Mean Field Game Master Equations with Anti-monotonicity Conditions

It is well known that the monotonicity condition, either in Lasry-Lions sense or in displacement sense, is crucial for the global well-posedness of mean field game master equations, as well as for the uniqueness of mean field equilibria and solutions to mean field game systems. In the literature, the monotonicity conditions are always taken in a fixed direction. In this paper we propose a new type of monotonicity condition in the opposite direction, which we call the anti-monotonicity condition, and establish the global well-posedness for mean field game master equations with nonseparable Hamiltonians. Our anti-monotonicity condition allows our data to violate both the Lasry-Lions monotonicity and the displacement monotonicity conditions.


Introduction
In this paper we consider the following second order master equation, arising from mean field games with common noise, with terminal condition V (T, x, µ) = G(x, µ): (1.1) Here β ≥ 0 is a constant, β 2 := 1 + β 2 , ∂ t , ∂ x , ∂ xx are standard temporal and spatial derivatives, ∂ µ , ∂ µµ are W 2 -Wasserstein derivatives, ξ and ξ are independent random variables with the same law µ and Ē is the expectation with respect to their joint law.The theory of Mean Field Games (MFGs, for short), initiated independently by Caines-Huang-Malhamé [15] and Lasry-Lions [40], studies the asymptotic behavior of stochastic differential games with a large number of players interacting in certain symmetric way.We refer to Lions [41], Cardaliaguet [16], Bensoussan-Frehse-Yam [7], Carmona-Delarue [21,22] and Cardaliaguet-Porretta [18] for a comprehensive exposition of the subject.First introduced by Lions [41], the master equation characterizes the value of the MFG, provided there is a unique mean field equilibrium.Roughly speaking, it plays the role of the HJB equation in the stochastic control theory.
We emphasize that the two inequalities in (1.4) share the same direction.Our goal of this paper is to propose a new type of monotonicity condition in the opposite direction, which we call anti-monotonicity condition, and establish the global well-posedness for the master equation (1.1), with possibly nonseparable Hamiltonian H.We remark that the mean field equilibrium is a fixed point, and the monotonicity conditions (1.4) were used to ensure the uniqueness of the fixed point.To motivate our anti-monotonicity condition, let us use a very simple example to illustrate the idea.Suppose that f : R 1 → R 1 is a continuously differentiable function and we are interested in its fixed point x * : f (x * ) = x * .When f is decreasing, i.e., f ′ ≤ 0, clearly f admits a unique fixed point x * .When f is increasing, in general neither the existence nor the uniqueness of x * is guaranteed.However, if f is sufficiently monotone, in the sense that f ′ ≥ 1 + ε for some ǫ > 0, then again f has a unique fixed point x * .While in complete different contexts, our conditions follow the same spirit.Roughly speaking, the standard monotonicity conditions (1.4) correspond to the case that f is decreasing, while our new anti-monotonicity condition corresponds to the case f is increasing, and for the same reason we will need to require our data to be sufficiently anti-monotone in appropriate sense.
To be precise, our anti-monotonicity condition takes the following form: for some appropriate constants λ 0 > 0, λ 1 ∈ R, λ 2 > 0, λ 3 ≥ 0. We remark that the inequality here takes the opposite direction to those in (1.4).In particular, the displacement monotonicity requires the convexity of G in x, while here G is typically concave in x, due to the first term in (1.5).This justifies the name of anti-monotonicity (and to have a better comparison with (1.4), we may also set λ 1 = 1).We also note that, considering the case λ 3 = 0, the second line of (1.5) is positive, this means that the first line of (1.5) should be sufficiently negative, which is exactly in the spirit that G to be sufficiently anti-monotone.
To establish the global well-posedness of the master equation (1.1), we follow the strategy in [32], which consists of three steps.The key step of this approach is to show a priori that the anti-monotonicity propagates along the solution V .That is, under appropriate conditions, as long as V (T, •) = G is anti-monotone, then V (t, •) is anti-monotone for all t.The second step is to show that the anti-monotonicity of V implies ∂ x V is uniformly Lipschitz continuous in (x, µ), under W 2 in µ.This, together with a representation formula established in [43], implies further the Lipschitz continuity under W 1 .In the final step we show that the uniform Lipschitz continuity under W 1 enables us to extend a local classical solution to a global one.
There is a major technical difference from [32] though.The assumptions we impose for the propagation of anti-monotonicity prevents us from assuming uniform Lipschitz continuity of the data G and H. Instead, we can only assume ∂ x G, ∂ x H are uniformly Lipschitz.This has two consequences.First, the a priori estimate for the boundedness of ∂ xx V , which is crucial for the global well-posedness of the master equation and is pretty easy to obtain under the conditions in [32], becomes very subtle.In fact, we need some serious efforts to obtain this estimate.Moreover, unlike in [32], under our conditions the solution V will not be Lipschitz continuous.Instead, we can only expect the Lipschitz continuity of ∂ x V .Therefore, we will actually consider the vector master equation of U := ∂ x V and establish its global well-posedness first.Once we obtain U , then it is immediate to solve the original master equation (1.1) for V .
The rest of the paper is organized as follows.In Section 2 we review the setting in [32] and introduce our problem.In Section 3 we introduce the new notion of anti-monotonicity and present the technical conditions used in the paper.In Section 4 we show a priori the crucial propagation of the anti-monotonicity.Section 5 is devoted to the a priori uniform Lipschitz estimate of ∂ x V in µ, first under W 2 and then under W 1 .In Section 6 we provide the a priori estimate for Finally in Section 7 we establish the global well-posedness of the master equation (1.1).

The setting
Throughout the paper we will use the setting in [32].We review it briefly in this section and refer to [32] for more details.
We next introduce the Wasserstein space and differential calculus on Wasserstein space.Let P := P(R d ) be the set of all probability measures on R d and, for any q ≥ 1, let P q denote the set of µ ∈ P with finite q-th moment.For any sub-σ-field G ⊂ F T and µ ∈ P q , we denote the set of R d -valued, G-measurable, and q-integrable random variables ξ by L q (G); and the set of ξ ∈ L q (G) such that the law L ξ = µ by L q (G; µ).For any µ, ν ∈ P q , the W q -Wasserstein distance between them is defined as follows: 1 q : for all ξ ∈ L q (F T ; µ), η ∈ L q (F T ; ν) .
For a W 2 -continuous functions U : P 2 → R, its Wasserstein gradient, also called Lions-derivative, takes the form ∂ µ U : (µ, x) ∈ P 2 × R d → R d and satisfies: for our master equation, and let C 1,2 (Θ) denote the set of continuous functions U ∈ Θ → R which has the following continuous derivatives: One crucial property of U ∈ C 1,2 (Θ) functions is the Itô formula.For i = 1, 2, let dX i t := b i t dt+σ i t dB t +σ i,0 t dB 0 t , where b i : [0, T ]×Ω → R d and σ i , σ i,0 : [0, T ]×Ω → R d×d are F-progressively measurable and bounded (for simplicity), and ρ t := L X 2 t |F 0 t , then we have See, e.g., [22,Theorem 4.17], [14,23]).Here L X 2 t |F 0 t stands for the conditional law of X 2 t given F 0 t , and ẼFt and ĒFt are the conditional expectations given F t corresponding to the probability measures P and P respectively.Throughout the paper, the elements of R d are viewed as column vectors; where ⊤ denotes the transpose, and similarly for the other second order derivatives; both the notations "•" and •, • denote the inner product of column vectors.
We finally introduce the mean field system related to the master equation (1.1).It either takes the form of forward backward McKean-Vlasov SDEs on [t 0 , T ]: given t 0 and ξ ∈ L 2 (F t 0 ), where L(x, µ, p) or take the form of forward backward stochastic PDE system on [t 0 , T ]: denoting where the solution triple (ρ, u, v) is F 0 -progressively measurable and ρ(t, •, ω) is a (random) probability measure.The systems (2.3) and (2.4) connect to the master equation (1.1) as follows: provided all the equations are well-posed and in particular (1.1) has a classical solution V , then It is already well known that, c.f. [22], if the master equation (1.1) has a classical solution V with bounded derivatives, then we can get existence and uniqueness of the mean field equilibrium, and the equilibrium of the corresponding N -player game will converge to the mean field equilibrium.
Therefore, we shall only focus on the global well-posedness of the master equation (1.1).
We conclude this section with the strategy in [32] for the global well-posedness of (1.1).
We will follow the same strategy in this paper, except that we shall replace the monotonicity condition with the anti-monotonicity condition: Step 1. Introduce appropriate monotonicity condition on data which ensure the propagation of the monotonicity along any classical solution to the master equation.
Step 2. Show that the monotonicity of V (t, •, •) implies an (a priori) uniform Lipschitz continuity of V in the measure variable µ.
Step 3. Combine the local well-posedness of classical solutions and the above uniform Lipschitz continuity to obtain the global well-posedness of classical solutions.

Assumptions and anti-monotonicity conditions
In this section, we introduce the following notations.For any A ∈ R d×d , (3.1)

It is obvious that, for any
and, when

Regularity assumptions
We first specify some technical assumptions on G and H.
where all the second and higher order derivatives of H involved above are uniformly bounded.
, and there exist constants , and all the second and higher order derivatives of G involved here are uniformly bounded.
Here the spaces C 2 , C 3 are defined in the same manner as C 1,2 (Θ).Note that at above we do not require the first order derivatives to be uniformly bounded.In fact, the condition (3.18) below does not allow ∂ x H to be bounded.
This implies further the Lipschitz continuity of ∂ x G in µ under W 2 on R d × P 2 , and we denote the Lipschitz constant by LG

Monotonicity and anti-monotoncity conditions
Under the above regularity conditions on the data G and H, the MFG may still have multiple mean field equilibria over a long time duration and thus the global well-posedness of classical solutions for the master equations can fail.Therefore, some structural conditions on G, H are needed in order to guarantee its global well-posedness.The typical structural conditions assumed in the literature are two types of monotonicity conditions, i.e., the Lasry-Lions monotonicity condition and the displacement monotonicity condition.
Here, as in Section 2, ( ξ, η) is an independent copy of (ξ, η).We remark that the displacement semi-monotonicity is obviously weaker than the displacement monotonicity (3.6), and when is bounded, it is also weaker than the Lasry-Lions monotonicity (3.5).We next turn to the monotonicity conditions for the Hamiltonian H.In the literature, the Lasry-Lions monotonicity has only been proposed for the separable Hamiltonians, i.e., H(x, µ, p) = H 0 (x, p) − F (x, µ) and F satisfies (1.2).In [32], a notion of displacement monotonicity for nonseparable H was proposed to study the well-posedness of the master equation (1.1).(ii) The function ϕ(ξ) in (3.8) is chosen to be ∂ x V (t, ξ, L ξ ) in the proof of the propagation of the displacement monotonicity (3.6) along V (t, •) in [32].Since ∂ x V is not priorily known, the displacement monotonicity (3.8) is made for any desirable function ϕ.
(iii) When H is non-separable, it still remains a challenge to find appropriate conditions on H so that the Lasry-Lions monotonicity (3.5) could propagate along the solution V (t, •).
Finally we introduce the anti-monotonicity condition, which is the main structural condition in this paper and serves as an alternative sufficient condition for the global wellposedness of the master equation.Denote (3.10) Remark 3.9 (i) The main feature of (3.10) is that the direction of the inequality is opposite to those in Definition 3.4.In particular, (3.10) implies the Lasry-Lions anti-monotonicity, i.e.
for the case that λ 0 = λ 3 = 0 and λ 1 = λ 2 = 1.In fact, in this case the condition (3.10) is stronger than (3.11) and we interpret it as U is sufficiently Lasry-Lions anti-monotone: Similarly, in the case λ 0 = λ 1 = λ 2 = 1 and λ 3 = 0, we see that (3.10) implies U is sufficiently displacement anti-monotone: Note that the concavity of U in x could help in (3.13), while in (3.6) its convexity is helpful.
(ii) The inequality (3.10) implies the displacement semi-anti-monotonicity, i.e.We next provide an example which is λ-anti-monotone.
Example 3.10 Let d = 1 and consider the function: for some constants a 0 , a 1 , It is clear that So U is Lasry-Lions monotone if a 1 ≥ 0, and Lasry-Lions anti-monotone if a 1 ≤ 0.
(ii) Similarly we have Then one can easily check that U is displacement monotone if a 0 ≥ 0, a 1 ≥ −a 0 , and displacement anti-monotone if a 0 ≤ 0, a 1 ≤ −a 0 .
(ii) When λ 0 = λ 1 , (3.10) is equivalent to the following integral form: for any (iii) In general, (3.10) is equivalent to the following integral form: for any Assumption 3.12 (i) G satisfies Assumption 3.2-(i) and is λ-anti-monotone for some λ ∈ D 4 ; (ii) H satisfies Assumption 3.1-(i) and there exist constants ) Note that we do not require structural conditions on ∂ xµ H here, and ∂ pp H can be degenerate.

Propagation of anti-monotonicity
In this section we show that any classical solution V to the master equation (1.1) could propagate the anti-monotonicity under appropriate conditions.
Theorem 4.1 Let Assumption 3.12 hold and V be a classical solution of the master equation and all the second and higher order derivatives of V involved above are also continuous in the time variable and are uniformly bounded.Assume further that there exist a constant L V xx > 0 such that and Introduce the following symmetric matrices, which depend only on γ, γ, λ, and L V xx : Then, whenever V (t, •) is λ-anti-monotone in the sense of (3.10) for all t ∈ [0, T ].
Proof.Without loss of generality, we shall prove the theorem only for t 0 = 0.
Fix ξ ∈ L 2 (F 0 ) and η ∈ L 2 (F 0 ).Given the desired regularity of V and H, the following system of McKean-Vlasov SDEs has a unique solution (X, δX): In the sequel, for simplicity of notation, we omit the variables (t, µ t ) as well as the dependence on ∂ x V , and denote and similarly for We remark that, ( Xt , δ Xt ) is a conditionally independent copy of (X t , δX t ) and µ t is F 0 t -measurable.

The Lipschitz continuity
We first show that the anti-monotonicity of V implies the uniformly Lipschitz continuity of ∂ x V in µ under W 2 .Unlike in [32], since we do not require the first order derivatives of G, H to be bounded, here we do not expect the Lipschitz continuity of V itself.
and all the second and higher order derivatives of V involved above are also continuous in the Proof.In this proof, C > 0 denotes a generic constant depending only on quantities mentioned in the statement of the theorem.As in the proof of Theorem 4.1, without loss of generality we show the theorem only for t 0 = 0. First, by (3.10) we have, for any ξ, η ∈ L 2 (F 1 t ), Next, applying Hölder's inequality to (5.1) we have From now on we fix ξ ∈ L 2 (F 0 ) and η ∈ L 2 (F 0 ) and continue to use the notation as in the proof of Theorem 4.1.In particular, X, δX, µ t , Υ, Ῡ are defined by (4.5).Applying (5.2) by replacing E with E F 0 t and noting that X t is F t -measurable, we have Using Hölder's inequality on (4.5) and noting in particular Taking expectation on (5.4) and using (5.3), we derive Then it follows from Grönwall's inequality that sup (5.5) Next, by (4.8), we have Take conditional expectation ẼFt , we have Then by (5.6) and the required regularity of G, H and V , we have Now take conditional expectation ẼF 0 , we get Thus, by the Grönwall inequality we have (5.7) Note that, recalling the setting in Section 2, δ Xt is measurable with respect to F 0 t ∨ F1 t , which is independent of F 0 under P. Then the conditional expectation in the right side of (5.7) is actually an expectation.Plug (5.5) into (5.7),we have (5.8) (5.9) Since ∂ µ V is continuous, then (5.9) actually holds for all x.In particular, this implies that there exists a constant C µ 0 2 > 0 such that We emphasize that the above Lipschitz continuity is under W 2 , while the global wellposedness of the master equation requires the W 1 -Lipschitz continuity.As in [32], we shall derive the desired W 1 -Lipschitz continuity from the W 2 -Lipschitz continuity by utilizing the pointwise representation for the Wasserstein derivative developed in [43].Note again that in Theorem 5.1 we only have the Lipschitz continuity for ∂ x V , but not for V , so at below we shall also consider U (t, x, µ) := ∂ x V (t, x, µ), which formally should satisfy the following vectorial master equation (5.11) To be precise, fix t 0 , ξ, we first consider the following McKean-Vlasov SDE on [t 0 , T ]: (5.12) Next, given ρ as above, for fixed x ∈ R d and letting (e 1 , • • • , e d ) denote the natural basis of R d , we introduce a series of FBSDEs, possibly McKean-Vlasov type: (5.13) (5.14) (5.15) (5.16) The following local (in time) result provides the crucial W 1 -Lipschitz continuity of U .
(ii) Define U (t 0 , x, µ) := ∇Y x,ξ t 0 .Then we have the pointwise representation: (5.17) Moreover, there exists a constant C µ 1 > 0, depending only on d, L (iii) Assume further that Assumptions 3.1-(ii) and 3.2-(ii) hold true.Then the vectorial master equation (5.11) has a unique classical solution U .Moreover, and all their derivatives in the state and probability measure variables are continuous in the time variable and are uniformly bounded.
(iv) The following decoupled McKean-Vlasov FBSDE is well-posed on [t 0 , T ] for any x ∈ R d .Define V (t 0 , x, µ) := Y x,ξ t 0 .Then V is the unique classical solution of the master equation (1.1) and We emphasize that at above C µ 1 depends on L G 2 in (3.4), but the δ depends only on LG 2 in Remark 3.3, not on L G 2 . Proof.
The proof of (i)-(iii) is very lengthy, but essentially identical to as that of [32, Proposition 6.2], except that [32] considers both ∂ µ V and ∂ xµ V = ∂ µ U .Therefore we omit it here.
(iv) By the smoothness of U obtained in (iii), clearly the V defined in (iv) is smooth and . By applying Itô's formula (2.2) we see that V satisfies the PDE: (5.20) Differentiate it with respect to x, we obtain the PDE for ( Compare this with (5.11), we see that U also satisfies (5.21).Thus, by the uniqueness we have . Plug this into (5.20)we verify that V satisfies (1.1).

Uniform estimates of ∂ xx V
We note that all the above results rely on the bound L V xx of ∂ xx V in (4.1).In particular, in Theorem 4.1 the L H xp depends on L V xx .Then it is crucial to obtain an a priori uniform estimate of L V xx which is independent of L H xp .Recall (2.5), we have ∂ xx V = ∂ xx u, so it suffices to establish the a priori estimate for the solution u to the backward SPDE in (2.4), for an arbitrarily given ρ (not necessarily satisfying the forward SPDE in (2.4)).
For this purpose we consider a special form of H. Assumption 6.1 H takes the following form: where A 0 ∈ R d×d is a constant matrix and H 0 : (ii) H 0 satisfies Assumption 3.1-(ii).
Given A 0 , consider its Jordan decomposition: where J 0 ∈ C d×d is the Jordan normal form of A 0 and Q 0 ∈ C d×d is invertible.Let Q0 denote the conjugate of Q 0 and thus Q 0 Q⊤ 0 is positive definite.The following estimate will be crucial.
Proof.Fix J 0 , Q 0 as in (6.4).It is obvious that Then, for any x, y ∈ R d with |x| = |y| = 1, we have .
To see (6.6), assume the Jordan normal form are all the eigenvalues of A 0 ; and U d i is the matrix whose (j, j + 1)-component is 1, j = 1, • • • , d i − 1, and all other components are 0. It is straightforward to see that Note that, for each i, since I d i and U d i can commute, and For any x (i) , y (i) ∈ C d i , it is clear that Then, for x = (x (1) , • • • , x (k) ), y = (y (1) , This implies (6.6) immediately.Remark 6.3 (i) The form (6.1) is assumed for the estimate (6.5) and for the property required in the proof of Theorem 6.4 below.In general e − t 0 ∂xpHds does not enjoy these properties.When d = 1, however, e − t 0 ∂xpHds obviously satisfies similar properties and thus we do not need the special form (6.1).Moreover, we remark that any alternative structures which could ensure a uniform a priori bound for ∂ xx u can serve our purpose.
(ii) It is clear that, under (6.1), (6.2), and (6.3), we may set Then (3.3) and (3.18) hold true.We shall remark though that the term κ(A 0 ) and the condition κ(∂ xx H 0 ) ≥ L H 0 xx are not used in Theorem 6.4 below.(iii) When A 0 is symmetric, one can easily see that L A 0 = 1, and in this case (6.5) can be improved: |e −A 0 t | ≤ e −κ ′ (A 0 )t .
Then we have the following uniform a priori estimate. where (6.9) Then the following estimate holds: (6.10) We note that (6.9) implies L u xx (θ) is well-defined for θ ≥ θ 3 , and we emphasize that the bound L u xx (θ 3 ) depends only on L H 0 2 , L G xx and L A , in particular not on T , κ ′ (A 0 ), or L Proof.Fix (t 0 , x).First, under our conditions it is clear that the following FBSDE on [t 0 , T ] has a unique solution (X x , ∇Y x , ∇Z x , ∇Z 0,x ): In particular, ∂ x u serves as the decoupling field: Next, denote L 0 := L u xx (κ ′ (A 0 )), and consider the following BSDE on [t 0 , T ]: Here A ∧ L 0 := [(−L 0 ) ∨ a ij ∧ L 0 ] i,j is the truncated matrix.The above BSDE has a Lipschitz continuous driver and thus is well-posed.Recalling (6.7) and applying Itô's formula we have Take conditional expectation E Ft on both sides, we obtain Recall (3.2) and apply Lemma 6.2, we have Taking the conditional expectation E Ft 0 and noting that κ + 1, we derive Then by Grönwall's inequality we have Recall (6.10), one can check straightforwardly that , by (6.9) and (6.10) we have Thus (6.14) implies where the last equality is due to the straightforward calculation.In particular, by setting t = t 0 , we have s and thus (6.13) becomes )) for all (t, x).This, together with (6.15), implies (6.10).

Global well-posedness
In this section we establish the global well-posedness of the master equation.We shall first construct the global well-posedness of the vectorial master equation (5.11).Following the idea in [23,22,43,32], the key is to extend a local classical solution to a global one through an a priori uniform Lipschitz continuity estimate of the solution in µ.We note that Theorem 6.4 implies the uniform a priori bound of ∂ xx V .Then, by applying Theorem 4.1 and 5.1, we obtain the uniform a priori Lipschitz continuity of U = ∂ x V with respect to µ under W 2 .Moreover, by Proposition 5.2 we derive the desired uniform a priori Lipschitz continuity of U with respect to µ under W 1 .
We now present the main well-posedness result.Note that the dependence on the parameters is quite subtle, so we will introduce them carefully.Following the order of the assumptions below, one can easily construct a class of G and H satisfying all of them, see e.g.Example 7.2.
In particular, in light of Lemma 6.3 (iii), we may set L A = 1 and consider symmetric A 0 .
Theorem 7.1 Let Assumption 3.2 with L G xx , L G 2 and Assumption 3.12 (i) with λ ∈ D 4 hold true, and H takes the form (6.1) such that Assumption 6.1 (ii) holds and there exists L H 0 2 satisfying the requirements in (6.3).Fix an arbitrary L A ≥ 1 and set θ 3 as in (6.9) and L V xx := L u xx (θ 3 ) as in (6.10).Assume further the following hold true.
Proof.The uniqueness as well as the wellposedness of the involved FBSDEs and the representation formula (5.17) follow exactly the same arguments as in [32,Theorem 6.3].Thus we shall only prove the existence.
as in (6.8).Then clearly Assumptions 3.1 and 3.12 hold true.By (7.1) and (7.2) we see that (6.9) holds true and thus we have the a priori estimate (6.10).Moreover, by (7.1) we have We conclude the paper by providing an example which satisfies all the assumptions in Theorem 7.1.We emphasize that there is no smallness assumption imposed here.Then, for M 0 large enough, which may depend on α, α, γ, γ, (λ 1 , λ 2 , λ 3 ), and L G 2 , L H 0 2 , one can choose appropriate λ 0 such that all the conditions in Theorem 7.1 hold true.
Proof.We first emphasize that (7.3) and (7.4) involve only ∂ xx G and ∂ xx H 0 .Note that the parameters L G 2 , L H 0 2 , which M 0 will depend on, do not involve these derivatives.So it is rather easy to construct G and H 0 satisfy both Assumptions 3.2, 6.1, and (7.3), (7.4) with arbitrarily large M 0 .Moreover, recall (3.4) and (6.2), by (7.3) and (7.4) it is clear that Then the θ 3 in (6.9) and L u xx (θ 3 ) in (6.10) become: recalling L A = 1, We now show that the following λ 0 satisfies all the requirements: First, by the choice of λ 0 , it is obvious that , which verifies (4.2).
Next, let O(M ) denote a generic positive function of M such that O(M ) M is bounded both from above and away from 0. Then we see that Since λ 0 M 0 = O(M 3 0 ), it is clear that G is λ-anti-monotone when M 0 is large enough.Moreover, since d = 1, we have κ(A 0 ) = κ(A 0 ) = κ ′ (A 0 ) = A 0 and L A 0 = 1 ≤ L A = 1.Recall (4.2) and (4.3).When M 0 is large, it is clear that 1 − θ 1 is uniformly away from 0 and then it follows from (7.8) that κ(A −1 1 A 2 ) = O(M 2 0 ).Thus, since A 0 = M 3 0 and γ > 1, for M 0 sufficiently large we have the following inequalities which verify (7.1): Finally, since ] for M 0 large enough.Then (7.4) implies (7.2).

Remark 3 . 5
The above formulations of the monotonicity conditions are convenient for our purpose.For U ∈ C 2 (R d × P 2 ), (3.5) and (3.6) are equivalent to (1.2) and (1.3), respectively, which appear more often in the literature.See[32, Remark 2.4].

8 )
Remark 3.7 (i)  The above definition of displacement monotonicity for non-separable Hamiltonians is not really used in the rest of the paper except for the comparison with the new notion of anti-monotonicity introduced below.We refer to[32, Proposition 3.7]  for another equivalent definition of the above one.
1 and λ 3 ≥ 0. Note that the condition (3.14) is weaker than (3.13) for the case.We recall that in the literature a function u : R d → R is said to be semi-concave, or λ-concave, if ∂ xx u ≤ λI d for some constant λ > 0, where I d is the d × d identity matrix.We follow the same spirit to call U λ-anti-monotone if U satisfies (3.10).

Theorem 5 . 1
Let Assumptions 3.1-(i), 3.2-(i) hold and V be a classical solution of the master equation (1.1) such that

. 2 )
Then the master equation (1.1) on [0, T ] admits a unique classical solution V with bounded ∂ x V ,∂ xx V and ∂ xµ V .Furthermore, the McKean-Vlasov FBSDEs (5.12), (5.13), (5.14), (5.15), (5.16) and(5.19) and thus the result of Theorem 4.1 holds true.We now let C µ 2 be the a priori (global) uniform Lipschitz estimate of ∂ x V with respect to µ under W 2 , as established by Theorems 4.1 and 5.1.Let δ > 0 be the constant in Proposition5.2,but withL G xx replaced with L V xx and L G 2 replaced with C µ 2 .Let 0 = T 0 < • • • < T n = T be a partition such that T i+1 − T i ≤ δ 2 , i = 0, • • • , n − 1.First, since T n − T n−2 ≤ δ,by Proposition 5.2 the master equation (1.1) on [T n−2 , T n ] with terminal condition G has a unique classical solution V .For each t ∈ [T n−2 , T n ], applying Theorem 6.4 we have|∂ xx V (T n−1 , •, •)| ≤ L V xx .Note that by Proposition 5.2-(iii)(iv) V (t,•, •) has further regularities, this enables us to apply Theorems 4.1 and 5.1 and obtain that∂ x V (t, •, •) is uniform Lipschitz continuous in µ under W 2 with Lipschitz constant C µ 2 .Moreover, by Proposition 5.2-(ii) ∂ x V (T n−1 , •, •) is also uniformly Lipschitz continuous in µ under W 1 .We next consider the master equation (1.1) on [T n−3 , T n−1 ] with terminal condition V (T n−1 , •, •).We emphasize that ∂ x V (T n−1 , •, •) has the above uniform regularity with the same constantsL V xx , C µ 2 ,then we may apply Proposition 5.2 with the same δ and obtain a classical solution V on [T n−3 , T n−1 ] with the additional regularities specified in Proposition 5.2-(iii)(iv).Clearly this extends the classical solution of the master equation to [T n−3 , T n ].We emphasize again that, while the bound of∂ xµ V (t, •) may become larger for t ∈ [T n−3 , T n−2 ] because the C µ 1 in (5.18) now depends on ∂ xµ V (T n−1 , •) L ∞ instead of ∂ xµ V (T n , •) L ∞ ,by the global a priori estimates in Theorems 4.1 and 5.1 we see that ∂ x V (t, •) corresponds to the same L V xx and C µ 2 for all t ∈ [T n−3 , T n ].This enables us to consider the master equation (1.1) on [T n−4 , T n−2 ] with terminal condition V (T n−2 , •, •), and then we obtain a classical solution on [T n−4 , T n ] with the desired uniform estimates and additional regularities.Repeat the arguments backwardly in time, we may construct a classical solution V for the original master equation (1.1) on [0, T ] with terminal condition G.Moreover, since this procedure is repeated only n times, by applying (5.18) repeatedly we see that (5.18) indeed holds true on [0, T ].