] , which is an R n -valued process denoting the trading speedof the n securities via the exchange. Note that, in addition to the random initial states ( ξ i ) Ni =1 ,we have d -dimensional common noise W and N d -dimensional idiosyncratic noises ( W i ) Ni =1 .Since we impose no restriction on the size among ( n, d , d, N ), we have an incomplete securitiesmarket in general.When the number of agents N is sufficiently large, it is natural to assume that each minoragent consider himself as a price taker . Throughout the paper, we assume that this is thecase. This means that each minor agent tries to solve the optimization problem by treating( ̟ t ) t ≥ as an exogenous process. Suppose that the trading strategy of the major agent isgiven by ( β t ) t ∈ [0 ,T ] , which denotes her trading speed. For given order flow ( β t ) t ∈ [0 ,T ] , thefinancial market is expected to produce an equilibrium price process ( ̟ t ) t ∈ [0 ,T ] which equalizethe demand and supply among all the agents. Our first goal is to find such a price process( ̟ t ) t ∈ [0 ,T ] which achieves N X i =1 b α it + β t = 0 (3.2) dt ⊗ d P -a.e., where (cid:0) ( b α it ) t ∈ [0 ,T ] (cid:1) Ni =1 are the optimal trading strategies of the minor agents solving(3 .
1) based on this price process ( ̟ t ) t ∈ [0 ,T ] .We shall show that the resultant equilibrium price process becomes dependent on ( β t ) t ∈ [0 ,T ] i.e., we have (cid:0) ̟ t ( β ) (cid:1) t ∈ [0 ,T ] . Note that the minor agents do not directly care about ( β t ) t ∈ [0 ,T ] .They are just destined to face, as price takers, the exogenous market price process, whichhappens to depend on the major’s strategy when it clears the market. The problem of themajor agent is now to solve inf β ∈ A J ( β ) (3.3)with the cost functional depending on f ( N )0 and g ( N )0 : J ( β ) := E hZ T f ( N )0 ( t, X t , β t , ̟ t ( β ) , Λ t , c t ) dt + g ( N )0 ( X T , c T ) i , with her own feedback effects taken into account. (Λ t ) t ∈ [0 ,T ] is an F -adapted process related6o the trading fee to be paid to the exchange. The state dynamics of the major agent describingher position size assumed to follow dX t = (cid:0) β t + l ( N )0 ( t, c t ) (cid:1) dt + σ ( N )0 ( t, c t ) dW t , t ∈ [0 , T ] (3.4)with some initial condition X ∈ R n . The superscript ( N ) of the coefficient functions is addedto indicate that there are ( N ) minor agents. It becomes useful when we take the large- N limitin later sections. We assume that the space of admissible strategies for the major agent is givenby A := H ( F ; R n ) ∩ { β T = 0 } , where the constraint β T = 0 is added in order to forbid thelast-time price manipulation.In our framework, the price process including the feedback effects from the major’s actionis determined endogenously. This is a clear contrast to the existing literature dealing with theoptimal execution strategy, where the form of the price impact as well as the fundamental priceprocess are exogenously given.Before going to the details, let us comment on the information structure for the agents. Remark 3.1.
If possible, we naturally want to restrict the space of admissible strategies foreach minor agent to A i = H ( F i ; R n ) , ≤ i ≤ N and that for the major agent to A = H ( F ; R n ) ∩ { β T = 0 } . In other words, we want to realize a market in which each agent onlycares about the common market shocks adapted to F and his/her own idiosyncratic shocksadapted to F i . This would be a much plausible model for the real financial market than our setupgiven above. Unfortunately, this looks impossible in the market consisting of finite number ofagents since, in general, the market-clearing price does not solely adapted to F but is dependenton the idiosyncratic shocks, too.As already mentioned in [28, 29], we shall observe that this ideal situation is actually realizedin the large population limit. There, we can restrict the admissible strategy of the i th minoragent to A i = H ( F i ; R n ) , and that of the major agent to A = H ( F ; R n ) ∩ { β T = 0 } . In fact,we can find ( ̟ t ) t ∈ [0 ,T ] is an F -adapted process, i.e. the market-clearing price is dependentonly on the common market shocks. By the convergence analysis, we shall see that this isapproximately true when the population size is large enough. Let us solve the problem for each minor agent with given order flow ( β t ) t ∈ [0 ,T ] ∈ A of themajor agent. This is done in a completely parallel manner with our previous work [29]. Wefirst specify the details of the functions introduced in the last section. For each 1 ≤ i ≤ N , weconsider the following measurable functions:( l i , σ i , σ i ) : [0 , T ] × R n × R n ∋ ( t, c , c i ) ( l i ( t, c , c i ) , σ i ( t, c , c i ) , σ i ( t, c , c i )) ∈ ( R n , R n × d , R n × d ) ,f i : [0 , T ] × ( R n ) ∋ ( t, x, ̟, c , c i ) f i ( t, x, ̟, c , c i ) ∈ R ,g i : ( R n ) ∋ ( x, c , c i ) g i ( x, c , c i ) ∈ R ,
7s well as f i : [0 , T ] × ( R n ) × S n + × ( R n ) → R and g i : ( R n ) → R defined by f i ( t, x, α, ̟, Λ , c , c i ) := h ̟, α i + 12 h α, Λ α i + f i ( t, x, ̟, c , c i ) ,g i ( x, ̟, c , c i ) := − δ h ̟, x i + g i ( x, c , c i ) , where δ ∈ [0 ,
1) is a given constant representing some discount factor. The first term h ̟, α i of f i denotes the direct cost incurred by the sales and purchase of the securities, and the secondterm h α, Λ α i denotes the (rate of) fee to be paid to the exchange based on the trading spreed.Let us also introduce the following measurable functions ( c fi , c gi , h fi , h gi ) for each 1 ≤ i ≤ N : c fi : [0 , T ] × ( R n ) ∋ ( t, c , c i ) c fi ( t, c , c i ) ∈ S n + ,c gi : ( R n ) ∋ ( c , c i ) c gi ( c , c i ) ∈ S n + ,h fi : [0 , T ] × ( R n ) h fi ( t, c , c i ) ∈ R n ,h gi : ( R n ) h gi ( c , c i ) ∈ R n . We assume the following conditions:
Assumption 3.1. (Minor-A)
Uniformly in ≤ i ≤ N , the functions satisfy the followings: (i) (Λ t ) t ∈ [0 ,T ] is an F -progressively measurable S n + -valued process such that there exist somepositive constants < λ ≤ λ < ∞ satisfying λ | θ | ≤ h θ, Λ t θ i ≤ λ | θ | for every ( ω, t, θ ) ∈ Ω × [0 , T ] × R n . (ii) For any ( t, c , c i ) ∈ [0 , T ] × ( R n ) , | l i ( t, c , c i ) | + | σ i ( t, c , c i ) | + | σ i ( t, c , c i ) | ≤ L (1 + | c | + | c i | ) . (iii) For any ( t, x, ̟, c , c i ) ∈ [0 , T ] × ( R n ) , | f i ( t, x, ̟, c , c i ) | + | g i ( x, c , c i ) | ≤ L (1 + | x | + | ̟ | + | c | + | c i | ) . (iv) For any ( t, x, ̟, c , c i ) ∈ [0 , T ] × ( R n ) , f i and g i are once continuously differentiable in x with ̟ -independent derivatives, and the functions ∂ x f i and ∂ x g i have the following affine-formin x : ∂ x f i ( t, x, ̟, c , c i ) (cid:0) =: ∂ x f i ( t, x, c , c i ) (cid:1) = c fi ( t, c , c i ) x + h fi ( t, c , c i ) ,∂ x g i ( x, c , c i ) = c gi ( c , c i ) x + h gi ( c , c i ) . Moreover, the functions ( c fi , c gi , h fi , h gi ) satisfy | h fi ( t, c , c i ) | + | h gi ( c , c i ) | ≤ L (1 + | c | + | c i | ) , | c fi ( t, c , c i ) | + | c gi ( c , c i ) | ≤ L, (cid:10) θ, c fi ( t, c , c i ) θ (cid:11) ≥ γ f | θ | , (cid:10) θ, c gi ( c , c i ) θ (cid:11) ≥ γ g | θ | , ∀ θ ∈ R n , with some positive constants γ f , γ g > . This is a special situation studied in Section 3.1 of [29]. In fact, the conditions in Assumption(Minor-A) are significantly more stringent than those used in [29]. We do this in order to8void introducing many sets of assumptions incrementally in later sections. In particular, theaffine-form condition in (iv) is to be used when we verify the optimality condition for themajor agent based on Theorem A.1. The associated (reduced) Hamiltonian for the i th agent H i : [0 , T ] × ( R n ) × S n + × ( R n ) → R is given by H i ( t, x, y, α, ̟, Λ , c , c i ) := (cid:10) y, α + l i ( t, c , c i ) (cid:11) + f i ( t, x, α, ̟, Λ , c , c i ) , which is jointly convex in ( x, y, α ) and strictly so in ( x, α ). The unique minimizer α of H i isgiven by b α ( y, ̟ ) := − Λ( y + ̟ ) , with Λ := Λ − . Therefore, the adjoint equation associated with the problem (3 .
1) for the i thagent arising from the stochastic maximum principle is given by, for t ∈ [0 , T ], ( dX it = (cid:0) − Λ t ( Y it + ̟ t ) + l i ( t, c t , c it ) (cid:1) dt + σ i ( t, c t , c it ) dW t + σ i ( t, c t , c it ) dW it ,dY it = − ∂ x f i ( t, X it , c t , c it ) dt + Z i, t dW t + P Nj =1 Z i,jt dW jt , (3.5)with X i = ξ i and Y iT = − δ̟ T + ∂ x g i ( X iT , c T , c iT ). Theorem 3.1.
Let Assumption (Minor-A) be in force. Then, for any ( ̟ t ) t ∈ [0 ,T ] ∈ H ( F ; R n ) satisfying ̟ T ∈ L ( F T ; R n ) , the problem (3 . for each agent ≤ i ≤ N is uniquely character-ized by the FBSDE (3 . which is strongly solvable with a unique solution ( X i , Y i , Z i, , ( Z i,j ) Nj =1 ) ∈ S ( F ; R n ) × S ( F ; R n ) × H ( F ; R n × d ) × ( H ( F ; R n × d )) N .Proof. This is the direct result of Theorem 3.1 in [29]. One can easily check Assumption 3.1 in[29] is satisfied under (Minor-A). Although (Λ t ) t ∈ [0 ,T ] is now stochastic, it does not introduceany additional difficulty. The existence of the unique solution to the FBSDE (3 .
5) can also beproved by the direct application of Theorem 2.6 in [51] (with β , µ > ( β t ) t ∈ [0 ,T ] From Theorem 3.1, we find that the optimal trading speed of each minor agent 1 ≤ i ≤ N isgiven by b α it = − Λ t ( Y it + ̟ t ) , t ∈ [0 , T ] , for any exogenous input ( ̟ t ) t ∈ [0 ,T ] . Since the market clearing condition requires P Ni =1 b α it + β t =0, dt ⊗ d P -a.e. the market price process needs to satisfy ̟ t = − m (cid:0) ( Y it ) (cid:1) + Λ t β t N , t ∈ [0 , T ] . (3.6)This relation suggests a large system of fully-coupled FBSDEs given below: for 1 ≤ i ≤ N , dX it = n − Λ t (cid:0) Y it − m (( Y jt )) (cid:1) − β t N + l i ( t, c t , c it ) o dt + σ i ( t, c t , c it ) dW t + σ i ( t, c t , c it ) dW it ,dY it = − ∂ x f i ( t, X it , c t , c it ) dt + Z i, t dW t + P Nj =1 Z i,jt dW jt , (3.7)9ith ( X i = ξ i ,Y iT = δ − δ m (cid:16)(cid:0) c gj ( c T , c jT ) X jT + h gj ( c T , c jT ) (cid:1)(cid:17) + c gi ( c T , c iT ) X iT + h gi ( c T , c iT ) . (3.8)The terminal condition for Y i is implied from Y iT = − δ̟ T + ∂ x g i ( X iT , c T , c iT )and the fact that ̟ T = − m (( Y iT )) (note that β T = 0). We have the following result. Theorem 3.2. (Theorem 3.2 [29])
Under Assumption (Minor-A), the market is cleared if and only if the market price process ( ̟ t ) t ∈ [0 ,T ] is given by (3 . with the solutions ( Y i ) Ni =1 to the N -coupled system of FBSDEs (3 . with (3 . .Proof. From Theorem 3.1, the necessity is obvious. On the other hand, suppose that thereexists a solution (cid:0) ( Y it ) t ∈ [0 ,T ] (cid:1) Ni =1 to (3 .
7) with (3 .
8) and that ( ̟ t ) t ∈ [0 ,T ] is given by (3 .
6) usingits solution. Then, with this ̟ as inputs, the solution ( y it ) t ∈ [0 ,T ] to (3 . y i = Y i due to the uniqueness of the solution to(3 . Assumption 3.2. (Minor-B)
There exists some F T -measurable S n -valued random variable c such that a := δ − δ || c − c gi ( c T , c iT ) || ∞ < γ g , ≤ i ≤ N. Theorem 3.3.
Let Assumptions (Minor-A, B) be in force. Then, for any given ( β t ) t ∈ [0 ,T ] ∈ A ,the N -coupled system of FBSDEs (3 . with (3 . has a unique strong solution ( X i , Y i , Z i, , ( Z i,j ) Nj =1 ) ∈ S ( F ; R n ) × S ( F ; R n ) × H ( F ; R n × d ) × ( H ( F ; R n × d )) N , ≤ i ≤ N .Proof. Let x i , y i ∈ R n be arbitrary constants. For notational simplicity, we write x = ( x i ) Ni =1 and y = ( y i ) Ni =1 . Putdrift[ x i ]( t, y ) := − Λ t (cid:0) y i − m (( y j )) (cid:1) − β t N + l i ( t, c t , c it ) , drift[ y i ]( t, x ) := − ∂ x f i ( t, x i , c t , c it ) , terminal[ y i ]( x ) := δ − δ m (cid:0) ( c gj ( c T , c jT ) x j + h gj ( c T , c jT )) (cid:1) + c gi ( c T , c iT ) x i + h gi ( c T , c iT ) . For two inputs ( x, y ) and ( x ′ , y ′ ), with the conventions ∆ x i := x i − x i ′ , ∆ y i := y i − y i ′ ,∆drift[ x i ]( t ) := drift[ x i ]( t, y ) − drift[ x i ]( t, y ′ ) , ∆drift[ y i ]( t ) := drift[ y i ]( t, x ) − drift[ y i ]( t, x ′ ) , ∆terminal[ y i ] := terminal[ y i ]( x ) − terminal[ y i ]( x ′ ) ,
10e have N X i =1 (cid:10) ∆drift[ x i ]( t ) , ∆ y i (cid:11) = − N X i =1 (cid:10) Λ t ∆ y i , ∆ y i (cid:11) + N (cid:10) Λ t m ((∆ y i )) , m ((∆ y i )) (cid:11) ≤ , N X i =1 (cid:10) ∆drift[ y i ]( t ) , ∆ x i (cid:11) = − (cid:10) c fi ( t, c t , c it )∆ x i , ∆ x i (cid:11) ≤ − γ f N X i =1 | ∆ x i | , N X i =1 (cid:10) ∆terminal[ y i ] , ∆ x i (cid:11) = δN − δ (cid:10) m (( c gi ( c T , c iT )∆ x i )) , m ((∆ x i )) (cid:11) + N X i =1 (cid:10) c gi ( c T , c iT )∆ x i , ∆ x i (cid:11) ≥ δN − δ (cid:10) c m ((∆ x i )) , m ((∆ x i )) (cid:11) + ( γ g − a ) N X i =1 | ∆ x i | ≥ ( γ g − a ) N X i =1 | ∆ x i | . (3.9)Thus we can apply Theorem 2.6 in [51] with ( β , µ ) = ( γ f , γ g − a ) and G = I . See also theproof for Theorem 3.3 in [29], which can be applied in essentially the same way for the currentproblem. We now investigate the optimization problem for the major agent. From Theorems 3.2 and3.3, her problem is given by inf β ∈ A J ( β ) with J ( β ) := E hZ T f ( N )0 (cid:16) t, X t , β t , − m (( Y it )) + Λ t β t N , Λ t , c t (cid:17) dt + g ( N )0 ( X T , c T ) i , subject to the dynamic constraints with 1 ≤ i ≤ N : dX t = (cid:0) β t + l ( N )0 ( t, c t ) (cid:1) dt + σ ( N )0 ( t, c t ) dW t ,dX it = n − Λ t (cid:0) Y it − m (( Y jt )) (cid:1) − β t N + l i ( t, c t , c it ) o dt + σ i ( t, c t , c it ) dW t + σ i ( t, c t , c it ) dW it ,dY it = − ∂ x f i ( t, X it , c t , c it ) dt + Z i, t dW t + P Nj =1 Z i,jt dW jt , t ∈ [0 , T ] (3.10)with X = N χ , χ ∈ R n ,X i = ξ i ,Y iT = δ − δ m (cid:16)(cid:0) c gj ( c T , c jT ) X jT + h gj ( c T , c jT ) (cid:1)(cid:17) + c gi ( c T , c iT ) X iT + h gi ( c T , c iT ) . (3.11)As we can see, the problem for the major agent turns out to be an optimization with respectto the system of controlled-FBSDEs instead of controlled-SDEs. See, for relevant information,Appendix A and the references therein. Remark 3.2.
At first glance, it may seem to be a linear price impact model popular in the iterature dealing with the optimal execution problem. However, notice that the term − m (( Y it )) is also dependent on the major agent’s strategy in a complicated fashion. Since we want to study the large population limit N → ∞ in later sections, it is convenientto define the normalized measurable functions:( l , s ) : [0 , T ] × R n ∋ ( t, c ) ( l ( t, c ) , s ( t, c )) ∈ ( R n , R n × d ) , f : [0 , T ] × ( R n ) ∋ ( t, x, c ) f ( t, x, c ) ∈ R , g : ( R n ) ∋ ( x, c ) g ( x, c ) ∈ R , and f : [0 , T ] × ( R n ) × S n × R n → R by f ( t, x, β, ̟, Λ , c ) := h β, ̟ i + 12 (cid:10) β, Λ β (cid:11) + f ( t, x, c ) . We then define the unnormalized functions by l ( N )0 ( t, c ) := N l ( t, c ) ,σ ( N )0 ( t, c ) := N s ( t, c ) ,f ( N )0 ( t, x, β, ̟, Λ , c ) := N f (cid:0) t, x/N, β/N, ̟, Λ , c (cid:1) ,f ( N )0 ( t, x, c ) := N f (cid:0) t, x/N, c (cid:1) ,g ( N )0 ( x, c ) := N g (cid:0) x/N, c (cid:1) . (3.12)Note that, we have f ( N )0 ( t, x, β, ̟, Λ , c ) = h β, ̟ i + 12 D β, Λ N β E + f ( N )0 ( t, x, c ) . Let us introduce the following assumptions.
Assumption 3.3. (Major)(i) (Λ t ) t ∈ [0 ,T ] is an F -progressively measurable S n -valued process such that there exist somepositive constants < λ ≤ λ < ∞ satisfying λ ≤ (cid:10) θ, (Λ t + 2Λ t ) θ (cid:11) ≤ λ | θ | for every ( ω, t, θ ) ∈ [0 , T ] × Ω × R n . (ii) For any ( t, c ) ∈ [0 , T ] × R n , | l ( t, c ) | + | s ( t, c ) | ≤ L (1 + | c | ) . (iii) For any ( t, x , c ) ∈ [0 , T ] × ( R n ) , | f ( t, x , c ) | + | g ( x , c ) | ≤ L (1 + | x | + | c | ) . (iv) f and g are once continuously differentiable in x and satisfy | ∂ x f ( t, x , c ) | + | ∂ x g ( x , c ) | ≤ L (1 + | x | + | c | ) , | ∂ x f ( t, x ′ , c ) − ∂ x f ( t, x , c ) | + | ∂ x g ( x ′ , c ) − ∂ x g ( x , c ) | ≤ L | x ′ − x | , for any ( t, x , x ′ , c ) ∈ [0 , T ] × ( R n ) . (v) f and g are strictly convex in the sense that there exist some positive constants γ f , γ g > nd f ( t, x ′ , c ) − f ( t, x , c ) − (cid:10) x ′ − x , ∂ x f ( t, x , c ) (cid:11) ≥ γ f | x ′ − x | , g ( x ′ , c ) − g ( x , c ) − (cid:10) x ′ − x , ∂ x g ( x , c ) (cid:11) ≥ γ g | x ′ − x | , hold for any ( t, x , x ′ , c ) ∈ [0 , T ] × ( R n ) . For later use, let us put γ f ( N )0 := γ f N , γ g ( N )0 := γ g N .
Remark 3.3.
With the above definition, we have ∂ x f ( N )0 ( t, x, c ) = N ∂∂x f ( t, x/N, c )= N ∂ ( x/N ) ∂x ∂ x f ( t, x/N, c ) = ∂ x f ( t, x/N, c ) . and similar relation for ∂ x g . Remark 3.4.
For the analysis with a fixed N , such a scaling is arbitrary and irrelevant.However, it plays an important role when we study the large population limit N → ∞ . Inparticular, the market share of the major agent must grow proportionally to the population size N . For example, if the cost functions contains (cid:10) β, Λ β (cid:11) instead of D β, Λ N β E , the market shareof the major agent becomes negligible in the large population limit. In this case, we obtains thesame market price as in [28, 29]. Following the analysis done in Appendix A, let us introduce the adjoint variables ( p , ( p i ) Ni =1 , ( r i ) Ni =1 )for ( x , ( x i ) Ni =1 , ( y i ) Ni =1 ), respectively. The (reduced) Hamiltonian H : [0 , T ] × R n × ( R n ) N × ( R n ) N × R n × ( R n ) N × ( R n ) N × R n × S n × S n + × R n × ( R n ) N → R of the system is defined by H ( t, x , ( x i ) Ni =1 , ( y i ) Ni =1 , p , ( p i ) Ni =1 , ( r i ) Ni =1 , β, Λ , Λ , c , ( c i ) Ni =1 ):= (cid:10) p , β + l ( N )0 ( t, c ) (cid:11) + N X i =1 D p i , − Λ (cid:0) y i − m (( y i )) (cid:1) − βN + l i ( t, c , c i ) E + N X i =1 (cid:10) r i , − ∂ x f i ( t, x i , c , c i ) (cid:11) + D β, − m (( y i )) + Λ βN E + 12 D β, Λ N β E + f ( N )0 ( t, x , c ) . (3.13)For a given set of p , ( p i ) Ni =1 , ( r i ) Ni =1 (and also (Λ , Λ , c , ( c i ) Ni =1 )), it is straightforward to checkthat H is jointly convex in ( x , ( x i ) Ni =1 , ( y i ) Ni =1 , β ) and strictly convex in ( x , β ). Here, recallthat ∂ x f i is affine in x i by Assumption (Minor-A, (iv)). For given inputs, the minimizer of the13amiltonian b β := argmin H ( β ) is given by b β = N V (cid:0) − p + m (( y i )) + m (( p i )) (cid:1) (3.14)where V := (Λ + 2Λ) − .The adjoint equations for ( p , ( p i ) Ni =1 , ( r i ) Ni =1 ) can be found from ( A. ≤ i ≤ N , dP t = − ∂ x f ( N )0 ( t, X t , c t ) dt + Q , t dW t + P Nj =1 Q ,jt dW jt ,dP it = c fi ( t, c t , c it ) R it dt + Q i, t dW t + P Nj =1 Q i,jt dW jt ,dR it = n Λ t (cid:0) P it − m (( P jt )) (cid:1) + β t N o dt, (3.15)with P T = ∂ x g ( N )0 ( X T , c T ) ,P iT = − c gi ( c T , c iT ) (cid:16) R iT + δ − δ m (( R jT )) (cid:17) ,R i = 0 . (3.16) Theorem 3.4.
Let Assumptions (Minior-A, B) and (Major) be in force. Suppose that thesystem of FBSDEs (3 . and (3 . with boundary conditions (3 . and (3 . has a so-lution X , Y i , P , P i , R i ∈ S ( F ; R n ) , Z i, , Q , , Q i, ∈ H ( F ; R n × d ) , and Z i,j , Q ,j , Q i,j ∈ H ( F ; R n × d ) , ≤ i, j ≤ N , with the control process β t = b β t , t ∈ [0 , T ] i.e., b β t = N V t (cid:0) − P t + m (( Y it )) + m (( P it )) (cid:1) , V t := (Λ t + 2Λ t ) − . Then, ( b β t ) t ∈ [0 ,T ] is the unique optimal control for the major agent.Proof. This is the direct result of Theorem A.1. Note that Assumption (Minor-A) (iv) plays acrucial role to guarantee the joint convexity of H and the affine property of Φ required in thetheorem. From Theorem 3.4, the crucial target of our analysis is the following coupled system of FBSDEs: dX t = (cid:0) b β t + l ( N )0 ( t, c t ) (cid:1) dt + σ ( N )0 ( t, c t ) dW t ,dX it = n − Λ t (cid:0) Y it − m (( Y jt )) (cid:1) − b β t N + l i ( t, c t , c it ) o dt + σ i ( t, c t , c it ) dW t + σ i ( t, c t , c it ) dW it ,dR it = n Λ t (cid:0) P it − m (( P jt )) (cid:1) + b β t N o dt,dP t = − ∂ x f ( N )0 ( t, X t , c t ) dt + Q , t dW t + P Nj =1 Q ,jt dW jt ,dY it = − ∂ x f i ( t, X it , c t , c it ) dt + Z i, t dW t + P Nj =1 Z i,jt dW jt ,dP it = c fi ( t, c t , c it ) R it dt + Q i, t dW t + P Nj =1 Q i,jt dW jt , (3.17)14ith X = N χ , X i = ξ i , R i = 0 P T = ∂ x g ( N )0 ( X T , c T ) ,Y iT = δ − δ m (cid:16)(cid:0) c gj ( c T , c jT ) X jT + h gj ( c T , c jT ) (cid:1)(cid:17) + c gi ( c T , c iT ) X iT + h gi ( c T , c iT ) ,P iT = − c gi ( c T , c iT ) (cid:16) R iT + δ − δ m (( R jT )) (cid:17) , (3.18)for 1 ≤ i ≤ N . Here, ( b β t ) t ∈ [0 ,T ] is defined by b β t = N V t (cid:0) − P t + m (( Y it )) + m (( P it )) (cid:1) , t ∈ [0 , T ] . The main result of this section is the next theorem.
Theorem 3.5.
Under Assumptions (Minor-A, B) and (Major), there exists a unique strongsolution X , Y i , P , P i , R i ∈ S ( F ; R n ) , Z i, , Q , , Q i, ∈ H ( F ; R n × d ) , and Z i,j , Q ,j , Q i,j ∈ H ( F ; R n × d ) , ≤ i, j ≤ N to the coupled system of FBSDEs (3 . with (3 . .Proof. We shall show that the monotone conditions used in Theorem 2.6 in [51] are actuallysatisfied. Let x , p and x i , y i , p i , r i , ≤ i ≤ N be arbitrary constants in R n . We put x =( x i ) Ni =1 , y = ( y i ) Ni =1 , p = ( p i ) Ni =1 , r = ( r i ) Ni =1 , and u = ( x , x, r, p , y, p ). We write b β ( t, u ) := V t (cid:0) − p + m (( y )) + m (( p )) (cid:1) . As in Theorem 3.3, we introduce the quantities:drift[ x ]( t, u ) := b β ( t, u ) + l ( N )0 ( t, c t ) , drift[ x i ]( t, u ) := − Λ t (cid:0) y i − m (( y )) (cid:1) − b β ( t, u ) N + l i ( t, c t , c it ) , drift[ r i ]( t, u ) := Λ t (cid:0) p i − m (( p )) (cid:1) + b β ( t, u ) N , drift[ p ]( t, u ) := − ∂ x f ( N )0 ( t, x , c t ) , drift[ y i ]( t, u ) := − ∂ x f i ( t, x i , c t , c it ) , drift[ p i ]( t, u ) := c fi ( t, c t , c it ) r i , andterminal[ p ]( u ) := ∂ x g ( N )0 ( x , c T ) , terminal[ y i ]( u ) := δ − δ m (cid:0) ( c gj ( c T , c jT ) x j + h gj ( c T , c jT )) (cid:1) + c gi ( c T , c iT ) x i + h gi ( c T , c iT ) , terminal[ p i ]( u ) := − c gi ( c T , c iT ) (cid:16) r i + δ − δ m (( r )) (cid:17) . With two inputs ( u, u ′ ), we define ∆ u := u − u ′ ,∆drift[ x ]( t ) := drift[ x ]( t, u ) − drift[ x ]( t, u ′ ) , ∆terminal[ p ] := terminal[ p ]( u ) − temrinal[ p ]( u ′ ) , (cid:10) ∆drift[ p ]( t ) , ∆ x (cid:11) = − (cid:10) ∂ x f ( t, x /N, c t ) − ∂ x f ( t, x ′ /N, c t ) , ∆ x t (cid:11) ≤ − N γ f | ∆ x /N | = − γ f ( N )0 | ∆ x | . It is then straightforward to get (cid:10) ∆drift[ p ]( t ) , ∆ x (cid:11) + N X i =1 (cid:10) ∆drift[ y i ]( t ) , ∆ x i (cid:11) + N X i =1 (cid:10) ( − I )∆drift[ p i ]( t ) , ∆ r i (cid:11) ≤ − γ f ( N )0 | ∆ x | − γ f N X i =1 (cid:0) | ∆ x i | + | ∆ r i | (cid:1) , where I = I n × n is the identity matrix. Next, with ∆ b β t := b β ( t, u ) − b β ( t, u ′ ), we have N X i =1 (cid:10) ∆drift[ x i ]( t ) , ∆ y i (cid:11) = N X i =1 D − Λ t (cid:0) ∆ y i − m ((∆ y )) (cid:1) − ∆ b β t N , ∆ y i E = − N X i =1 (cid:10) Λ t ∆ y i , ∆ y i (cid:11) + N (cid:10) Λ t m ((∆ y )) , m ((∆ y )) (cid:11) − N D ∆ b β t N , m ((∆ y )) E ≤ − N D ∆ b β t N , m ((∆ y )) E . By similar calculation, we get (cid:10) ∆drift[ x ]( t ) , ∆ p (cid:11) + N X i =1 (cid:10) ∆drift[ x i ]( t ) , ∆ y i (cid:11) + N X i =1 (cid:10) ( − I )∆drift[ r i ]( t ) , ∆ p i (cid:11) ≤ − N D ∆ b β t N , − ∆ p + m ((∆ y )) + m ((∆ p )) E = − N D ∆ b β t N , (Λ t + 2Λ t ) ∆ b β t N E ≤ . Therefore, from the drift contribution, we eventually have (cid:10) ∆drift[ p ]( t ) , ∆ x (cid:11) + N X i =1 (cid:10) ∆drift[ y i ]( t ) , ∆ x i (cid:11) + N X i =1 (cid:10) ( − I )∆drift[ p i ]( t ) , ∆ r i (cid:11) + (cid:10) ∆drift[ x ]( t ) , ∆ p (cid:11) + N X i =1 (cid:10) ∆drift[ x i ]( t ) , ∆ y i (cid:11) + N X i =1 (cid:10) ( − I )∆drift[ r i ]( t ) , ∆ p i (cid:11) ≤ − γ f ( N )0 | ∆ x | − γ f N X i =1 (cid:0) | ∆ x i | + | ∆ r i | (cid:1) . (3.19)16or the terminal conditions, by the similar calculation done in (3 . (cid:10) ∆terminal[ p ] , ∆ x (cid:11) + N X i =1 (cid:10) ∆terminal[ y i ] , ∆ x i (cid:11) + N X i =1 (cid:10) ( − I )∆terminal[ p i ] , ∆ r i (cid:11) ≥ γ g ( N )0 | ∆ x | + ( γ g − a ) N X i =1 (cid:0) | ∆ x i | + | ∆ r i | (cid:1) . (3.20)Using (3 .
19) and (3 . A ( t, u ) = I n × n I n × n ) N − I n × n ) N I n × n I n × n ) N
00 0 0 0 0 ( − I n × n ) N drift[ p ]drift[ y ]drift[ p ]drift[ x ]drift[ x ]drift[ r ] ( t, u )and G = I n × n I n × n ) N
00 0 ( − I n × n ) N . In particular, we have β := min( γ f ( N )0 , γ f ) > µ := min( γ g ( N )0 , γ g − a ) >
0. Note that thecoefficients of the Brownian motions ( σ i etc.) are irrelevant since they are uncontrolled andstate-independent. In fact, one can repeat the proof for Theorem 3.3 in [29] in essentially thesame way by simply replacing the analysis for d (cid:10) ∆ y t , ∆ x t (cid:11) with that for d * ∆ p t ∆ y t ∆ p t , G ∆ x t ∆ x t ∆ r t + using the above estimates.Thanks to Theorem 3.5, we now find the market-clearing price process is given by ̟ t = − m (( Y it )) + Λ t V t (cid:16) − P t + m (( Y it )) + m (( P it )) (cid:17) , t ∈ [0 , T ] (3.21)using the solutions to the coupled system of FBSDEs (3 .
17) with (3 . Mean-field Equilibrium
Let us work on the probability space with N = 1 in Section 2, i.e. (Ω , F , P , F ) = (Ω , F , P , F ).In the following, we use the notation: E t (cid:2) · (cid:3) := E (cid:2) · |F t (cid:3) . Let us first introduce the following assumptions.
Assumption 4.1. (MFG)(i) ( l, σ , σ, f , g, c f , c g , h f , h g ) satisfy the same conditions corresponding to those for ( l i , σ i , σ i , f i , g i ,c fi , c gi , h fi , h gi ) in Assumption (Minor-A). (ii) There exists some F T -measurable S n -valued random variable c such that a := δ − δ || c − c g ( c T , c T ) || ∞ < γ g . (iii) For the other variables and functions, we assume the same conditions as those in Assump-tions (Minor-A) and (Major).
For the space of admissible strategies A := H ( F ; R n ) ∩ { β T = 0 } , we suppose that themajor agent tries to solve inf β ∈ A J ( β ) (4.1)where J ( β ) := E hZ T f (cid:16) t, x t , β t , − E t [ y t ] + Λ t β t , Λ t , c t (cid:17) dt + g ( x T , c T ) i subject to the following dynamic constraints: dx t = (cid:0) β t + l ( t, c t ) (cid:1) dt + s ( t, c t ) dW t ,dx t = n − Λ t (cid:0) y t − E t [ y t ] (cid:1) − β t + l ( t, c t , c t ) o dt + σ ( t, c t , c t ) dW t + σ ( t, c t , c t ) dW t ,dy t = − ∂ x f ( t, x t , c t , c t ) dt + z , t dW t + z , t dW t , (4.2)with x = χ , x = ξ ,y T = δ − δ E T (cid:2) c g ( c T , c T ) x T + h g ( c T , c T ) (cid:3) + c g ( c T , c T ) x T + h g ( c T , c T ) . (4.3)Here, the problem for the major agent is the optimization with respect to the controlled-FBSDE of conditional McKean-Vlasov type. One can naturally expect the above formulationof the problem in the mean-field limit from the McKean-Vlasov FBSDEs given in [28] and theexpression in (3 . Remark 4.1.
Notice that, the above problem is well posed in the sense that for a given β ∈ A , there exists a unique strong solution to (4 . and the corresponding cost J ( β ) is finite. In particular, the unique existence for ( x , y ) can be proved by a simple modificationof Theorem 4.2 in [28]. . H : [0 , T ] × ( R n ) × S n × S n + × ( R n ) → R by H (cid:0) t, x , x , y , y , p , p , p , r , β, Λ , Λ , c , c (cid:1) := (cid:10) p , β + l ( t, c ) (cid:11) + (cid:10) p , − Λ( y − y ) + l ( t, c , c ) (cid:11) + (cid:10) p , − β (cid:11) + (cid:10) r , − ∂ x f ( t, x , c , c ) (cid:11) + (cid:10) β, − y + Λ β (cid:11) + 12 (cid:10) β, Λ β (cid:11) + f ( t, x , c ) . (4.4)It is important to observe that the map( x , x , y , y , β ) H ( t, x , x , y , y , p , p , p , r , β, Λ , Λ , c , c )is jointly convex and strictly convex in β (and x ). It is easy to find b β = V (cid:0) − p + y + p (cid:1) , with V := (Λ + 2Λ) − gives the minimizer of H with respect to β .The relevant set of adjoint equations can be inferred from Appendix A combined withChapter 6 in [9], or from (3 .
17) and (3 . dr t = n Λ t (cid:0) p t − E t [ p t ] (cid:1) + β t o dt,dp t = − ∂ x f ( t, x t , c t ) dt + q , t dW t ,dp t = c f ( t, c t , c t ) r t dt + q , t dW t + q , t dW t , (4.5)with r = 0 ,p T = ∂ x g ( x T , c T ) ,p T = − c g ( c T , c T ) (cid:16) r T + δ − δ E T [ r T ] (cid:17) . (4.6)Although the results in Appendix A and Chapter 6 in [9] seem to provide most of the necessaryingredients, we left the general analysis on the controlled-FBSDEs of (conditional) McKean-Vlasov type for the future research. For our current purpose, it suffices to prove the nextverification theorem: Theorem 4.1.
Let Assumption (MFG) be in force. Suppose that there exists a solution ( b x , b x , b r , b p , b p , b y ) to ((4 . , (4 . and ((4 . , (4 . with the control process β satisfying b β t = V t (cid:0) − b p t + E t [ b y t ] + E t [ b p t ] (cid:1) dt ⊗ d P -a.e., then ( b β t ) t ∈ [0 ,T ] is the unique optimal control for the problem (4 . .Proof. For a given β ∈ A , we denote the associated solution to (4 .
2) by ( x , x , y ). We19hall study the difference: J ( β ) − J ( b β ) = E h g ( x T , c T ) − g ( b x T , c T )+ Z T (cid:16) f (cid:0) t, x t , β t , − E t [ y t ] + Λ t β t , Λ t , c t (cid:1) − f (cid:0) t, b x t , b β t , − E t [ b y t ] + Λ t b β t , Λ t , c t (cid:1)(cid:17) dt i . First, observe that E h(cid:10)b p T , x T − b x T (cid:11) + (cid:10)b r T , y T − b y T (cid:11)i = δ − δ E h − (cid:10) E T [ b r T ] , c g ( c T , c T )( x T − b x T ) (cid:11) + (cid:10)b r T , E T [ c g ( c T , c T )( x T − b x T )] (cid:11)i = 0 . Thus, from the convexity of g , we have E h g ( x T , c T ) − g ( b x T , c T ) i ≥ E h(cid:10)b p T , x T − b x T (cid:11) + (cid:10)b p T , x T − b x T (cid:11) + (cid:10)b r T , y T − b y T (cid:11)i . (4.7)Let us use b Θ t := (cid:0)b x t , b x t , b y , E t [ b y t ] , b p t , b p t , E t [ b p t ] , b r t (cid:1) , b θ t := (cid:0)b p t , b p t , E t [ b p t ] , b r t (cid:1) and omit thecommon arguments (Λ t , Λ t , c t , c t ) in the Hamiltonian. Since Λ , b β are F -adapted, we have E (cid:2)(cid:10) Λ t ( b p t − E t [ b p t ]) + b β t , y t − b y t (cid:11)(cid:3) = E (cid:2)(cid:10) Λ t b p t , y t − b y t (cid:11)(cid:3) + E (cid:2)(cid:10) − Λ t E t [ b p t ] + b β t , y t − b y t (cid:11)(cid:3) = E (cid:2)(cid:10) Λ t b p t , y t − b y t (cid:11)(cid:3) + E (cid:2)(cid:10) − Λ t b p t + b β t , E t [ y t ] − E t [ b y t ] (cid:11)(cid:3) = E (cid:2) − (cid:10) ∂ y H ( t, b Θ t , b β t ) , y t − b y t (cid:11) − (cid:10) ∂ y H ( t, b Θ t , b β t ) , E t [ y t ] − E t [ b y t ] (cid:11)(cid:3) . With these results and (4 . J ( β ) − J ( b β ) ≥ E Z T h H (cid:0) t, x t , x t , y t , E t [ y t ] , b θ t , β t (cid:1) − H (cid:0) t, b Θ t , b β t (cid:1) − (cid:10) ∂ x H ( t, b Θ t , b β t ) , x t − b x t (cid:11) − (cid:10) ∂ x H ( t, b Θ t , b β t ) , x t − b x t (cid:11) − (cid:10) ∂ y H ( t, b Θ t , b β t ) , y t − b y t (cid:11) − (cid:10) ∂ y H ( t, b Θ t , b β t ) , E t [ y t ] − E t [ b y t ] (cid:11)i dt ≥ E Z T h H (cid:0) t, x t , x t , y t , E t [ y t ] , b θ t , β t (cid:1) − H (cid:0) t, b Θ t , b β t (cid:1) − (cid:10) ∂ x H ( t, b Θ t , b β t ) , x t − b x t (cid:11) − (cid:10) ∂ x H ( t, b Θ t , b β t ) , x t − b x t (cid:11) − (cid:10) ∂ y H ( t, b Θ t , b β t ) , y t − b y t (cid:11) − (cid:10) ∂ y H ( t, b Θ t , b β t ) , E t [ y t ] − E t [ b y t ] (cid:11) − (cid:10) ∂ β H ( t, b Θ t , b β t ) , β t − b β t (cid:11)i dt ≥ , where the second inequality follows from the fact that b β t = argmin β H ( t, b Θ t , β ). The equalityholds only when β = b β due to the strict convexity.20rom Theorem 4.1, it is clear that the relevant set of equations is given by dx t = (cid:0) b β t + l ( t, c t ) (cid:1) dt + s ( t, c t ) dW t ,dx t = n − Λ t (cid:0) y t − E t [ y t ] (cid:1) − b β t + l ( t, c t , c t ) o dt + σ ( t, c t , c t ) dW t + σ ( t, c t , c t ) dW t ,dr t = n Λ t (cid:0) p t − E t [ p t ] (cid:1) + b β t o dt,dp t = − ∂ x f ( t, x t , c t ) dt + q , t dW t ,dy t = − ∂ x f ( t, x t , c t , c t ) dt + z , t dW t + z , t dW t ,dp t = c f ( t, c t , c t ) r t dt + q , t dW t + q , t dW t , (4.8)with x = χ , x = ξ , r = 0 ,p T = ∂ x g ( x T , c T ) ,y T = δ − δ E T (cid:2) c g ( c T , c T ) x T + h g ( c T , c T ) (cid:3) + c g ( c T , c T ) x T + h g ( c T , c T ) ,p T = − c g ( c T , c T ) (cid:16) r T + δ − δ E T [ r T ] (cid:17) , (4.9)where b β t , t ∈ [0 , T ] is defined by b β t := V t (cid:0) − p t + E t [ y t ] + E t [ p t ] (cid:1) . The next theorem guarantee the existence of the solution to the above FBSDE and hence theoptimal control for the major agent in the mean-field limit.
Theorem 4.2.
Under the Assumption (MFG), there exists a unique strong solution x , p ∈ S ( F ; R n ) , x , r , y , p ∈ S ( F ; R n ) , q , ∈ H ( F ; R n × d ) , z , , q , ∈ H ( F ; R n × d ) and z , , q , ∈ H ( F ; R n × d ) to the system of FBSDEs of conditional McKean-Vlasov type (4 . with (4 . .Proof. As we have done in the proof for Theorem 3.5, we introduce u := ( x , x , r , p , y , p )as arbitrary square integrable random variables with appropriate dimensions satisfying that( x , p ) are F t -measurable, and the others are F t -measurable. For these inputs, we definedrift[ x ]( t, u ) := b β ( t, u ) + l ( t, c t ) , drift[ x ]( t, u ) := − Λ t ( y − E t [ y ]) − b β ( t, u ) + l ( t, c t , c t ) , drift[ r ]( t, u ) := Λ t ( p − E t [ p ]) + b β ( t, u ) , drift[ p ]( t, u ) := − ∂ x f ( t, x , c t ) , drift[ y ]( t, u ) := − ∂ x f ( t, x , c t , c t ) , drift[ p ]( t, u ) := c f ( t, c t , c t ) r , where b β ( t, u ) := V t ( − p + E t [ y ] + E t [ p ]). For two different inputs u, u ′ , we set ∆ u := u − u ′ ,∆drift[ x ]( t ) := drift[ x ]( t, u ) − drift[ x ]( t, u ′ ) and similarly for the other quantities, too. Since21 t and p are F t -measurable, we see E t (cid:2)(cid:10) ∆drift[ x ]( t ) , ∆ p (cid:11) + (cid:10) ∆drift[ x ]( t ) , ∆ y (cid:11) + (cid:10) ( − I )∆drift[ r ]( t ) , ∆ p (cid:11)(cid:3) = − E t (cid:2)(cid:10) V t ( − ∆ p + E t [∆ y + ∆ p ]) , − ∆ p + ∆ y + ∆ p (cid:11)(cid:3) = − (cid:10) V t ( − ∆ p + E t [∆ y + ∆ p ] , − ∆ p + E t [∆ y + ∆ p ] (cid:11) ≤ . Now, it is easy to obtain E (cid:2)(cid:10) ∆drift[ p ]( t ) , ∆ x (cid:11) + (cid:10) ∆drift[ y ]( t ) , ∆ x (cid:11) + (cid:10) ( − I )∆drift[ p ]( t ) , ∆ r (cid:11) + (cid:10) ∆drift[ x ]( t ) , ∆ p (cid:11) + (cid:10) ∆drift[ x ]( t ) , ∆ y (cid:11) + (cid:10) ( − I )∆drift[ r ]( t ) , ∆ p (cid:11)(cid:3) ≤ − γ f E | ∆ x | − γ f (cid:0) E | ∆ x | + E | ∆ r | (cid:1) . (4.10)Now we set v := ( x , x , r ) as arbitrary square integrable random variables with appropri-ate dimensions satisfying that x are F T -measurable, and the others are F T -measurable. Forthese inputs, let us defineterminal[ p ]( v ) := ∂ x g ( x , c T ) , terminal[ y ]( v ) := δ − δ E T (cid:2) c g ( c T , c T ) x + h g ( c T , c T ) (cid:3) + c g ( c T , c T ) x + h g ( c T , c T ) , terminal[ p ]( v ) := − c g ( c T , c T ) (cid:16) r + δ − δ E T [ r ] (cid:17) , and with two different input v, v ′ , we denote by ∆ v := v − v ′ ,∆terminal[ p ] := terminal[ p ]( v ) − terminal[ p ]( v ′ )and similarly for the other quantities. Observe that E T (cid:2)(cid:10) ∆terminal[ y ] , ∆ x (cid:11)(cid:3) = E T hD δ − δ E T [ c g ( c T , c T )∆ x ] + c g ( c T , c T )∆ x , ∆ x Ei ≥ γ g E T | ∆ x | + δ − δ (cid:10) c E T [∆ x ] , E T [∆ x ] (cid:11) + δ − δ (cid:10) E T [( c g ( c T , c T ) − c )∆ x ] , E T [∆ x ] (cid:11) ≥ ( γ g − a ) E T | ∆ x | . Similar calculation yields E (cid:2)(cid:10) ∆terminal[ p ] , ∆ x (cid:11) + (cid:10) ∆terminal[ y ] , ∆ x (cid:11) + (cid:10) ( − I )∆terminal[ p ] , ∆ r (cid:11)(cid:3) ≥ γ g E | ∆ x | + ( γ g − a ) E (cid:2) | ∆ x | + | ∆ r | (cid:3) . (4.11)Using the estimates (4 .
10) and (4 . d (cid:10) ∆ y t , ∆ x t (cid:11) with that for d * ∆ p t ∆ y t ∆ p t , G ∆ x t ∆ x t ∆ r t + with G = I n × n I n × n ) 00 0 ( − I n × n ) . From Theorem 4.2, the market-clearing price in the mean-field limit is naturally expected tobe ̟ mfg t := − E t [ y t ] + Λ t V t (cid:0) − p t + E t [ y t ] + E t [ p t ] (cid:1) , t ∈ [0 , T ] . (5.1)In this section, we shall show that this is indeed the case for the homogeneous minor agents.Lastly, we also provide the estimate on the difference of the equilibrium price between the twomarkets; one is the homogeneous mean-field limit and the other is the heterogeneous marketof finite population. We now go back to the original setup of probability space given in Section 2. We first assumethat the minor agents are homogeneous.
Assumption 5.1. (Minor-Homogeneous)
The conditions in Assumption (MFG) hold true.Moreover, every minor agents ≤ i ≤ N is subject to the common coefficient functions ( l, σ , σ, f , g, c f , c g , h f , h g ) given there. For each 1 ≤ i ≤ N , let us construct F i -adapted processes, corresponding to those given by(4 .
8) and (4 . dx t = n V t ( − p t + E t [ y it ] + E t [ p it ]) + l ( t, c t ) o dt + s ( t, c t ) dW t ,dx it = n − Λ t ( y it − E t [ y it ]) − V t ( − p t + E t [ y it ] + E [ p it ]) + l ( t, c t , c it ) o dt + σ ( t, c t , c it ) dW t + σ ( t, c t , c it ) dW it ,dr it = n Λ t ( p it − E t [ p it ]) + V t ( − p t + E t [ y it ] + E t [ p it ]) o dt,dp t = − ∂ x f ( t, x t , c t ) dt + q , t dW t ,dy it = − ∂ x f ( t, x it , c t , c it ) dt + z i, t dW t + z i,it dW it ,dp it = c f ( t, c t , c it ) r it dt + q i, t dW t + q i,it dW it , (5.2)23ith x = χ , x i = ξ i , r i = 0 ,p T = ∂ x g ( x T , c T ) ,y iT = δ − δ E T (cid:2) c g ( c T , c iT ) x iT + h g ( c T , c iT ) (cid:3) + c g ( c T , c iT ) x iT + h g ( c T , c iT ) ,p iT = − c g ( c T , c iT ) (cid:16) r iT + δ − δ E T [ r iT ] (cid:17) . (5.3)By construction of the probability space and the fact that ( c i , ξ i , W i ) are independently andidentically distributed (i.i.d.), F i -adapted processes ( x i , r i , y i , p i ) are F -conditionally i.i.d. Inparticular, for any ϕ i = x i , r i , y i , p i , we have E t [ ϕ it ] = E t [ ϕ t ] and also E T (cid:2) c g ( c T , c iT ) x iT + h g ( c T + c iT ) (cid:3) = E T (cid:2) c g ( c T , c T ) x T + h g ( c T + c T ) (cid:3) . Therefore, ( x , p ) defined in (5 .
2) and (5 .
3) is indeed indistinguishable for every copy.We are going to compare (cid:0) x , p , ( x i ) Ni =1 , ( r i ) Ni =1 , ( y i ) Ni =1 , ( p i ) Ni =1 (cid:1) given above with the dy-namics (cid:0) X /N, P , ( X i ) Ni =1 , ( R i ) Ni =1 , ( Y i ) Ni =1 , ( P i ) Ni =1 (cid:1) given by (3 .
17) and (3 .
18) with homoge-neous coefficients. Using the scaling rule in (3 .
12) and Remark 3.3, we have for 1 ≤ i ≤ N , d X t N = n V t (cid:0) − P t + m (( Y it )) + m (( P it )) (cid:1) + l ( t, c t ) o dt + s ( t, c t ) dW t ,dX it = n − Λ t (cid:0) Y it − m (( Y jt )) (cid:1) − V t (cid:0) − P t + m (( Y it )) + m (( P it )) (cid:1) + l ( t, c t , c it ) o dt + σ ( t, c t , c it ) dW t + σ ( t, c t , c it ) dW it ,dR it = n Λ t (cid:0) P it − m (( P jt )) (cid:1) + V t (cid:0) − P t + m (( Y it )) + m (( P it )) (cid:1)o dt,dP t = − ∂ x f ( t, X t /N, c t ) dt + Q , t dW t + P Nj =1 Q ,jt dW jt ,dY it = − ∂ x f ( t, X it , c t , c it ) dt + Z i, t dW t + P Nj =1 Z i,jt dW jt ,dP it = c f ( t, c t , c it ) R it dt + Q i, t dW t + P Nj =1 Q i,jt dW jt , (5.4)with X = N χ , X i = ξ i , R i = 0 ,P T = ∂ x g ( X T /N, c T ) ,Y iT = δ − δ m (cid:16)(cid:0) c g ( c T , c jT ) X jT + h g ( c T , c jT ) (cid:1)(cid:17) + c g ( c T , c iT ) X iT + h g ( c T , c iT ) ,P iT = − c g ( c T , c iT ) (cid:16) R iT + δ − δ m (( R jT )) (cid:17) . (5.5)Thanks to the symmetry, ( X i , R i , Y i , P i ) have the same distribution for every 1 ≤ i ≤ N ,24lthough they are not independent due to their interactions. Let us introduce the notation:∆ x t := X t N − x t , ∆ x it := X it − x it , ∆ r it := R it − r it , ∆ p t := P t − p t , ∆ y it := Y it − y it , ∆ p it := P it − p it , ∆ q , t := Q , t − q , t , ∆ q ,jt := Q ,jt , ∆ z i, t := Z i, t − z i, t , ∆ z i,jt := Z i,jt − δ i,j z i,it , ∆ q i, t := Q i, t − q i, t , ∆ q i,jt := Q i,jt − δ i,j q i,it , where δ i,j stands for Kronecker delta. We also define µ r,Nt := 1 N N X i =1 δ r it , µ y,Nt := 1 N N X i =1 δ y it , µ p,Nt := 1 N N X i =1 δ p it , µ g,N := 1 N N X i =1 δ c g ( c T ,c iT ) x iT + h g ( c T ,c iT ) ,µ rt := L ( r t |F t ) , µ yt := L ( y t |F t ) , µ pt := L ( p t |F t ) , µ g := L ( c g ( c T , c T ) x T + h g ( c T , c T ) |F T ) . Here, µ r,N , µ y,N , µ p,N and µ g,N denote the empirical measures, and the others conditionaldistributions. When the filtration defined on the product space is completed, there appearssome subtle issue on the conditional distribution about its measurability. However, we canalways construct a measurable version by modifying it only on the null sets. We always supposethat ( µ r , µ y , µ p , µ g ) are measurable versions constructed in such a way. See Section 2.1.3 in[10] for details. Since (cid:0) r i , y i , p i , c g ( c T , c iT ) x iT + h g ( c T , c iT ) (cid:1) , ≤ i ≤ N are F conditionally i.i.d.and also ( r i , y i , p i ) are continuous processes, we have the following convergence properties. Lemma 5.1.
Let Assumption (Minor-Homogeneous) be in force. Then we have lim N →∞ sup t ∈ [0 ,T ] E h W ( µ r,Nt , µ rt ) + W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) i = 0 , lim N →∞ E (cid:2) W ( µ g,N , µ g ) (cid:3) = 0 . Moreover, if there exist some positive constants Γ and Γ g such that sup t ∈ [0 ,T ] (cid:0) E [ | r t | k ] k + E [ | y t | k ] k + E [ | p t | k ] k (cid:1) ≤ Γ and E (cid:2) | c g ( c T , c T ) x T + h g ( c T , c T ) | k (cid:3) k ≤ Γ g for some k > , thenthere exists some constant C independent of N such that sup t ∈ [0 ,T ] E h W ( µ r,Nt , µ rt ) + W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) i ≤ C Γ ǫ N , E (cid:2) W ( µ g,N , µ g ) (cid:3) ≤ C Γ g ǫ N , with ǫ N := N − / max( n, (1 + log( N ) N =4 ) .Proof. See Lemma 4.1 in [29] and the proof for Theorem 5.1 in [28]. More details on theGlivenko-Cantelli convergence in the Wasserstein distance are available from Section 5.1 in [9]and references therein.The next property of the Wasserstein distance is important for our purpose. For any25 , ν ∈ P ( R n ), it is easy to check (cid:12)(cid:12)(cid:12)Z R n xµ ( dx ) − Z R n yν ( dy ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z R n × n ( x − y ) π ( dx, dy ) (cid:12)(cid:12)(cid:12) ≤ Z R n × n | x − y | π ( dx, dy )for any coupling π ∈ Π ( µ, ν ) with marginals µ and ν . Taking infimum over π ∈ Π ( µ, ν ), weget (cid:12)(cid:12)(cid:12)Z R n xµ ( dx ) − Z R n yν ( dy ) (cid:12)(cid:12)(cid:12) ≤ W ( µ, ν ) ≤ W ( µ, ν ) . (5.6)We are now ready to prove the main result of this section. Theorem 5.1.
Let Assumption (Minor-Homogeneous) be in force. Then, for every ≤ i ≤ N ,there exists an N -independent constant C such that E h sup t ∈ [0 ,T ] (cid:0) | ∆ x t | + | ∆ x it | + | ∆ r it | + | ∆ p t | + | ∆ y it | + | ∆ p it | (cid:1) + N X j =0 Z T (cid:0) | ∆ q ,jt | + | ∆ z i,jt | + | ∆ q i,jt | (cid:1) dt i ≤ C E h W ( µ g,N , µ g ) + W ( µ r,NT , µ rT ) + Z T (cid:16) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:17) dt i . Proof.
Let us define γ > γ := min (cid:8) γ f , γ f , γ g , γ g − a (cid:9) . First step : We want to apply Itˆo-formula to (cid:16)(cid:10) ∆ p t , ∆ x t (cid:11) + 1 N N X i =1 (cid:10) ∆ y it , ∆ x it (cid:11) + 1 N N X i =1 (cid:10) ∆ p it , ( − I )∆ r it (cid:11)(cid:17) . (5.7)With this in mind, we check the following estimates. It is easy to see, with obvious notation, (cid:10) drift[∆ p t ] , ∆ x t (cid:11) + 1 N N X i =1 (cid:10) drift[∆ y it ] , ∆ x it (cid:11) + 1 N N X i =1 (cid:10) drift[∆ p it ] , ( − I )∆ r it (cid:11) ≤ − γ f | ∆ x t | − γ f N N X i =1 (cid:0) | ∆ x it | + | ∆ r it | (cid:1) (5.8)Using (5 . (cid:10) drift[∆ x t ] , ∆ p t (cid:11) = (cid:10) V t (cid:0) − ∆ p t + m (( Y it )) − E t [ y t ] + m (( P it )) − E t [ p t ] (cid:1) , ∆ p t (cid:11) = (cid:10) V t (cid:0) − ∆ p t + m ((∆ y it )) + m ((∆ p it )) (cid:1) , ∆ p t (cid:11) + (cid:10) V t (cid:0) m (( y it )) − E t [ y t ] + m (( p it )) − E t [ p t ] (cid:1) , ∆ p t (cid:11) ≤ (cid:10) V t (cid:0) − ∆ p t + m ((∆ y it )) + m ((∆ p it )) (cid:1) , ∆ p t (cid:11) + C (cid:0) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:1) | ∆ p t | . (cid:10) drift[∆ x t ] , ∆ p t (cid:11) + 1 N N X i =1 (cid:10) drift[∆ x it ] , ∆ y it (cid:11) + 1 N N X i =1 (cid:10) drift[∆ r it ] , ( − I )∆ p it (cid:11) ≤ − (cid:10) V t (cid:0) − ∆ p t + m ((∆ y it )) + m ((∆ p it )) (cid:1) , − ∆ p t + m ((∆ y it )) + m ((∆ p it )) (cid:11) + C (cid:0) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:1)(cid:0) | ∆ p t | + | m ((∆ y it )) | + | m ((∆ p it )) | (cid:1) ≤ C (cid:0) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:1)(cid:0) | ∆ p t | + m (( | ∆ y it | )) | + m (( | ∆ p it | )) (cid:1) . (5.9)Now, let us check the terminal parts. Similar analysis used in (3 .
9) yields (cid:10) ∆ p T , ∆ x T (cid:11) + 1 N N X i =1 (cid:10) ∆ y iT , ∆ x iT (cid:11) + 1 N N X i =1 (cid:10) ∆ p iT , ( − I )∆ r iT (cid:11) ≥ γ g | ∆ x T | + ( γ g − a ) 1 N N X i =1 (cid:0) | ∆ x iT | + | ∆ r iT | (cid:1) − CW ( µ g,N , µ g ) m (( | ∆ x iT | )) − CW ( µ r,NT , µ rT ) m (( | ∆ r iT | )) . (5.10)Using (5 . .
9) and (5 . .
7) gives γ E h | ∆ x T | + 1 N N X i =1 (cid:0) | ∆ x iT | + | ∆ r iT | (cid:1) + Z T (cid:16) | ∆ x t | + 1 N N X i =1 (cid:0) | ∆ x it | + | ∆ r it | (cid:1)(cid:17) dt i ≤ C E h W ( µ g,N , µ g ) m (( | ∆ x iT | )) + W ( µ r,NT , µ rT ) m (( | ∆ r iT | ))+ Z T (cid:0) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:1)(cid:0) | ∆ p t | + m (( | ∆ y it | )) + m (( | ∆ p it | )) (cid:1) dt i . By Young’s inequality and the symmetry of the distribution, we find that the following in-equality holds for every 1 ≤ i ≤ N : E h | ∆ x T | + | ∆ x iT | + | ∆ r iT | + Z T (cid:0) | ∆ x t | + | ∆ x it | + | ∆ r it | (cid:1) dt i ≤ C E h W ( µ g,N , µ g ) + W ( µ r,NT , µ rT ) + Z T (cid:0) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:1)(cid:0) | ∆ p t | + m (( | ∆ y it | )) + m (( | ∆ p it | )) (cid:1) dt i . (5.11) Second step E h sup t ∈ [0 ,T ] | ∆ p t | + Z T (cid:16) | ∆ q , t | + N X j =1 | ∆ q ,jt | (cid:17) dt i ≤ C E h | ∆ p T | + Z T | ∂ x f ( t, X t /N, c t ) − f ( t, x t , c t ) | dt i ≤ C E h | ∆ x T | + Z T | ∆ x t | dt i . Carrying out the similar analysis for (∆ y i , ∆ p i ) and using the symmetry among 1 ≤ i ≤ N , weobtain for any 1 ≤ i ≤ N , E h sup t ∈ [0 ,T ] (cid:0) | ∆ p t | + | ∆ y it | + | ∆ p it | (cid:1) + N X j =0 Z T (cid:0) | ∆ q ,jt | + | ∆ z i,jt | + | ∆ q i,jt | (cid:1) dt i ≤ C E h | ∆ x T | + | ∆ x iT | + | ∆ r iT | + Z T (cid:0) | ∆ x t | + | ∆ x it | + | ∆ r it | (cid:1) dt i + C E h W ( µ g,n , µ g ) + W ( µ r,NT , µ rT ) i ≤ C E h W ( µ g,N , µ g ) + W ( µ r,NT , µ rT ) + Z T (cid:0) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:1)(cid:0) | ∆ p t | + m (( | ∆ y it | )) + m (( | ∆ p it | )) (cid:1) dt i , where we have used (5 .
11) in the second inequality.From (5 . .
12) and the symmetry among 1 ≤ i ≤ N , Young’s inequality yields E h | ∆ x T | + | ∆ x iT | + | ∆ r iT | + Z T (cid:0) | ∆ x t | + | ∆ x it | + | ∆ r it | (cid:1) dt i + E h sup t ∈ [0 ,T ] (cid:0) | ∆ p t | + | ∆ y it | + | ∆ p it | (cid:1) + N X j =0 Z T (cid:0) | ∆ q ,jt | + | ∆ z i,jt | + | ∆ q i,jt | (cid:1) dt i ≤ C E h W ( µ g,N , µ g ) + W ( µ r,NT , µ rT ) + Z T (cid:16) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:17) dt i . Now the desired estimate follows from a simple application of Burkholder-Davis-Gundy (BDG)inequality to the forward variables (∆ x , ∆ x i , ∆ r i ). For understanding the implications of Lemma 5.1 and Theorem 5.1, let us denote the market-clearing price for the N homogeneous minor agents by ̟ Ho ,Nt := − m (( Y it )) + V t (cid:0) − P t + m (( Y it )) + m (( P it )) (cid:1) , t ∈ [0 , T ] (5.12)28sing the solution to (5 .
4) with (5 . ̟ mfg t in (5 . Theorem 5.2.
Under Assumption (Minor-Homogeneous), the following inequality holds: E Z T (cid:12)(cid:12) ̟ Ho ,Nt − ̟ mfg t (cid:12)(cid:12) dt ≤ C E h W ( µ g,N , µ g ) + W ( µ r,NT , µ rT ) + Z T (cid:16) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:17) dt i , where C is some positive constant independent of N .Proof. Using the symmetry, we have E (cid:2) | ̟ Ho ,Nt − ̟ mfg t | (cid:3) ≤ C E h | ∆ p t | + | ∆ y t | + | ∆ p t | + W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) i . Hence Theorem 5.1 gives the desired estimate.From Lemma 5.1, we observe that ( ̟ Ho ,Nt ) t ∈ [0 ,T ] converges to ( ̟ mfg t ) t ∈ [0 ,T ] in the largepopulation limit of homogeneous minor agents. In this limit, the optimization problem for each i th minor agent given in (3 .
5) is solved within (Ω i , F i , P i ; F i ) since the market price process ̟ mfg is now F -adapted i.e. dependent only on the common market information. One canobserve that the natural information structure mentioned in Remark 3.1 is actually achievedin the mean-field limit.Before closing the paper, let us briefly discuss about the stability relation between theheterogeneous and the homogeneous market. Let ( X , ( X i ) Ni =1 , ( R i ) Ni =1 , P , ( P i ) Ni =1 ) denote theunique solution to (3 .
17) with (3 .
18) given by Theorem 3.5 in the market with heterogeneousminor agents, and ( X , ( X i ) Ni =1 , ( R i ) Ni =1 , P , ( P i ) Ni =1 ) the unique solution to (5 .
4) with (5 . ≤ i ≤ N , δl i ( t ) := l i ( t, c t , c it ) − l ( t, c t , c it ) ,δσ i ( t ) := σ i ( t, c t , c it ) − σ ( t, c t , c it ) , δσ i ( t ) := σ i ( t, c t , c it ) − σ ( t, c t , c it ) ,δ∂ x f i ( t ) := ∂ x f i ( t, X it , c t , c it ) − ∂ x f ( t, X it , c t , c it ) ,δc fi ( t ) := c fi ( t, c t , c it ) − c f ( t, c t , c it ) ,δh gi = h gi ( c T , c iT ) − h g ( c T , c iT ) . Denoting the market-clearing price (3 .
21) in the market with N heterogeneous agents by( ̟ He ,Nt ) t ∈ [0 ,T ] , we have the next stability result. Corollary 5.1.
Let Assumptions (Minor-A, B) and (MFG) be in force. Then the following nequality holds: E Z T (cid:12)(cid:12) ̟ He ,Nt − ̟ mfg t | dt ≤ C E h W ( µ g,N , µ g ) + W ( µ r,NT , µ rT ) + Z T (cid:16) W ( µ y,Nt , µ yt ) + W ( µ p,Nt , µ pt ) (cid:17) dt i + C N N X i =1 E Z T (cid:16) | ∂ x f i ( t ) | + | δc fi ( t ) R it | + | δl i ( t ) | + | δσ i ( t ) | + | δσ i ( t ) | (cid:17) dt + C N N X i =1 E h | δc gi X iT + δh gi | + (cid:12)(cid:12)(cid:12) δc gi (cid:16) R iT + δ − δ m (( R iT )) (cid:17)(cid:12)(cid:12)(cid:12) i . Proof.
Let us put ∆ X t := X t − X t , ∆ Y it = Y it − Y it , and similarly for the others. Thanks tothe stability of fully-coupled FBSDEs, see for example Proposition 3.1 in [29] or more generallyProposition 3.4 in [55], we have1 N N X i =1 E h sup t ∈ [0 ,T ] (cid:16)(cid:12)(cid:12)(cid:12) ∆ X t N (cid:12)(cid:12)(cid:12) + | ∆ X it | + | ∆ R it | + | ∆ P t | + | ∆ Y it | + | ∆ P it | (cid:17) + N X j =0 Z T (cid:0) | ∆ Q ,jt | + | ∆ Z i,jt | + | ∆ Q i,jt | (cid:1) dt i ≤ C N N X i =1 E Z T (cid:16) | δ∂ x f i ( t ) | + | δc fi ( t ) R it | + | δl i ( t ) | + | δσ i ( t ) | + | δσ i ( t ) | (cid:17) dt + C N N X i =1 E h | δc gi X iT + δh gi | + (cid:12)(cid:12)(cid:12) δc gi (cid:16) R iT + δ − δ m (( R iT )) (cid:17)(cid:12)(cid:12)(cid:12) i . (5.13)Since E (cid:12)(cid:12) ̟ He ,Nt − ̟ Ho ,Nt (cid:12)(cid:12) ≤ C E h | ∆ P t | + 1 N N X i =1 ( | ∆ Y it | + | ∆ P it | ) i , the estimate (5 .
13) and Theorem 5.2 give the desired inequality.
A Sufficient maximum conditions for controlled-FBSDEs
Our optimization problem for the major agent requires the maximum principle for a system ofcontrolled-FBSDEs. The general issues of controlled-FBSDEs have been studied, in particular,by Yong [54, 55], where the second-order necessary conditions are given for non-convex con-trol domain. In the current paper, we actually need the sufficient conditions (i.e. verificationtheorem) rather than the necessary conditions. On the other hand, we only need the convexcontrol domain. Since we cannot find a useful summary in the existing literature, we providethe relevant theorem in this appendix. For the readers’ convenience, we provide the theoremunder the setup more general than what is actually needed for our purpose.We let (Ω , F , P , F ) be a complete filtered probability space satisfying the usual conditions.30t supports a d -dimensional Brownian motion W and F may be non-trivial. Let the controldomain A ⊂ R k be closed and convex and the space of admissible controls is denoted by A = H ( F ; A ). For a given T >
0, we introduce the following measurable functions: b : Ω × [0 , T ] × R n × R m × R m × d × A → R n ,σ : Ω × [0 , T ] × R n × R m × R m × d × A → R n × d ,f : Ω × [0 , T ] × R n × R m × R m × d × A → R m ,γ : Ω × R m → R n , Φ : Ω × R n → R m , φ : Ω × R m → R m . With these coefficient functions, we consider the following controlled system of FBSDEs: dx t = b ( t, x t , y t , z t , u t ) dt + σ ( t, x t , y t , z t , u t ) dW t ,dy t = f ( t, x t , y t , z t , u t ) dt + z t dW t ,x = γ ( y ) + ξ,y T = Φ( x T ) + φ ( y ) , (A.1)where ξ ∈ L ( F ; R n ) is given. See [54, 55] for various motivations to include the mixedinitial-terminal conditions.We study an optimization problem, inf u ∈ A J ( u ), with J ( u ) := E hZ T F ( t, x t , y t , z t , u t ) dt + G ( x T ) + g ( y ) i , under the dynamic constraints ( A. F : Ω × [0 , T ] × R n × R m × R m × d × A → R ,G : Ω × R n → R , g : Ω × R m → R are measurable functions representing the cost for the agent. The Hamiltonian H : Ω × [0 , T ] × R n × R m × R m × d × R n × R n × d × R m × A → R is defined by H ( t, x, y, z, p, q, r, u ) := (cid:10) p, b ( t, x, y, z, u ) (cid:11) + (cid:10) q, σ ( t, x, y, z, u ) (cid:11) + (cid:10) r, f ( t, x, y, z, u ) (cid:11) + F ( t, x, y, z, u ) , where the brackets in the second term in the right-hand side denotes a trace operation. Assumption A.1. (i)
For any ( x, y, z, u ) ∈ R n × R m × R m × d × A , ( b, σ, f, F ) are F -progressivelymeasurable, ( γ, g ) are F -measurable and Φ , φ, G are F T -measurable. (ii) For any ( u t ) t ∈ [0 ,T ] ∈ A , there exists a unique strong solution ( x t , y t , z t ) t ∈ [0 ,T ] ∈ S ( F ; R n ) × S ( F ; R m ) × H ( F ; R m × d ) to the controlled FBSDE ( A. . (iii) ( b, σ, f, γ, Φ , φ ) are one-time continuously differentiable in ( x, y, z, u ) with bounded deriva-tives. (iv) ( F, G, g ) are one-time continuously differentiable in ( x, y, z, u ) with uniformly Lipschitzcontinuous derivatives. Moreover, for any given ( x, y, z, u ) , these derivatives are square inte- For the existence of unique solutions to fully-coupled FBSDEs, see [51, 55]. In particular, the latter dealswith the mixed initial-terminal conditions. rable. (v) For any ( u t ) t ∈ [0 ,T ] ∈ A , J ( u ) is finite. (vi) ( G, g ) are convex and ( γ, Φ , φ ) are affine functions in ( x, y ) . Remark A.1.
For a scalar-valued function f ( x ) ∈ R , we use the convention f x ( x ) = ( ∂ x i f ( x )) ni =1 ∈ R n . For a vector-valued function f ( x ) ∈ R m , we use f x ( x ) ∈ R m × n with ( f x ( x )) i,j = ( ∂ x j f i ( x )) . The adjoint equations are given as follows: dr t = − H y ( t, x t , y t , z t , p t , q t , r t , u t ) dt − H z ( t, x t , y t , z t , p t , q t , r t , u t ) dW t ,dp t = − H x ( t, x t , y t , z t , p t , q t , r t , u t ) dt + q t dW t ,r = E (cid:2) φ y ( y ) ⊤ r T |F (cid:3) − γ y ( y ) ⊤ p − g y ( y ) ,p T = − Φ x ( x T ) ⊤ r T + G x ( x T ) . (A.2) Theorem A.1.
Let Assumption A.1 be in force. Suppose that ( b x t , b y t , b z t ) t ∈ [0 ,T ] ∈ S × S × H is a unique solution to the FBSDE ( A. with some admissible control process ( b u t ) t ∈ [0 ,T ] ∈ A . Assume that there exists a solution ( b p t , b q t , b r t ) t ∈ [0 ,T ] ∈ S × H × S to ( A. with inputs ( b x t , b y t , b z t , b u t ) t ∈ [0 ,T ] , and that the map R n × R m × R m × d × A ∋ ( x, y, z, u ) H ( t, x, y, z, b p t , b q t , b r t , u ) ∈ R is jointly convex in ( x, y, z, u ) and strictly convex in u , dt ⊗ d P -a.e. Moreover, the equality H ( t, b x t , b y t , b z t , b p t , b q t , b r t , b u t ) = inf u ∈ A H ( t, b x t , b y t , b z t , b p t , b q t , b r t , u ) holds dt ⊗ d P -a.e. Then, ( b u t ) t ∈ [0 ,T ] is a unique optimal solution.Proof. Let us denote by ( x t , y t , z t ) t ∈ [0 ,T ] the unique solution to ( A.
1) with a given controlprocess ( u t ) t ∈ [0 ,T ] ∈ A . For notational convenience, let us introduce θ t := ( x t , y t , z t ) , b θ t := ( b x t , b y t , b z t ) , b ̺ t := ( b p t , b q t , b r t ) , b Θ t := ( b x t , b y t , b z t , b p t , b q t , b r t ) , t ∈ [0 , T ] . Since ( γ, Φ , φ ) are affine, we have E h(cid:10) G x ( b x T ) , x T − b x T (cid:11) + (cid:10) g y ( b y ) , y − b y (cid:11)i = E h(cid:10)b p T + Φ x ( b x T ) ⊤ b r T , x T − b x T (cid:11) + (cid:10) − b r + E [ φ y ( b y ) ⊤ b r T |F ] − γ y ( b y ) ⊤ b p , y − b y (cid:11)i = E h(cid:10)b p T , x T − b x T (cid:11) − (cid:10)b p , x − b x (cid:11) + (cid:10)b r T , y T − b y T (cid:11) − (cid:10)b r , y − b y (cid:11)i , where we have used the relation, for example, Φ x ( b x T )( x T − b x T ) = Φ( x T ) − Φ( b x T ).32ow, Itˆo-formula gives E h(cid:10)b p T , x T − b x T (cid:11) − (cid:10)b p , x − b x (cid:11) + (cid:10)b r T , y T − b y T (cid:11) − (cid:10)b r , y − b y (cid:11)i = E Z T h h b p t , b ( t, θ t , u t ) − b ( t, b θ t , b u t ) (cid:11) − (cid:10) H x ( t, b Θ t , b u t ) , x t − b x t (cid:11) + (cid:10)b q t , σ ( t, θ t , u t ) − σ ( t, b θ t , b u t ) (cid:11) + (cid:10)b r t , f ( t, θ t u t ) − f ( t, b θ t , b u t ) (cid:11) − (cid:10) H y ( t, b Θ t , b u t ) , y t − b y t (cid:11) − (cid:10) H z ( t, b Θ t , b u t ) , z t − b z z (cid:11)i dt. It is easy to check that the stochastic integration part becomes a true martingale. Using theconvexity of G and g , we have J ( u ) − J ( b u ) ≥ E h(cid:10) G x ( b x T ) , x T − b x T (cid:11) + (cid:10) g y ( b y ) , y − b y (cid:11) + Z T (cid:2) F ( t, θ t , u t ) − F ( t, b θ t , b u t ) (cid:3) dt i = E Z T h H ( t, θ t , b ̺ t , u t ) − H ( t, b Θ t , b u t ) − (cid:10) H x ( t, b Θ t , b u t ) , x t − b x t (cid:11) − (cid:10) H y ( t, b Θ t , b u t ) , y t − b y t (cid:11) − (cid:10) H z ( t, b Θ t , b u t ) , z t − b z t (cid:11)i dt ≥ E Z T h H ( t, θ t , b ̺ t , u t ) − H ( t, b Θ t , b u t ) − (cid:10) H x ( t, b Θ t , b u t ) , x t − b x t (cid:11) − (cid:10) H y ( t, b Θ t , b u t ) , y t − b y t (cid:11) − (cid:10) H z ( t, b Θ t , b u t ) , z t − b z t (cid:11) − (cid:10) H u ( t, b Θ t , b u t ) , u t − b u t (cid:11)i dt ≥ , where the equality hods if and only if ( u = b u ) due to the strict convexity. References [1] Alasseur, C., Ben Taher, I., Matoussi, A., 2020,
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